TPTP Problem File: SLH0910^1.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 : Khovanskii_Theorem/0008_Khovanskii/prob_00457_015577__13450868_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1354 ( 757 unt; 77 typ; 0 def)
% Number of atoms : 3089 (1585 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10078 ( 216 ~; 77 |; 135 &;8599 @)
% ( 0 <=>;1051 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 319 ( 319 >; 0 *; 0 +; 0 <<)
% Number of symbols : 74 ( 71 usr; 11 con; 0-3 aty)
% Number of variables : 3247 ( 213 ^;2998 !; 36 ?;3247 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:14:17.371
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
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__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 (71)
thf(sy_c_Bernoulli_Obernpoly_001t__Real__Oreal,type,
bernpoly_real: nat > real > real ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
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_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__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_Oplus__class_Oplus_001t__Set__Oset_It__Int__Oint_J,type,
plus_plus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
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_Otimes__class_Otimes_001t__Set__Oset_It__Int__Oint_J,type,
times_times_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
times_times_set_real: set_real > set_real > set_real ).
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__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__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_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
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_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $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__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__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
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_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Weierstrass__Theorems_OBernstein,type,
weiers7429072931691461095nstein: nat > nat > real > real ).
thf(sy_c_Weierstrass__Theorems_Oreal__polynomial__function_001t__Real__Oreal,type,
weiers3457258110322917882n_real: ( real > real ) > $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_N,type,
n: nat ).
thf(sy_v_j,type,
j: nat ).
thf(sy_v_n,type,
n2: nat ).
% Relevant facts (1269)
thf(fact_0_assms,axiom,
ord_less_eq_nat @ n @ n2 ).
% assms
thf(fact_1_of__nat__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3539618377306564664at_int
@ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
@ A ) ) ).
% of_nat_sum
thf(fact_2_of__nat__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3542108847815614940at_nat
@ ^ [X: nat] : ( semiri1316708129612266289at_nat @ ( F @ X ) )
@ A ) ) ).
% of_nat_sum
thf(fact_3_of__nat__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups6591440286371151544t_real
@ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
@ A ) ) ).
% of_nat_sum
thf(fact_4_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_5_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_6_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_7_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_8_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_9_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_10_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X2 )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_11_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X2 )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_12_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X2 )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_13_atMost__eq__iff,axiom,
! [X2: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X2 )
= ( set_ord_atMost_nat @ Y ) )
= ( X2 = Y ) ) ).
% atMost_eq_iff
thf(fact_14_lessThan__eq__iff,axiom,
! [X2: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X2 )
= ( set_ord_lessThan_nat @ Y ) )
= ( X2 = Y ) ) ).
% lessThan_eq_iff
thf(fact_15_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_16_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_17_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_18_sum__subtractf,axiom,
! [F: nat > real,G: nat > real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X: nat] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
@ A )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups6591440286371151544t_real @ G @ A ) ) ) ).
% sum_subtractf
thf(fact_19_real__polynomial__function__sum__of__powers,axiom,
! [J: nat] :
? [P: real > real] :
( ( weiers3457258110322917882n_real @ P )
& ! [N2: nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( power_power_real @ ( semiri5074537144036343181t_real @ I ) @ J )
@ ( set_ord_atMost_nat @ N2 ) )
= ( P @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% real_polynomial_function_sum_of_powers
thf(fact_20_sum__delta__notmem_I4_J,axiom,
! [X2: nat,S: set_nat,P2: nat > real,Q: nat > real] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [Y2: nat] : ( if_real @ ( X2 = Y2 ) @ ( P2 @ Y2 ) @ ( Q @ Y2 ) )
@ S )
= ( groups6591440286371151544t_real @ Q @ S ) ) ) ).
% sum_delta_notmem(4)
thf(fact_21_sum__delta__notmem_I4_J,axiom,
! [X2: nat,S: set_nat,P2: nat > nat,Q: nat > nat] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y2: nat] : ( if_nat @ ( X2 = Y2 ) @ ( P2 @ Y2 ) @ ( Q @ Y2 ) )
@ S )
= ( groups3542108847815614940at_nat @ Q @ S ) ) ) ).
% sum_delta_notmem(4)
thf(fact_22_sum__delta__notmem_I3_J,axiom,
! [X2: nat,S: set_nat,P2: nat > real,Q: nat > real] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [Y2: nat] : ( if_real @ ( Y2 = X2 ) @ ( P2 @ Y2 ) @ ( Q @ Y2 ) )
@ S )
= ( groups6591440286371151544t_real @ Q @ S ) ) ) ).
% sum_delta_notmem(3)
thf(fact_23_sum__delta__notmem_I3_J,axiom,
! [X2: nat,S: set_nat,P2: nat > nat,Q: nat > nat] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y2: nat] : ( if_nat @ ( Y2 = X2 ) @ ( P2 @ Y2 ) @ ( Q @ Y2 ) )
@ S )
= ( groups3542108847815614940at_nat @ Q @ S ) ) ) ).
% sum_delta_notmem(3)
thf(fact_24_lessThan__subset__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X2 ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X2 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_25_lessThan__subset__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X2 ) @ ( set_or5984915006950818249n_real @ Y ) )
= ( ord_less_eq_real @ X2 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_26_lessThan__subset__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X2 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_27_atMost__subset__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X2 ) @ ( set_ord_atMost_int @ Y ) )
= ( ord_less_eq_int @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_28_atMost__subset__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X2 ) @ ( set_ord_atMost_real @ Y ) )
= ( ord_less_eq_real @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_29_atMost__subset__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_30_atMost__iff,axiom,
! [I2: int,K: int] :
( ( member_int @ I2 @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I2 @ K ) ) ).
% atMost_iff
thf(fact_31_atMost__iff,axiom,
! [I2: real,K: real] :
( ( member_real @ I2 @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I2 @ K ) ) ).
% atMost_iff
thf(fact_32_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_33_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_34_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_35_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_36_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_37_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_38_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_39_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_40_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_41_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_42_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_43_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_44_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_45_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_46_int__sum,axiom,
! [F: nat > nat,A: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3539618377306564664at_int
@ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
@ A ) ) ).
% int_sum
thf(fact_47_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_48_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_49_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_50_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_51_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_52_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_53_bounded__Max__nat,axiom,
! [P2: nat > $o,X2: nat,M2: nat] :
( ( P2 @ X2 )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P2 @ M3 )
=> ~ ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_54_sum__subtractf__nat,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
@ A )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_55_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_56_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_57_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_58_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_59_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_60_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_61_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_62_sum__diff__distrib,axiom,
! [Q: nat > nat,P2: nat > nat,N: nat] :
( ! [X3: nat] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P2 @ X3 ) )
=> ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P2 @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X: nat] : ( minus_minus_nat @ ( P2 @ X ) @ ( Q @ X ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_63_sum__mono,axiom,
! [K2: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K2 ) @ ( groups6591440286371151544t_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_64_sum__mono,axiom,
! [K2: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K2 ) @ ( groups3542108847815614940at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_65_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U: int] :
( collect_int
@ ^ [X: int] : ( ord_less_eq_int @ X @ U ) ) ) ) ).
% atMost_def
thf(fact_66_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U: real] :
( collect_real
@ ^ [X: real] : ( ord_less_eq_real @ X @ U ) ) ) ) ).
% atMost_def
thf(fact_67_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X: nat] : ( ord_less_eq_nat @ X @ U ) ) ) ) ).
% atMost_def
thf(fact_68_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_69_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_70_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_71_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I2: nat > nat,J: nat > nat,T: set_nat,H: nat > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_nat @ ( I2 @ B2 ) @ S2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_72_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I2: nat > nat,J: nat > nat,T: set_nat,H: nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_nat @ ( I2 @ B2 ) @ S2 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S2 )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_73_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ( ( member_nat @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_74_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ( ( member_nat @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_75_sum_Oeq__general,axiom,
! [B3: set_nat,A: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A )
= ( groups6591440286371151544t_real @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_76_sum_Oeq__general,axiom,
! [B3: set_nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_77_sum_Ocong,axiom,
! [A: set_nat,B3: set_nat,G: nat > real,H: nat > real] :
( ( A = B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= ( groups6591440286371151544t_real @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_78_sum_Ocong,axiom,
! [A: set_nat,B3: set_nat,G: nat > nat,H: nat > nat] :
( ( A = B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_79_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_80_sum_Oswap,axiom,
! [G: nat > nat > real,B3: set_nat,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups6591440286371151544t_real @ ( G @ I ) @ B3 )
@ A )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ I @ J2 )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_81_sum_Oswap,axiom,
! [G: nat > nat > nat,B3: set_nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( G @ I ) @ B3 )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [J2: nat] :
( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ I @ J2 )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_82_sum__delta__notmem_I1_J,axiom,
! [X2: nat,S: set_nat,P2: nat > real,Q: nat > real] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [Y2: nat] : ( if_real @ ( Y2 = X2 ) @ ( P2 @ X2 ) @ ( Q @ Y2 ) )
@ S )
= ( groups6591440286371151544t_real @ Q @ S ) ) ) ).
% sum_delta_notmem(1)
thf(fact_83_sum__delta__notmem_I1_J,axiom,
! [X2: nat,S: set_nat,P2: nat > nat,Q: nat > nat] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y2: nat] : ( if_nat @ ( Y2 = X2 ) @ ( P2 @ X2 ) @ ( Q @ Y2 ) )
@ S )
= ( groups3542108847815614940at_nat @ Q @ S ) ) ) ).
% sum_delta_notmem(1)
thf(fact_84_sum__delta__notmem_I2_J,axiom,
! [X2: nat,S: set_nat,P2: nat > real,Q: nat > real] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [Y2: nat] : ( if_real @ ( X2 = Y2 ) @ ( P2 @ X2 ) @ ( Q @ Y2 ) )
@ S )
= ( groups6591440286371151544t_real @ Q @ S ) ) ) ).
% sum_delta_notmem(2)
thf(fact_85_sum__delta__notmem_I2_J,axiom,
! [X2: nat,S: set_nat,P2: nat > nat,Q: nat > nat] :
( ~ ( member_nat @ X2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y2: nat] : ( if_nat @ ( X2 = Y2 ) @ ( P2 @ X2 ) @ ( Q @ Y2 ) )
@ S )
= ( groups3542108847815614940at_nat @ Q @ S ) ) ) ).
% sum_delta_notmem(2)
thf(fact_86_real__polynomial__function__diff,axiom,
! [F: real > real,G: real > real] :
( ( weiers3457258110322917882n_real @ F )
=> ( ( weiers3457258110322917882n_real @ G )
=> ( weiers3457258110322917882n_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% real_polynomial_function_diff
thf(fact_87_real__polynomial__function__power,axiom,
! [F: real > real,N: nat] :
( ( weiers3457258110322917882n_real @ F )
=> ( weiers3457258110322917882n_real
@ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N ) ) ) ).
% real_polynomial_function_power
thf(fact_88_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_89_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_90_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_91_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_92_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_93_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_94_real__arch__simple,axiom,
! [X2: real] :
? [N3: nat] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_95_diff__eq__diff__less__eq,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A2 @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_96_diff__eq__diff__less__eq,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A2 @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_97_diff__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_98_diff__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_99_diff__left__mono,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_100_diff__left__mono,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_101_diff__mono,axiom,
! [A2: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_102_diff__mono,axiom,
! [A2: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_103_real__polynomial__function_Ointros_I2_J,axiom,
! [C: real] :
( weiers3457258110322917882n_real
@ ^ [X: real] : C ) ).
% real_polynomial_function.intros(2)
thf(fact_104_real__polynomial__function__separable,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
=> ? [F2: real > real] :
( ( weiers3457258110322917882n_real @ F2 )
& ( ( F2 @ X2 )
!= ( F2 @ Y ) ) ) ) ).
% real_polynomial_function_separable
thf(fact_105_real__polynomial__function__imp__sum,axiom,
! [F: real > real] :
( ( weiers3457258110322917882n_real @ F )
=> ? [A3: nat > real,N3: nat] :
( F
= ( ^ [X: real] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( A3 @ I ) @ ( power_power_real @ X @ I ) )
@ ( set_ord_atMost_nat @ N3 ) ) ) ) ) ).
% real_polynomial_function_imp_sum
thf(fact_106_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_107_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_108_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_109_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_110_int__diff__cases,axiom,
! [Z: int] :
~ ! [M3: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_111_mult_Oleft__commute,axiom,
! [B: real,A2: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A2 @ C ) )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_112_mult_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_113_mult_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A2 @ C ) )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_114_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_115_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_116_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_117_mult_Oassoc,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_118_mult_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_119_mult_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_120_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_121_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_122_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_123_real__polynomial__function_Ointros_I4_J,axiom,
! [F: real > real,G: real > real] :
( ( weiers3457258110322917882n_real @ F )
=> ( ( weiers3457258110322917882n_real @ G )
=> ( weiers3457258110322917882n_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% real_polynomial_function.intros(4)
thf(fact_124_power__commutes,axiom,
! [A2: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A2 @ N ) @ A2 )
= ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_125_power__commutes,axiom,
! [A2: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 )
= ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_126_power__commutes,axiom,
! [A2: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 )
= ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).
% power_commutes
thf(fact_127_power__mult__distrib,axiom,
! [A2: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A2 @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_128_power__mult__distrib,axiom,
! [A2: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A2 @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_129_power__mult__distrib,axiom,
! [A2: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A2 @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_130_power__commuting__commutes,axiom,
! [X2: real,Y: real,N: nat] :
( ( ( times_times_real @ X2 @ Y )
= ( times_times_real @ Y @ X2 ) )
=> ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_131_power__commuting__commutes,axiom,
! [X2: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X2 @ Y )
= ( times_times_nat @ Y @ X2 ) )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_132_power__commuting__commutes,axiom,
! [X2: int,Y: int,N: nat] :
( ( ( times_times_int @ X2 @ Y )
= ( times_times_int @ Y @ X2 ) )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_133_mult__of__nat__commute,axiom,
! [X2: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_134_mult__of__nat__commute,axiom,
! [X2: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_135_mult__of__nat__commute,axiom,
! [X2: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_136_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_137_sum__distrib__left,axiom,
! [R: real,F: nat > real,A: set_nat] :
( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A ) )
= ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( times_times_real @ R @ ( F @ N4 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_138_sum__distrib__left,axiom,
! [R: nat,F: nat > nat,A: set_nat] :
( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A ) )
= ( groups3542108847815614940at_nat
@ ^ [N4: nat] : ( times_times_nat @ R @ ( F @ N4 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_139_sum__distrib__right,axiom,
! [F: nat > real,A: set_nat,R: real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A ) @ R )
= ( groups6591440286371151544t_real
@ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_140_sum__distrib__right,axiom,
! [F: nat > nat,A: set_nat,R: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ R )
= ( groups3542108847815614940at_nat
@ ^ [N4: nat] : ( times_times_nat @ ( F @ N4 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_141_sum__product,axiom,
! [F: nat > real,A: set_nat,G: nat > real,B3: set_nat] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A ) @ ( groups6591440286371151544t_real @ G @ B3 ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] :
( groups6591440286371151544t_real
@ ^ [J2: nat] : ( times_times_real @ ( F @ I ) @ ( G @ J2 ) )
@ B3 )
@ A ) ) ).
% sum_product
thf(fact_142_sum__product,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat,B3: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] :
( groups3542108847815614940at_nat
@ ^ [J2: nat] : ( times_times_nat @ ( F @ I ) @ ( G @ J2 ) )
@ B3 )
@ A ) ) ).
% sum_product
thf(fact_143_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_144_order__antisym__conv,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_145_order__antisym__conv,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_146_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_147_linorder__le__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_148_linorder__le__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_149_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_150_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_151_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_152_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_153_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_154_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_155_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_156_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_157_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_158_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_159_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_160_ord__eq__le__subst,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_161_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_162_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_163_ord__eq__le__subst,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_164_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_165_ord__eq__le__subst,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_166_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_167_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_168_linorder__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_169_linorder__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_170_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_171_order__eq__refl,axiom,
! [X2: int,Y: int] :
( ( X2 = Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_172_order__eq__refl,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_173_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_174_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_175_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_176_order__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_177_order__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_178_order__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_179_order__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_180_order__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_181_order__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_182_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_183_order__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_184_order__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_185_order__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_186_order__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_187_order__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_188_order__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_189_order__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_190_order__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_191_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_192_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_193_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_194_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_195_antisym,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_196_antisym,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_197_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_198_dual__order_Otrans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_199_dual__order_Otrans,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_200_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_201_dual__order_Oantisym,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_202_dual__order_Oantisym,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_203_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_204_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_205_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_206_linorder__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P2 @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P2 @ B2 @ A3 )
=> ( P2 @ A3 @ B2 ) )
=> ( P2 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_207_linorder__wlog,axiom,
! [P2: int > int > $o,A2: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( P2 @ A3 @ B2 ) )
=> ( ! [A3: int,B2: int] :
( ( P2 @ B2 @ A3 )
=> ( P2 @ A3 @ B2 ) )
=> ( P2 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_208_linorder__wlog,axiom,
! [P2: real > real > $o,A2: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( P2 @ A3 @ B2 ) )
=> ( ! [A3: real,B2: real] :
( ( P2 @ B2 @ A3 )
=> ( P2 @ A3 @ B2 ) )
=> ( P2 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_209_order__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_210_order__trans,axiom,
! [X2: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_211_order__trans,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_212_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_213_order_Otrans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_214_order_Otrans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_215_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_216_order__antisym,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_217_order__antisym,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_218_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_219_ord__le__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_220_ord__le__eq__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_221_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_222_ord__eq__le__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_223_ord__eq__le__trans,axiom,
! [A2: real,B: real,C: real] :
( ( A2 = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_224_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_225_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
& ( ord_less_eq_int @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_226_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
& ( ord_less_eq_real @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_227_le__cases3,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_228_le__cases3,axiom,
! [X2: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_229_le__cases3,axiom,
! [X2: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_230_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_231_nle__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_232_nle__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B ) )
= ( ( ord_less_eq_real @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_233_diff__eq__diff__eq,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A2 = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_234_diff__eq__diff__eq,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A2 = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_235_diff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_236_diff__right__commute,axiom,
! [A2: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_237_diff__right__commute,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_238_real__polynomial__function__iff__sum,axiom,
( weiers3457258110322917882n_real
= ( ^ [F3: real > real] :
? [A4: nat > real,N4: nat] :
( F3
= ( ^ [X: real] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( A4 @ I ) @ ( power_power_real @ X @ I ) )
@ ( set_ord_atMost_nat @ N4 ) ) ) ) ) ) ).
% real_polynomial_function_iff_sum
thf(fact_239_polyfun__eq__coeffs,axiom,
! [C: nat > real,N: nat,D: nat > real] :
( ( ! [X: real] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ X @ I ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( D @ I ) @ ( power_power_real @ X @ I ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( C @ I )
= ( D @ I ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_240_sumr__diff__mult__const2,axiom,
! [F: nat > int,N: nat,R: int] :
( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R ) )
= ( groups3539618377306564664at_int
@ ^ [I: nat] : ( minus_minus_int @ ( F @ I ) @ R )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sumr_diff_mult_const2
thf(fact_241_sumr__diff__mult__const2,axiom,
! [F: nat > real,N: nat,R: real] :
( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R ) )
= ( groups6591440286371151544t_real
@ ^ [I: nat] : ( minus_minus_real @ ( F @ I ) @ R )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sumr_diff_mult_const2
thf(fact_242_sum__k__Bernstein,axiom,
! [N: nat,X2: real] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( weiers7429072931691461095nstein @ N @ K3 @ X2 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) ).
% sum_k_Bernstein
thf(fact_243_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_244_set__times__intro,axiom,
! [A2: real,C2: set_real,B: real,D2: set_real] :
( ( member_real @ A2 @ C2 )
=> ( ( member_real @ B @ D2 )
=> ( member_real @ ( times_times_real @ A2 @ B ) @ ( times_times_set_real @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_245_set__times__intro,axiom,
! [A2: nat,C2: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_set_nat @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_246_set__times__intro,axiom,
! [A2: int,C2: set_int,B: int,D2: set_int] :
( ( member_int @ A2 @ C2 )
=> ( ( member_int @ B @ D2 )
=> ( member_int @ ( times_times_int @ A2 @ B ) @ ( times_times_set_int @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_247_vector__space__over__itself_Oscale__sum__left,axiom,
! [F: nat > real,A: set_nat,X2: real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A ) @ X2 )
= ( groups6591440286371151544t_real
@ ^ [A4: nat] : ( times_times_real @ ( F @ A4 ) @ X2 )
@ A ) ) ).
% vector_space_over_itself.scale_sum_left
thf(fact_248_vector__space__over__itself_Oscale__sum__right,axiom,
! [A2: real,F: nat > real,A: set_nat] :
( ( times_times_real @ A2 @ ( groups6591440286371151544t_real @ F @ A ) )
= ( groups6591440286371151544t_real
@ ^ [X: nat] : ( times_times_real @ A2 @ ( F @ X ) )
@ A ) ) ).
% vector_space_over_itself.scale_sum_right
thf(fact_249_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_250_int__ops_I7_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_251_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_252_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_253_verit__la__generic,axiom,
! [A2: int,X2: int] :
( ( ord_less_eq_int @ A2 @ X2 )
| ( A2 = X2 )
| ( ord_less_eq_int @ X2 @ A2 ) ) ).
% verit_la_generic
thf(fact_254_power__mult,axiom,
! [A2: real,M: nat,N: nat] :
( ( power_power_real @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_255_power__mult,axiom,
! [A2: nat,M: nat,N: nat] :
( ( power_power_nat @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_256_power__mult,axiom,
! [A2: int,M: nat,N: nat] :
( ( power_power_int @ A2 @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A2 @ M ) @ N ) ) ).
% power_mult
thf(fact_257_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_258_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_259_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_260_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_261_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_262_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_263_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_264_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_265_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_266_verit__comp__simplify1_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_267_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_268_verit__la__disequality,axiom,
! [A2: int,B: int] :
( ( A2 = B )
| ~ ( ord_less_eq_int @ A2 @ B )
| ~ ( ord_less_eq_int @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_269_verit__la__disequality,axiom,
! [A2: real,B: real] :
( ( A2 = B )
| ~ ( ord_less_eq_real @ A2 @ B )
| ~ ( ord_less_eq_real @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_270_vector__space__over__itself_Oscale__scale,axiom,
! [A2: real,B: real,X2: real] :
( ( times_times_real @ A2 @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ ( times_times_real @ A2 @ B ) @ X2 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_271_vector__space__over__itself_Oscale__left__commute,axiom,
! [A2: real,B: real,X2: real] :
( ( times_times_real @ A2 @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ B @ ( times_times_real @ A2 @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_272_set__times__elim,axiom,
! [X2: real,A: set_real,B3: set_real] :
( ( member_real @ X2 @ ( times_times_set_real @ A @ B3 ) )
=> ~ ! [A3: real,B2: real] :
( ( X2
= ( times_times_real @ A3 @ B2 ) )
=> ( ( member_real @ A3 @ A )
=> ~ ( member_real @ B2 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_273_set__times__elim,axiom,
! [X2: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ X2 @ ( times_times_set_nat @ A @ B3 ) )
=> ~ ! [A3: nat,B2: nat] :
( ( X2
= ( times_times_nat @ A3 @ B2 ) )
=> ( ( member_nat @ A3 @ A )
=> ~ ( member_nat @ B2 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_274_set__times__elim,axiom,
! [X2: int,A: set_int,B3: set_int] :
( ( member_int @ X2 @ ( times_times_set_int @ A @ B3 ) )
=> ~ ! [A3: int,B2: int] :
( ( X2
= ( times_times_int @ A3 @ B2 ) )
=> ( ( member_int @ A3 @ A )
=> ~ ( member_int @ B2 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_275_int__if,axiom,
! [P2: $o,A2: nat,B: nat] :
( ( P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_276_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_277_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A2: real,X2: real,Y: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ X2 @ Y ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ X2 ) @ ( times_times_real @ A2 @ Y ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_278_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A2: real,B: real,X2: real] :
( ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ X2 )
= ( minus_minus_real @ ( times_times_real @ A2 @ X2 ) @ ( times_times_real @ B @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_279_polyfun__linear__factor__root,axiom,
! [C: nat > int,A2: int,N: nat] :
( ( ( groups3539618377306564664at_int
@ ^ [I: nat] : ( times_times_int @ ( C @ I ) @ ( power_power_int @ A2 @ I ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int )
=> ~ ! [B2: nat > int] :
~ ! [Z3: int] :
( ( groups3539618377306564664at_int
@ ^ [I: nat] : ( times_times_int @ ( C @ I ) @ ( power_power_int @ Z3 @ I ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_int @ ( minus_minus_int @ Z3 @ A2 )
@ ( groups3539618377306564664at_int
@ ^ [I: nat] : ( times_times_int @ ( B2 @ I ) @ ( power_power_int @ Z3 @ I ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_280_polyfun__linear__factor__root,axiom,
! [C: nat > real,A2: real,N: nat] :
( ( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ A2 @ I ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real )
=> ~ ! [B2: nat > real] :
~ ! [Z3: real] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ Z3 @ I ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( minus_minus_real @ Z3 @ A2 )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( B2 @ I ) @ ( power_power_real @ Z3 @ I ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_281_polyfun__linear__factor,axiom,
! [C: nat > int,N: nat,A2: int] :
? [B2: nat > int] :
! [Z3: int] :
( ( groups3539618377306564664at_int
@ ^ [I: nat] : ( times_times_int @ ( C @ I ) @ ( power_power_int @ Z3 @ I ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_int
@ ( times_times_int @ ( minus_minus_int @ Z3 @ A2 )
@ ( groups3539618377306564664at_int
@ ^ [I: nat] : ( times_times_int @ ( B2 @ I ) @ ( power_power_int @ Z3 @ I ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups3539618377306564664at_int
@ ^ [I: nat] : ( times_times_int @ ( C @ I ) @ ( power_power_int @ A2 @ I ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_282_polyfun__linear__factor,axiom,
! [C: nat > real,N: nat,A2: real] :
? [B2: nat > real] :
! [Z3: real] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ Z3 @ I ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_real
@ ( times_times_real @ ( minus_minus_real @ Z3 @ A2 )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( B2 @ I ) @ ( power_power_real @ Z3 @ I ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ A2 @ I ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_283_right__diff__distrib_H,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_284_right__diff__distrib_H,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_285_right__diff__distrib_H,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_286_left__diff__distrib_H,axiom,
! [B: real,C: real,A2: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A2 )
= ( minus_minus_real @ ( times_times_real @ B @ A2 ) @ ( times_times_real @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_287_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A2 )
= ( minus_minus_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_288_left__diff__distrib_H,axiom,
! [B: int,C: int,A2: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A2 )
= ( minus_minus_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_289_right__diff__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_290_right__diff__distrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_291_left__diff__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_292_left__diff__distrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_293_inf__period_I1_J,axiom,
! [P2: real > $o,D2: real,Q: real > $o] :
( ! [X3: real,K4: real] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D2 ) ) ) )
=> ( ! [X3: real,K4: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D2 ) ) ) )
=> ! [X4: real,K5: real] :
( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ ( minus_minus_real @ X4 @ ( times_times_real @ K5 @ D2 ) ) )
& ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K5 @ D2 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_294_inf__period_I1_J,axiom,
! [P2: int > $o,D2: int,Q: int > $o] :
( ! [X3: int,K4: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D2 ) ) ) )
=> ( ! [X3: int,K4: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D2 ) ) ) )
=> ! [X4: int,K5: int] :
( ( ( P2 @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D2 ) ) )
& ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D2 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_295_inf__period_I2_J,axiom,
! [P2: real > $o,D2: real,Q: real > $o] :
( ! [X3: real,K4: real] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D2 ) ) ) )
=> ( ! [X3: real,K4: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D2 ) ) ) )
=> ! [X4: real,K5: real] :
( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ ( minus_minus_real @ X4 @ ( times_times_real @ K5 @ D2 ) ) )
| ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K5 @ D2 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_296_inf__period_I2_J,axiom,
! [P2: int > $o,D2: int,Q: int > $o] :
( ! [X3: int,K4: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D2 ) ) ) )
=> ( ! [X3: int,K4: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D2 ) ) ) )
=> ! [X4: int,K5: int] :
( ( ( P2 @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D2 ) ) )
| ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D2 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_297_lemma__termdiff1,axiom,
! [Z: int,H: int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [P3: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_power_int @ Z @ P3 ) ) @ ( power_power_int @ Z @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups3539618377306564664at_int
@ ^ [P3: nat] : ( times_times_int @ ( power_power_int @ Z @ P3 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P3 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_298_lemma__termdiff1,axiom,
! [Z: real,H: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [P3: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_power_real @ Z @ P3 ) ) @ ( power_power_real @ Z @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups6591440286371151544t_real
@ ^ [P3: nat] : ( times_times_real @ ( power_power_real @ Z @ P3 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P3 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_299_set__plus__intro,axiom,
! [A2: nat,C2: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A2 @ B ) @ ( plus_plus_set_nat @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_300_set__plus__intro,axiom,
! [A2: int,C2: set_int,B: int,D2: set_int] :
( ( member_int @ A2 @ C2 )
=> ( ( member_int @ B @ D2 )
=> ( member_int @ ( plus_plus_int @ A2 @ B ) @ ( plus_plus_set_int @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_301_set__plus__intro,axiom,
! [A2: real,C2: set_real,B: real,D2: set_real] :
( ( member_real @ A2 @ C2 )
=> ( ( member_real @ B @ D2 )
=> ( member_real @ ( plus_plus_real @ A2 @ B ) @ ( plus_plus_set_real @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_302_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_303_add__left__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_304_add__left__cancel,axiom,
! [A2: real,B: real,C: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_305_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_306_add__right__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_307_add__right__cancel,axiom,
! [B: real,A2: real,C: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_308_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_309_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_310_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_311_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_312_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_313_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_314_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_315_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_316_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_317_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_318_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_319_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_320_add__le__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_321_add__le__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_322_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_323_add__le__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_324_add__le__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_325_mult__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( ( times_times_real @ A2 @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_326_mult__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_327_mult__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_328_mult__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( ( times_times_real @ C @ A2 )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_329_mult__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_330_mult__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_331_mult__eq__0__iff,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_332_mult__eq__0__iff,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_333_mult__eq__0__iff,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_334_mult__zero__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_335_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_336_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_337_mult__zero__left,axiom,
! [A2: real] :
( ( times_times_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_338_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_339_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_340_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A2: real,X2: real,B: real] :
( ( ( times_times_real @ A2 @ X2 )
= ( times_times_real @ B @ X2 ) )
= ( ( A2 = B )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_341_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A2: real,X2: real,Y: real] :
( ( ( times_times_real @ A2 @ X2 )
= ( times_times_real @ A2 @ Y ) )
= ( ( X2 = Y )
| ( A2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_342_vector__space__over__itself_Oscale__zero__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_343_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_344_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A2: real,X2: real] :
( ( ( times_times_real @ A2 @ X2 )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_345_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_346_double__eq__0__iff,axiom,
! [A2: real] :
( ( ( plus_plus_real @ A2 @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_347_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_348_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_349_add_Oright__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% add.right_neutral
thf(fact_350_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_351_double__zero__sym,axiom,
! [A2: real] :
( ( zero_zero_real
= ( plus_plus_real @ A2 @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_352_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_353_add__cancel__left__left,axiom,
! [B: int,A2: int] :
( ( ( plus_plus_int @ B @ A2 )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_354_add__cancel__left__left,axiom,
! [B: real,A2: real] :
( ( ( plus_plus_real @ B @ A2 )
= A2 )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_355_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_356_add__cancel__left__right,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_357_add__cancel__left__right,axiom,
! [A2: real,B: real] :
( ( ( plus_plus_real @ A2 @ B )
= A2 )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_358_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_359_add__cancel__right__left,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ B @ A2 ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_360_add__cancel__right__left,axiom,
! [A2: real,B: real] :
( ( A2
= ( plus_plus_real @ B @ A2 ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_361_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_362_add__cancel__right__right,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ A2 @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_363_add__cancel__right__right,axiom,
! [A2: real,B: real] :
( ( A2
= ( plus_plus_real @ A2 @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_364_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_365_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_366_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_367_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_368_add__0,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% add_0
thf(fact_369_diff__self,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% diff_self
thf(fact_370_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_371_diff__0__right,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_0_right
thf(fact_372_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_373_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_374_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_375_diff__zero,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_zero
thf(fact_376_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_377_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_378_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_379_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_380_add__diff__cancel,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_381_add__diff__cancel,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_382_diff__add__cancel,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_383_diff__add__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_384_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_385_add__diff__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_386_add__diff__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_387_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_388_add__diff__cancel__left_H,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_389_add__diff__cancel__left_H,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_390_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_391_add__diff__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_392_add__diff__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_393_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_394_add__diff__cancel__right_H,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_395_add__diff__cancel__right_H,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_396_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_397_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_398_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_399_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_400_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_401_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_402_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_403_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_404_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_405_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_406_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_407_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_408_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_409_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_410_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_411_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_412_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_413_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_414_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_415_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_416_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_417_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_418_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_419_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_420_le__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_421_le__add__same__cancel2,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ B @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_422_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_423_le__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_424_le__add__same__cancel1,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ A2 @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_425_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_426_add__le__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_427_add__le__same__cancel2,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B ) @ B )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_428_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_429_add__le__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_430_add__le__same__cancel1,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A2 ) @ B )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_431_diff__ge__0__iff__ge,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_eq_int @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_432_diff__ge__0__iff__ge,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_eq_real @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_433_sum__squares__eq__zero__iff,axiom,
! [X2: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_434_sum__squares__eq__zero__iff,axiom,
! [X2: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_435_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_436_le__add__diff__inverse2,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_437_le__add__diff__inverse2,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_438_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_439_le__add__diff__inverse,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_440_le__add__diff__inverse,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_441_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_442_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_443_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_444_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_445_int__ops_I5_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_446_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_447_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_448_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_449_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_450_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_451_set__zero__plus2,axiom,
! [A: set_nat,B3: set_nat] :
( ( member_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_set_nat @ B3 @ ( plus_plus_set_nat @ A @ B3 ) ) ) ).
% set_zero_plus2
thf(fact_452_set__zero__plus2,axiom,
! [A: set_int,B3: set_int] :
( ( member_int @ zero_zero_int @ A )
=> ( ord_less_eq_set_int @ B3 @ ( plus_plus_set_int @ A @ B3 ) ) ) ).
% set_zero_plus2
thf(fact_453_set__zero__plus2,axiom,
! [A: set_real,B3: set_real] :
( ( member_real @ zero_zero_real @ A )
=> ( ord_less_eq_set_real @ B3 @ ( plus_plus_set_real @ A @ B3 ) ) ) ).
% set_zero_plus2
thf(fact_454_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_455_verit__sum__simplify,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% verit_sum_simplify
thf(fact_456_verit__sum__simplify,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% verit_sum_simplify
thf(fact_457_set__plus__elim,axiom,
! [X2: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ X2 @ ( plus_plus_set_nat @ A @ B3 ) )
=> ~ ! [A3: nat,B2: nat] :
( ( X2
= ( plus_plus_nat @ A3 @ B2 ) )
=> ( ( member_nat @ A3 @ A )
=> ~ ( member_nat @ B2 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_458_set__plus__elim,axiom,
! [X2: int,A: set_int,B3: set_int] :
( ( member_int @ X2 @ ( plus_plus_set_int @ A @ B3 ) )
=> ~ ! [A3: int,B2: int] :
( ( X2
= ( plus_plus_int @ A3 @ B2 ) )
=> ( ( member_int @ A3 @ A )
=> ~ ( member_int @ B2 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_459_set__plus__elim,axiom,
! [X2: real,A: set_real,B3: set_real] :
( ( member_real @ X2 @ ( plus_plus_set_real @ A @ B3 ) )
=> ~ ! [A3: real,B2: real] :
( ( X2
= ( plus_plus_real @ A3 @ B2 ) )
=> ( ( member_real @ A3 @ A )
=> ~ ( member_real @ B2 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_460_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_461_add__nonpos__eq__0__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_462_add__nonpos__eq__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X2 @ Y )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_463_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_464_add__nonneg__eq__0__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_465_add__nonneg__eq__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X2 @ Y )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_466_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_467_add__nonpos__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_468_add__nonpos__nonpos,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_469_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_470_add__nonneg__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_471_add__nonneg__nonneg,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_472_add__increasing2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_473_add__increasing2,axiom,
! [C: int,B: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_474_add__increasing2,axiom,
! [C: real,B: real,A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A2 )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_475_add__decreasing2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_476_add__decreasing2,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_477_add__decreasing2,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_478_add__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_479_add__increasing,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_480_add__increasing,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_481_add__decreasing,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_482_add__decreasing,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_483_add__decreasing,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_484_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_485_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_486_ring__class_Oring__distribs_I2_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_487_ring__class_Oring__distribs_I2_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_488_ring__class_Oring__distribs_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_489_ring__class_Oring__distribs_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_490_comm__semiring__class_Odistrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_491_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_492_comm__semiring__class_Odistrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_493_distrib__left,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_494_distrib__left,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_495_distrib__left,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_496_distrib__right,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_497_distrib__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_498_distrib__right,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_499_combine__common__factor,axiom,
! [A2: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_500_combine__common__factor,axiom,
! [A2: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_501_combine__common__factor,axiom,
! [A2: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_502_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_503_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_504_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_505_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_506_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I2 @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_507_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I2 @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_508_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_509_group__cancel_Oadd1,axiom,
! [A: int,K: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_510_group__cancel_Oadd1,axiom,
! [A: real,K: real,A2: real,B: real] :
( ( A
= ( plus_plus_real @ K @ A2 ) )
=> ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_511_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A2: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_512_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A2: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A2 @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_513_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A2: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A2 @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_514_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_515_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_516_comm__monoid__add__class_Oadd__0,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_517_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_518_add_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_519_add_Oassoc,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_520_add_Oleft__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_521_add_Oleft__cancel,axiom,
! [A2: real,B: real,C: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_522_add_Oright__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_523_add_Oright__cancel,axiom,
! [B: real,A2: real,C: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C @ A2 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_524_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_525_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_526_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_527_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_528_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_529_add_Ocomm__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% add.comm_neutral
thf(fact_530_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_531_add_Ogroup__left__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_532_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_533_add_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_534_add_Oleft__commute,axiom,
! [B: real,A2: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A2 @ C ) )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_535_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_536_add__left__imp__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_537_add__left__imp__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_538_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_539_add__right__imp__eq,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_540_add__right__imp__eq,axiom,
! [B: real,A2: real,C: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_541_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_542_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_543_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_544_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_545_mult__right__cancel,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C )
= ( times_times_real @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_546_mult__right__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_547_mult__right__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_548_mult__left__cancel,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A2 )
= ( times_times_real @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_549_mult__left__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_550_mult__left__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_551_no__zero__divisors,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A2 @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_552_no__zero__divisors,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_553_no__zero__divisors,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A2 @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_554_divisors__zero,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
= zero_zero_real )
=> ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_555_divisors__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_556_divisors__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
=> ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_557_mult__not__zero,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
!= zero_zero_real )
=> ( ( A2 != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_558_mult__not__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_559_mult__not__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
!= zero_zero_int )
=> ( ( A2 != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_560_conj__le__cong,axiom,
! [X2: int,X5: int,P2: $o,P4: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P2 = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_561_imp__le__cong,axiom,
! [X2: int,X5: int,P2: $o,P4: $o] :
( ( X2 = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P2 = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_562_sum__squares__ge__zero,axiom,
! [X2: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_563_sum__squares__ge__zero,axiom,
! [X2: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_564_lambda__zero,axiom,
( ( ^ [H2: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_565_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_566_lambda__zero,axiom,
( ( ^ [H2: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_567_sum__squares__le__zero__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_568_sum__squares__le__zero__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_569_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_570_add__le__imp__le__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_571_add__le__imp__le__right,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_572_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_573_add__le__imp__le__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_574_add__le__imp__le__left,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_575_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_576_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_577_add__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_578_add__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_579_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A2 @ C4 ) ) ) ).
% less_eqE
thf(fact_580_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_581_add__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_582_add__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_583_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_584_add__mono,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_585_add__mono,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_586_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_587_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_588_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_589_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_590_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_591_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( I2 = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_592_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_593_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_594_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_595_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A2: real,X2: real,Y: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ X2 @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ X2 ) @ ( times_times_real @ A2 @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_596_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A2: real,B: real,X2: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ X2 )
= ( plus_plus_real @ ( times_times_real @ A2 @ X2 ) @ ( times_times_real @ B @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_597_group__cancel_Osub1,axiom,
! [A: real,K: real,A2: real,B: real] :
( ( A
= ( plus_plus_real @ K @ A2 ) )
=> ( ( minus_minus_real @ A @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_598_group__cancel_Osub1,axiom,
! [A: int,K: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( minus_minus_int @ A @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_599_diff__eq__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ( minus_minus_real @ A2 @ B )
= C )
= ( A2
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_600_diff__eq__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( minus_minus_int @ A2 @ B )
= C )
= ( A2
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_601_eq__diff__eq,axiom,
! [A2: real,C: real,B: real] :
( ( A2
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_602_eq__diff__eq,axiom,
! [A2: int,C: int,B: int] :
( ( A2
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_603_add__diff__eq,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_604_add__diff__eq,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_605_diff__diff__eq2,axiom,
! [A2: real,B: real,C: real] :
( ( minus_minus_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_606_diff__diff__eq2,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_607_diff__add__eq,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_608_diff__add__eq,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_609_diff__add__eq__diff__diff__swap,axiom,
! [A2: real,B: real,C: real] :
( ( minus_minus_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_610_diff__add__eq__diff__diff__swap,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_611_add__implies__diff,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_612_add__implies__diff,axiom,
! [C: real,B: real,A2: real] :
( ( ( plus_plus_real @ C @ B )
= A2 )
=> ( C
= ( minus_minus_real @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_613_add__implies__diff,axiom,
! [C: int,B: int,A2: int] :
( ( ( plus_plus_int @ C @ B )
= A2 )
=> ( C
= ( minus_minus_int @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_614_diff__diff__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_615_diff__diff__eq,axiom,
! [A2: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_616_diff__diff__eq,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_617_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_618_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_619_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_620_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_621_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_622_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_623_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_624_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_625_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_626_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_627_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_628_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_629_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_630_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_631_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_632_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_633_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_634_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_635_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: real,A2: real,B: real] :
( ( X2 != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ X2 )
= ( times_times_real @ B @ X2 ) )
=> ( A2 = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_636_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A2: real,X2: real,Y: real] :
( ( A2 != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ X2 )
= ( times_times_real @ A2 @ Y ) )
=> ( X2 = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_637_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [A4: real,B4: real] :
( ( minus_minus_real @ A4 @ B4 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_638_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A4: int,B4: int] :
( ( minus_minus_int @ A4 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_639_power__not__zero,axiom,
! [A2: real,N: nat] :
( ( A2 != zero_zero_real )
=> ( ( power_power_real @ A2 @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_640_power__not__zero,axiom,
! [A2: nat,N: nat] :
( ( A2 != zero_zero_nat )
=> ( ( power_power_nat @ A2 @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_641_power__not__zero,axiom,
! [A2: int,N: nat] :
( ( A2 != zero_zero_int )
=> ( ( power_power_int @ A2 @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_642_sum_Oneutral,axiom,
! [A: set_nat,G: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_643_sum_Oneutral,axiom,
! [A: set_nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_644_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A )
!= zero_zero_real )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_645_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_646_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_647_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_648_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_649_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_650_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_651_add__le__add__imp__diff__le,axiom,
! [I2: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_652_add__le__add__imp__diff__le,axiom,
! [I2: real,K: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_653_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_654_add__le__imp__le__diff,axiom,
! [I2: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_655_add__le__imp__le__diff,axiom,
! [I2: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_656_eq__add__iff1,axiom,
! [A2: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_657_eq__add__iff1,axiom,
! [A2: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_658_eq__add__iff2,axiom,
! [A2: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A2 ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_659_eq__add__iff2,axiom,
! [A2: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_660_square__diff__square__factored,axiom,
! [X2: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X2 @ Y ) @ ( minus_minus_real @ X2 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_661_square__diff__square__factored,axiom,
! [X2: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X2 @ Y ) @ ( minus_minus_int @ X2 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_662_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_663_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_664_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_665_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_666_mult__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_667_mult__mono,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_668_mult__mono,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_669_mult__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_670_mult__mono_H,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_671_mult__mono_H,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_672_zero__le__square,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_673_zero__le__square,axiom,
! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_674_split__mult__pos__le,axiom,
! [A2: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_675_split__mult__pos__le,axiom,
! [A2: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_676_mult__left__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_677_mult__left__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_678_mult__nonpos__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_679_mult__nonpos__nonpos,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_680_mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_681_mult__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_682_mult__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_683_mult__right__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_684_mult__right__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_685_mult__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_686_mult__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_687_mult__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_688_mult__le__0__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_689_mult__le__0__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_690_split__mult__neg__le,axiom,
! [A2: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_691_split__mult__neg__le,axiom,
! [A2: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_692_split__mult__neg__le,axiom,
! [A2: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_693_mult__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_694_mult__nonneg__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_695_mult__nonneg__nonneg,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_696_mult__nonneg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_697_mult__nonneg__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_698_mult__nonneg__nonpos,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_699_mult__nonpos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_700_mult__nonpos__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_701_mult__nonpos__nonneg,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_702_mult__nonneg__nonpos2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_703_mult__nonneg__nonpos2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_704_mult__nonneg__nonpos2,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A2 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_705_zero__le__mult__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_706_zero__le__mult__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_707_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_708_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_709_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_710_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_711_sum_Odistrib,axiom,
! [G: nat > real,H: nat > real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H @ X ) )
@ A )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A ) @ ( groups6591440286371151544t_real @ H @ A ) ) ) ).
% sum.distrib
thf(fact_712_sum_Odistrib,axiom,
! [G: nat > nat,H: nat > nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H @ X ) )
@ A )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A ) @ ( groups3542108847815614940at_nat @ H @ A ) ) ) ).
% sum.distrib
thf(fact_713_real__polynomial__function_Ointros_I3_J,axiom,
! [F: real > real,G: real > real] :
( ( weiers3457258110322917882n_real @ F )
=> ( ( weiers3457258110322917882n_real @ G )
=> ( weiers3457258110322917882n_real
@ ^ [X: real] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% real_polynomial_function.intros(3)
thf(fact_714_ordered__ring__class_Ole__add__iff2,axiom,
! [A2: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_715_ordered__ring__class_Ole__add__iff2,axiom,
! [A2: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A2 ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_716_ordered__ring__class_Ole__add__iff1,axiom,
! [A2: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_717_ordered__ring__class_Ole__add__iff1,axiom,
! [A2: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_718_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C )
= ( B
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_719_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_720_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_721_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_722_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_723_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_724_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_725_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_726_le__add__diff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_727_diff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% diff_add
thf(fact_728_le__diff__eq,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_729_le__diff__eq,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_eq_real @ A2 @ ( minus_minus_real @ C @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_730_diff__le__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_731_diff__le__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( ord_less_eq_real @ A2 @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_732_power__add,axiom,
! [A2: real,M: nat,N: nat] :
( ( power_power_real @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) ) ) ).
% power_add
thf(fact_733_power__add,axiom,
! [A2: nat,M: nat,N: nat] :
( ( power_power_nat @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ).
% power_add
thf(fact_734_power__add,axiom,
! [A2: int,M: nat,N: nat] :
( ( power_power_int @ A2 @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) ) ) ).
% power_add
thf(fact_735_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_736_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_737_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_738_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_739_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_740_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_741_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_742_zero__le__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% zero_le_power
thf(fact_743_zero__le__power,axiom,
! [A2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).
% zero_le_power
thf(fact_744_zero__le__power,axiom,
! [A2: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) ) ) ).
% zero_le_power
thf(fact_745_power__mono,axiom,
! [A2: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_746_power__mono,axiom,
! [A2: int,B: int,N: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_747_power__mono,axiom,
! [A2: real,B: real,N: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_748_sum__nonpos,axiom,
! [A: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_749_sum__nonpos,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_750_sum__nonneg,axiom,
! [A: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_751_sum__nonneg,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_752_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_753_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_754_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_755_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N4: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_756_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_757_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_758_sum__power__add,axiom,
! [X2: int,M: nat,I4: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I: nat] : ( power_power_int @ X2 @ ( plus_plus_nat @ M @ I ) )
@ I4 )
= ( times_times_int @ ( power_power_int @ X2 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ I4 ) ) ) ).
% sum_power_add
thf(fact_759_sum__power__add,axiom,
! [X2: real,M: nat,I4: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( power_power_real @ X2 @ ( plus_plus_nat @ M @ I ) )
@ I4 )
= ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ I4 ) ) ) ).
% sum_power_add
thf(fact_760_polyfun__eq__0,axiom,
! [C: nat > real,N: nat] :
( ( ! [X: real] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ X @ I ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) )
= ( ! [I: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( C @ I )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_0
thf(fact_761_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > real,N: nat,K: nat] :
( ! [W3: real] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ W3 @ I ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( C @ K )
= zero_zero_real ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_762_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_763_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_764_nat__diff__add__eq2,axiom,
! [I2: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U2 ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_765_nat__diff__add__eq1,axiom,
! [J: nat,I2: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_766_nat__le__add__iff2,axiom,
! [I2: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U2 ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_767_nat__le__add__iff1,axiom,
! [J: nat,I2: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_768_nat__eq__add__iff2,axiom,
! [I2: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U2 ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_769_nat__eq__add__iff1,axiom,
! [J: nat,I2: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U2 ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_770_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_771_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_772_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_773_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_774_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_775_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_776_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_777_left__add__mult__distrib,axiom,
! [I2: nat,U2: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U2 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_778_polyfun__diff__alt,axiom,
! [N: nat,A2: nat > int,X2: int,Y: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_int
@ ( groups3539618377306564664at_int
@ ^ [I: nat] : ( times_times_int @ ( A2 @ I ) @ ( power_power_int @ X2 @ I ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups3539618377306564664at_int
@ ^ [I: nat] : ( times_times_int @ ( A2 @ I ) @ ( power_power_int @ Y @ I ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X2 @ Y )
@ ( groups3539618377306564664at_int
@ ^ [J2: nat] :
( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( A2 @ ( plus_plus_nat @ ( plus_plus_nat @ J2 @ K3 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ K3 ) ) @ ( power_power_int @ X2 @ J2 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J2 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_779_polyfun__diff__alt,axiom,
! [N: nat,A2: nat > real,X2: real,Y: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_real
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( A2 @ I ) @ ( power_power_real @ X2 @ I ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( A2 @ I ) @ ( power_power_real @ Y @ I ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X2 @ Y )
@ ( groups6591440286371151544t_real
@ ^ [J2: nat] :
( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( A2 @ ( plus_plus_nat @ ( plus_plus_nat @ J2 @ K3 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ K3 ) ) @ ( power_power_real @ X2 @ J2 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J2 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_780_mult__diff__mult,axiom,
! [X2: real,Y: real,A2: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ Y ) @ ( times_times_real @ A2 @ B ) )
= ( plus_plus_real @ ( times_times_real @ X2 @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X2 @ A2 ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_781_mult__diff__mult,axiom,
! [X2: int,Y: int,A2: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X2 @ Y ) @ ( times_times_int @ A2 @ B ) )
= ( plus_plus_int @ ( times_times_int @ X2 @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X2 @ A2 ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_782_add__scale__eq__noteq,axiom,
! [R: real,A2: real,B: real,C: real,D: real] :
( ( R != zero_zero_real )
=> ( ( ( A2 = B )
& ( C != D ) )
=> ( ( plus_plus_real @ A2 @ ( times_times_real @ R @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_783_add__scale__eq__noteq,axiom,
! [R: nat,A2: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A2 = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_784_add__scale__eq__noteq,axiom,
! [R: int,A2: int,B: int,C: int,D: int] :
( ( R != zero_zero_int )
=> ( ( ( A2 = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A2 @ ( times_times_int @ R @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_785_ge__iff__diff__ge__0,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A4: int] : ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A4 @ B4 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_786_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A4 @ B4 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_787_mult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% mult_1
thf(fact_788_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_789_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_790_mult_Oright__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.right_neutral
thf(fact_791_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_792_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_793_vector__space__over__itself_Oscale__one,axiom,
! [X2: real] :
( ( times_times_real @ one_one_real @ X2 )
= X2 ) ).
% vector_space_over_itself.scale_one
thf(fact_794_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_795_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_796_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_797_power__one__right,axiom,
! [A2: real] :
( ( power_power_real @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_798_power__one__right,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_799_power__one__right,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ one_one_nat )
= A2 ) ).
% power_one_right
thf(fact_800_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_801_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_802_mult__cancel__right2,axiom,
! [A2: real,C: real] :
( ( ( times_times_real @ A2 @ C )
= C )
= ( ( C = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_803_mult__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ( times_times_int @ A2 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_804_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_805_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_806_mult__cancel__left2,axiom,
! [C: real,A2: real] :
( ( ( times_times_real @ C @ A2 )
= C )
= ( ( C = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_807_mult__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ( times_times_int @ C @ A2 )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_808_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_809_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_810_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_811_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_812_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_813_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_814_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_815_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_816_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_817_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_818_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_819_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_820_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_821_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_822_comm__monoid__mult__class_Omult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_823_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_824_comm__monoid__mult__class_Omult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_825_mult_Ocomm__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.comm_neutral
thf(fact_826_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_827_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_828_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_829_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_830_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_831_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_832_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_833_lambda__one,axiom,
( ( ^ [X: real] : X )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_834_lambda__one,axiom,
( ( ^ [X: nat] : X )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_835_lambda__one,axiom,
( ( ^ [X: int] : X )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_836_power__eq__if,axiom,
( power_power_real
= ( ^ [P3: real,M4: nat] : ( if_real @ ( M4 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P3 @ ( power_power_real @ P3 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_837_power__eq__if,axiom,
( power_power_nat
= ( ^ [P3: nat,M4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_838_power__eq__if,axiom,
( power_power_int
= ( ^ [P3: int,M4: nat] : ( if_int @ ( M4 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_839_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_840_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_841_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_842_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_843_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_844_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_845_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_846_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_847_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_848_one__le__power,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% one_le_power
thf(fact_849_one__le__power,axiom,
! [A2: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ).
% one_le_power
thf(fact_850_one__le__power,axiom,
! [A2: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A2 @ N ) ) ) ).
% one_le_power
thf(fact_851_left__right__inverse__power,axiom,
! [X2: real,Y: real,N: nat] :
( ( ( times_times_real @ X2 @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_852_left__right__inverse__power,axiom,
! [X2: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X2 @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_853_left__right__inverse__power,axiom,
! [X2: int,Y: int,N: nat] :
( ( ( times_times_int @ X2 @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_854_power__0,axiom,
! [A2: real] :
( ( power_power_real @ A2 @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_855_power__0,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_856_power__0,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_857_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_858_mult__left__le__one__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_859_mult__left__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_860_mult__right__le__one__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_861_mult__right__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_862_mult__le__one,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_863_mult__le__one,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_864_mult__le__one,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_865_mult__left__le,axiom,
! [C: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_866_mult__left__le,axiom,
! [C: int,A2: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_867_mult__left__le,axiom,
! [C: real,A2: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_868_power__le__one,axiom,
! [A2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_869_power__le__one,axiom,
! [A2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_870_power__le__one,axiom,
! [A2: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ A2 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_871_square__diff__one__factored,axiom,
! [X2: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X2 @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_872_square__diff__one__factored,axiom,
! [X2: int] :
( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X2 @ one_one_int ) @ ( minus_minus_int @ X2 @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_873_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_874_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_875_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_876_power__increasing,axiom,
! [N: nat,N5: nat,A2: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N5 ) ) ) ) ).
% power_increasing
thf(fact_877_power__increasing,axiom,
! [N: nat,N5: nat,A2: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N5 ) ) ) ) ).
% power_increasing
thf(fact_878_power__increasing,axiom,
! [N: nat,N5: nat,A2: real] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ one_one_real @ A2 )
=> ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ A2 @ N5 ) ) ) ) ).
% power_increasing
thf(fact_879_convex__bound__le,axiom,
! [X2: int,A2: int,Y: int,U2: int,V: int] :
( ( ord_less_eq_int @ X2 @ A2 )
=> ( ( ord_less_eq_int @ Y @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U2 @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U2 @ X2 ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_880_convex__bound__le,axiom,
! [X2: real,A2: real,Y: real,U2: real,V: real] :
( ( ord_less_eq_real @ X2 @ A2 )
=> ( ( ord_less_eq_real @ Y @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U2 @ V )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U2 @ X2 ) @ ( times_times_real @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_881_power__decreasing,axiom,
! [N: nat,N5: nat,A2: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N5 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_882_power__decreasing,axiom,
! [N: nat,N5: nat,A2: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N5 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_883_power__decreasing,axiom,
! [N: nat,N5: nat,A2: real] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ A2 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A2 @ N5 ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_884_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_885_power__diff__1__eq,axiom,
! [X2: int,N: nat] :
( ( minus_minus_int @ ( power_power_int @ X2 @ N ) @ one_one_int )
= ( times_times_int @ ( minus_minus_int @ X2 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_886_power__diff__1__eq,axiom,
! [X2: real,N: nat] :
( ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ one_one_real )
= ( times_times_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_887_one__diff__power__eq,axiom,
! [X2: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_888_one__diff__power__eq,axiom,
! [X2: real,N: nat] :
( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) )
= ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_889_pth__d,axiom,
! [X2: real] :
( ( plus_plus_real @ X2 @ zero_zero_real )
= X2 ) ).
% pth_d
thf(fact_890_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_891_add__0__iff,axiom,
! [B: int,A2: int] :
( ( B
= ( plus_plus_int @ B @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_892_add__0__iff,axiom,
! [B: real,A2: real] :
( ( B
= ( plus_plus_real @ B @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% add_0_iff
thf(fact_893_pth__7_I1_J,axiom,
! [X2: real] :
( ( plus_plus_real @ zero_zero_real @ X2 )
= X2 ) ).
% pth_7(1)
thf(fact_894_crossproduct__eq,axiom,
! [W: real,Y: real,X2: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X2 @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X2 @ Y ) ) )
= ( ( W = X2 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_895_crossproduct__eq,axiom,
! [W: nat,Y: nat,X2: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X2 @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X2 @ Y ) ) )
= ( ( W = X2 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_896_crossproduct__eq,axiom,
! [W: int,Y: int,X2: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X2 @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X2 @ Y ) ) )
= ( ( W = X2 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_897_crossproduct__noteq,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( A2 != B )
& ( C != D ) )
= ( ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ D ) )
!= ( plus_plus_real @ ( times_times_real @ A2 @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_898_crossproduct__noteq,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( A2 != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A2 @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_899_crossproduct__noteq,axiom,
! [A2: int,B: int,C: int,D: int] :
( ( ( A2 != B )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A2 @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_900_add__diff__add,axiom,
! [A2: real,C: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_901_add__diff__add,axiom,
! [A2: int,C: int,B: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% add_diff_add
thf(fact_902_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_903_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_904_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_905_sum__unroll,axiom,
! [N: nat,F: nat > int] :
( ( ( N = zero_zero_nat )
=> ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ N ) )
= ( F @ zero_zero_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_int @ ( F @ N ) @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% sum_unroll
thf(fact_906_sum__unroll,axiom,
! [N: nat,F: nat > real] :
( ( ( N = zero_zero_nat )
=> ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ N ) )
= ( F @ zero_zero_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_real @ ( F @ N ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% sum_unroll
thf(fact_907_sum__unroll,axiom,
! [N: nat,F: nat > nat] :
( ( ( N = zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ N ) )
= ( F @ zero_zero_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_nat @ ( F @ N ) @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% sum_unroll
thf(fact_908_Bernoulli_Osum__diff,axiom,
! [F: nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( minus_minus_real @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) )
@ ( set_ord_atMost_nat @ N ) )
= ( minus_minus_real @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( F @ zero_zero_nat ) ) ) ).
% Bernoulli.sum_diff
thf(fact_909_linepath__le__1,axiom,
! [A2: int,B: int,U2: int] :
( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ U2 )
=> ( ( ord_less_eq_int @ U2 @ one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ one_one_int @ U2 ) @ A2 ) @ ( times_times_int @ U2 @ B ) ) @ one_one_int ) ) ) ) ) ).
% linepath_le_1
thf(fact_910_linepath__le__1,axiom,
! [A2: real,B: real,U2: real] :
( ( ord_less_eq_real @ A2 @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ U2 )
=> ( ( ord_less_eq_real @ U2 @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U2 ) @ A2 ) @ ( times_times_real @ U2 @ B ) ) @ one_one_real ) ) ) ) ) ).
% linepath_le_1
thf(fact_911_sum__Bernstein,axiom,
! [N: nat,X2: real] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( weiers7429072931691461095nstein @ N @ K3 @ X2 )
@ ( set_ord_atMost_nat @ N ) )
= one_one_real ) ).
% sum_Bernstein
thf(fact_912_sum__le__prod1,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A2 @ B ) ) ) ) ) ).
% sum_le_prod1
thf(fact_913_Bernstein__nonneg,axiom,
! [X2: real,N: nat,K: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ X2 @ one_one_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( weiers7429072931691461095nstein @ N @ K @ X2 ) ) ) ) ).
% Bernstein_nonneg
thf(fact_914_segment__bound__lemma,axiom,
! [B3: real,X2: real,Y: real,U2: real] :
( ( ord_less_eq_real @ B3 @ X2 )
=> ( ( ord_less_eq_real @ B3 @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ U2 )
=> ( ( ord_less_eq_real @ U2 @ one_one_real )
=> ( ord_less_eq_real @ B3 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U2 ) @ X2 ) @ ( times_times_real @ U2 @ Y ) ) ) ) ) ) ) ).
% segment_bound_lemma
thf(fact_915_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_916_int__ge__induct,axiom,
! [K: int,I2: int,P2: int > $o] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P2 @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_917_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_918_int__le__induct,axiom,
! [I2: int,K: int,P2: int > $o] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P2 @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_919_is__num__normalize_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_920_is__num__normalize_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_921_int__induct,axiom,
! [P2: int > $o,K: int,I2: int] :
( ( P2 @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_induct
thf(fact_922_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_923_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_924_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_925_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_926_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_927_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_928_affine__ineq,axiom,
! [X2: int,V: int,U2: int] :
( ( ord_less_eq_int @ X2 @ one_one_int )
=> ( ( ord_less_eq_int @ V @ U2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ V @ ( times_times_int @ X2 @ U2 ) ) @ ( plus_plus_int @ U2 @ ( times_times_int @ X2 @ V ) ) ) ) ) ).
% affine_ineq
thf(fact_929_affine__ineq,axiom,
! [X2: real,V: real,U2: real] :
( ( ord_less_eq_real @ X2 @ one_one_real )
=> ( ( ord_less_eq_real @ V @ U2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ V @ ( times_times_real @ X2 @ U2 ) ) @ ( plus_plus_real @ U2 @ ( times_times_real @ X2 @ V ) ) ) ) ) ).
% affine_ineq
thf(fact_930_linear__plus__1__le__power,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X2 @ one_one_real ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_931_mult__eq__1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ( ( times_times_nat @ A2 @ B )
= one_one_nat )
= ( ( A2 = one_one_nat )
& ( B = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_932_mult__eq__1,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ( ( times_times_int @ A2 @ B )
= one_one_int )
= ( ( A2 = one_one_int )
& ( B = one_one_int ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_933_mult__eq__1,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ A2 @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ( ( times_times_real @ A2 @ B )
= one_one_real )
= ( ( A2 = one_one_real )
& ( B = one_one_real ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_934_of__nat__code,axiom,
( semiri5074537144036343181t_real
= ( ^ [N4: nat] :
( semiri7260567687927622513x_real
@ ^ [I: real] : ( plus_plus_real @ I @ one_one_real )
@ N4
@ zero_zero_real ) ) ) ).
% of_nat_code
thf(fact_935_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N4: nat] :
( semiri8420488043553186161ux_int
@ ^ [I: int] : ( plus_plus_int @ I @ one_one_int )
@ N4
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_936_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N4: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I: nat] : ( plus_plus_nat @ I @ one_one_nat )
@ N4
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_937_real__of__nat__ge__one__iff,axiom,
! [N: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ one_one_nat @ N ) ) ).
% real_of_nat_ge_one_iff
thf(fact_938_power__le__one__iff,axiom,
! [A2: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ one_one_real )
= ( ( N = zero_zero_nat )
| ( ord_less_eq_real @ A2 @ one_one_real ) ) ) ) ).
% power_le_one_iff
thf(fact_939_diff__bernpoly,axiom,
! [N: nat,X2: real] :
( ( minus_minus_real @ ( bernpoly_real @ N @ ( plus_plus_real @ X2 @ one_one_real ) ) @ ( bernpoly_real @ N @ X2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% diff_bernpoly
thf(fact_940_mult__if__delta,axiom,
! [P2: $o,Q2: real] :
( ( P2
=> ( ( times_times_real @ ( if_real @ P2 @ one_one_real @ zero_zero_real ) @ Q2 )
= Q2 ) )
& ( ~ P2
=> ( ( times_times_real @ ( if_real @ P2 @ one_one_real @ zero_zero_real ) @ Q2 )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_941_mult__if__delta,axiom,
! [P2: $o,Q2: nat] :
( ( P2
=> ( ( times_times_nat @ ( if_nat @ P2 @ one_one_nat @ zero_zero_nat ) @ Q2 )
= Q2 ) )
& ( ~ P2
=> ( ( times_times_nat @ ( if_nat @ P2 @ one_one_nat @ zero_zero_nat ) @ Q2 )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_942_mult__if__delta,axiom,
! [P2: $o,Q2: int] :
( ( P2
=> ( ( times_times_int @ ( if_int @ P2 @ one_one_int @ zero_zero_int ) @ Q2 )
= Q2 ) )
& ( ~ P2
=> ( ( times_times_int @ ( if_int @ P2 @ one_one_int @ zero_zero_int ) @ Q2 )
= zero_zero_int ) ) ) ).
% mult_if_delta
thf(fact_943_root__polyfun,axiom,
! [N: nat,Z: int,A2: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_int @ Z @ N )
= A2 )
= ( ( groups3539618377306564664at_int
@ ^ [I: nat] : ( times_times_int @ ( if_int @ ( I = zero_zero_nat ) @ ( uminus_uminus_int @ A2 ) @ ( if_int @ ( I = N ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z @ I ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ) ).
% root_polyfun
thf(fact_944_root__polyfun,axiom,
! [N: nat,Z: real,A2: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_real @ Z @ N )
= A2 )
= ( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( times_times_real @ ( if_real @ ( I = zero_zero_nat ) @ ( uminus_uminus_real @ A2 ) @ ( if_real @ ( I = N ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z @ I ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ) ).
% root_polyfun
thf(fact_945_neg__equal__iff__equal,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_946_neg__equal__iff__equal,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_947_add_Oinverse__inverse,axiom,
! [A2: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_948_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_949_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_950_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_951_real__polynomial__function__minus,axiom,
! [F: real > real] :
( ( weiers3457258110322917882n_real @ F )
=> ( weiers3457258110322917882n_real
@ ^ [X: real] : ( uminus_uminus_real @ ( F @ X ) ) ) ) ).
% real_polynomial_function_minus
thf(fact_952_neg__le__iff__le,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_953_neg__le__iff__le,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_954_neg__equal__zero,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= A2 )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_955_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_956_equal__neg__zero,axiom,
! [A2: real] :
( ( A2
= ( uminus_uminus_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_957_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_958_neg__equal__0__iff__equal,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_959_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_960_neg__0__equal__iff__equal,axiom,
! [A2: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A2 ) )
= ( zero_zero_real = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_961_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_962_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_963_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_964_mult__minus__right,axiom,
! [A2: real,B: real] :
( ( times_times_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A2 @ B ) ) ) ).
% mult_minus_right
thf(fact_965_mult__minus__right,axiom,
! [A2: int,B: int] :
( ( times_times_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A2 @ B ) ) ) ).
% mult_minus_right
thf(fact_966_minus__mult__minus,axiom,
! [A2: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A2 @ B ) ) ).
% minus_mult_minus
thf(fact_967_minus__mult__minus,axiom,
! [A2: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A2 @ B ) ) ).
% minus_mult_minus
thf(fact_968_mult__minus__left,axiom,
! [A2: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A2 @ B ) ) ) ).
% mult_minus_left
thf(fact_969_mult__minus__left,axiom,
! [A2: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A2 @ B ) ) ) ).
% mult_minus_left
thf(fact_970_vector__space__over__itself_Oscale__minus__right,axiom,
! [A2: real,X2: real] :
( ( times_times_real @ A2 @ ( uminus_uminus_real @ X2 ) )
= ( uminus_uminus_real @ ( times_times_real @ A2 @ X2 ) ) ) ).
% vector_space_over_itself.scale_minus_right
thf(fact_971_vector__space__over__itself_Oscale__minus__left,axiom,
! [A2: real,X2: real] :
( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ X2 )
= ( uminus_uminus_real @ ( times_times_real @ A2 @ X2 ) ) ) ).
% vector_space_over_itself.scale_minus_left
thf(fact_972_add__minus__cancel,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ A2 @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_973_add__minus__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_974_minus__add__cancel,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( plus_plus_real @ A2 @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_975_minus__add__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_976_minus__add__distrib,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_977_minus__add__distrib,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_978_minus__diff__eq,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B ) )
= ( minus_minus_real @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_979_minus__diff__eq,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B ) )
= ( minus_minus_int @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_980_real__add__minus__iff,axiom,
! [X2: real,A2: real] :
( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real )
= ( X2 = A2 ) ) ).
% real_add_minus_iff
thf(fact_981_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_982_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_983_neg__0__le__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_984_neg__0__le__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_985_neg__le__0__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_986_neg__le__0__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_987_less__eq__neg__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_988_less__eq__neg__nonpos,axiom,
! [A2: real] :
( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_989_neg__less__eq__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_990_neg__less__eq__nonneg,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_991_ab__left__minus,axiom,
! [A2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_992_ab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_993_add_Oright__inverse,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_994_add_Oright__inverse,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_995_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_996_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_997_diff__0,axiom,
! [A2: real] :
( ( minus_minus_real @ zero_zero_real @ A2 )
= ( uminus_uminus_real @ A2 ) ) ).
% diff_0
thf(fact_998_diff__0,axiom,
! [A2: int] :
( ( minus_minus_int @ zero_zero_int @ A2 )
= ( uminus_uminus_int @ A2 ) ) ).
% diff_0
thf(fact_999_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_1000_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_1001_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_1002_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_1003_diff__minus__eq__add,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A2 @ B ) ) ).
% diff_minus_eq_add
thf(fact_1004_diff__minus__eq__add,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A2 @ B ) ) ).
% diff_minus_eq_add
thf(fact_1005_uminus__add__conv__diff,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( minus_minus_real @ B @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_1006_uminus__add__conv__diff,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( minus_minus_int @ B @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_1007_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_1008_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_1009_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_1010_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_1011_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_1012_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_1013_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_1014_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_1015_left__minus__one__mult__self,axiom,
! [N: nat,A2: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A2 ) )
= A2 ) ).
% left_minus_one_mult_self
thf(fact_1016_left__minus__one__mult__self,axiom,
! [N: nat,A2: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A2 ) )
= A2 ) ).
% left_minus_one_mult_self
thf(fact_1017_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1018_le__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_minus_iff
thf(fact_1019_le__minus__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A2 ) ) ) ).
% le_minus_iff
thf(fact_1020_minus__le__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_1021_minus__le__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_1022_le__imp__neg__le,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_1023_le__imp__neg__le,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_1024_minus__diff__minus,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B ) ) ) ).
% minus_diff_minus
thf(fact_1025_minus__diff__minus,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B ) ) ) ).
% minus_diff_minus
thf(fact_1026_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_1027_minus__equation__iff,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= B )
= ( ( uminus_uminus_real @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_1028_minus__equation__iff,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= B )
= ( ( uminus_uminus_int @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_1029_equation__minus__iff,axiom,
! [A2: real,B: real] :
( ( A2
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_1030_equation__minus__iff,axiom,
! [A2: int,B: int] :
( ( A2
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_1031_verit__negate__coefficient_I3_J,axiom,
! [A2: real,B: real] :
( ( A2 = B )
=> ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1032_verit__negate__coefficient_I3_J,axiom,
! [A2: int,B: int] :
( ( A2 = B )
=> ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1033_minus__diff__commute,axiom,
! [B: real,A2: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A2 )
= ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_1034_minus__diff__commute,axiom,
! [B: int,A2: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A2 )
= ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_1035_group__cancel_Oneg1,axiom,
! [A: real,K: real,A2: real] :
( ( A
= ( plus_plus_real @ K @ A2 ) )
=> ( ( uminus_uminus_real @ A )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_1036_group__cancel_Oneg1,axiom,
! [A: int,K: int,A2: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( uminus_uminus_int @ A )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_1037_add_Oinverse__distrib__swap,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1038_add_Oinverse__distrib__swap,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1039_square__eq__iff,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ A2 )
= ( times_times_real @ B @ B ) )
= ( ( A2 = B )
| ( A2
= ( uminus_uminus_real @ B ) ) ) ) ).
% square_eq_iff
thf(fact_1040_square__eq__iff,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ A2 )
= ( times_times_int @ B @ B ) )
= ( ( A2 = B )
| ( A2
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_1041_minus__mult__commute,axiom,
! [A2: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( times_times_real @ A2 @ ( uminus_uminus_real @ B ) ) ) ).
% minus_mult_commute
thf(fact_1042_minus__mult__commute,axiom,
! [A2: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( times_times_int @ A2 @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_1043_is__num__normalize_I8_J,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_1044_is__num__normalize_I8_J,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_1045_sum__negf,axiom,
! [F: nat > real,A: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X: nat] : ( uminus_uminus_real @ ( F @ X ) )
@ A )
= ( uminus_uminus_real @ ( groups6591440286371151544t_real @ F @ A ) ) ) ).
% sum_negf
thf(fact_1046_neg__eq__iff__add__eq__0,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= B )
= ( ( plus_plus_real @ A2 @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1047_neg__eq__iff__add__eq__0,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= B )
= ( ( plus_plus_int @ A2 @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1048_eq__neg__iff__add__eq__0,axiom,
! [A2: real,B: real] :
( ( A2
= ( uminus_uminus_real @ B ) )
= ( ( plus_plus_real @ A2 @ B )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1049_eq__neg__iff__add__eq__0,axiom,
! [A2: int,B: int] :
( ( A2
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A2 @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1050_add_Oinverse__unique,axiom,
! [A2: real,B: real] :
( ( ( plus_plus_real @ A2 @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A2 )
= B ) ) ).
% add.inverse_unique
thf(fact_1051_add_Oinverse__unique,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A2 )
= B ) ) ).
% add.inverse_unique
thf(fact_1052_ab__group__add__class_Oab__left__minus,axiom,
! [A2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1053_ab__group__add__class_Oab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1054_add__eq__0__iff,axiom,
! [A2: real,B: real] :
( ( ( plus_plus_real @ A2 @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_1055_add__eq__0__iff,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_1056_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_1057_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_1058_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_1059_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_1060_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_1061_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_1062_square__eq__1__iff,axiom,
! [X2: real] :
( ( ( times_times_real @ X2 @ X2 )
= one_one_real )
= ( ( X2 = one_one_real )
| ( X2
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_1063_square__eq__1__iff,axiom,
! [X2: int] :
( ( ( times_times_int @ X2 @ X2 )
= one_one_int )
= ( ( X2 = one_one_int )
| ( X2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_1064_group__cancel_Osub2,axiom,
! [B3: real,K: real,B: real,A2: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( minus_minus_real @ A2 @ B3 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A2 @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_1065_group__cancel_Osub2,axiom,
! [B3: int,K: int,B: int,A2: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A2 @ B3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_1066_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1067_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1068_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1069_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1070_pth__2,axiom,
( minus_minus_real
= ( ^ [X: real,Y2: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y2 ) ) ) ) ).
% pth_2
thf(fact_1071_real__eq__0__iff__le__ge__0,axiom,
! [X2: real] :
( ( X2 = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_1072_real__minus__mult__self__le,axiom,
! [U2: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U2 @ U2 ) ) @ ( times_times_real @ X2 @ X2 ) ) ).
% real_minus_mult_self_le
thf(fact_1073_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_1074_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_1075_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_1076_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X: real,Y2: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y2 ) ) ) ) ).
% minus_real_def
thf(fact_1077_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_1078_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_1079_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_1080_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_1081_power__minus,axiom,
! [A2: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A2 @ N ) ) ) ).
% power_minus
thf(fact_1082_power__minus,axiom,
! [A2: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A2 @ N ) ) ) ).
% power_minus
thf(fact_1083_real__0__le__add__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1084_real__add__le__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_le_0_iff
thf(fact_1085_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_1086_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1087_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1088_Bernoulli__inequality,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_1089_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1090_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1091_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_1092_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1093_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_1094_arsinh__minus__real,axiom,
! [X2: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X2 ) )
= ( uminus_uminus_real @ ( arsinh_real @ X2 ) ) ) ).
% arsinh_minus_real
thf(fact_1095_ln__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq_real @ one_one_real @ X2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% ln_ge_zero
thf(fact_1096_ln__add__one__self__le__self,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% ln_add_one_self_le_self
thf(fact_1097_binomial__deriv1,axiom,
! [N: nat,A2: real,B: real] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) ) @ ( power_power_real @ A2 @ ( minus_minus_nat @ K3 @ one_one_nat ) ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( plus_plus_real @ A2 @ B ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% binomial_deriv1
thf(fact_1098_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1099_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1100_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1101_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1102_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1103_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_1104_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_1105_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1106_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1107_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_1108_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_1109_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_1110_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_1111_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1112_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1121_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_1122_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1123_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1124_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M: nat] :
( ( ( power_power_nat @ X2 @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X2
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1125_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1126_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_1127_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_1128_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_1129_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_1130_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1131_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1132_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_1133_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_1134_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_1135_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1136_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_1137_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_1138_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_1139_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K4 )
=> ~ ( P2 @ I5 ) )
& ( P2 @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1140_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1141_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1142_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1143_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1144_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1145_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1146_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1147_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1148_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1149_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1150_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1151_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_1152_diff__induct,axiom,
! [P2: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P2 @ X3 @ Y3 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1153_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1154_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P2 @ I ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P2 @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1155_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1156_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1157_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1158_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1159_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1160_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1161_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1162_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_1163_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1164_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P2 @ M5 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_1165_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1166_exists__least__lemma,axiom,
! [P2: nat > $o] :
( ~ ( P2 @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P2 @ X_1 )
=> ? [N3: nat] :
( ~ ( P2 @ N3 )
& ( P2 @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1167_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1168_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_1169_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1170_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1171_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_1172_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1173_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_1174_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P2 @ M5 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_1175_nat__induct__at__least,axiom,
! [M: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P2 @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1176_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,Y3: nat,Z5: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z5 )
=> ( R2 @ X3 @ Z5 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1177_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1178_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_1179_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1180_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_1181_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_1182_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1183_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_1184_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1185_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_1186_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_1187_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_1188_inc__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P2 @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1189_dec__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P2 @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_1190_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_1191_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1192_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_1193_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1194_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1195_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1196_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1197_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1198_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1199_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1200_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_1201_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1202_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_1203_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_1204_strict__inc__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1205_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_1206_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P2 @ M5 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_1207_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P2 @ M5 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_1208_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1209_less__Suc__induct,axiom,
! [I2: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K4: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K4 )
=> ( ( P2 @ I3 @ J3 )
=> ( ( P2 @ J3 @ K4 )
=> ( P2 @ I3 @ K4 ) ) ) ) )
=> ( P2 @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1210_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1211_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_1212_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1213_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1214_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_1215_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1216_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_1217_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1218_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1219_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_1220_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1221_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_1222_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P2 @ I ) ) )
= ( ( P2 @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P2 @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1223_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_1224_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1225_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1226_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_1227_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1228_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1229_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1230_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_1231_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I2: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1232_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1233_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M4 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1234_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1235_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1236_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_1237_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_1238_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1239_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1240_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1241_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_1242_binomial__unroll,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( K = zero_zero_nat )
=> ( ( binomial @ N @ K )
= one_one_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( binomial @ N @ K )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ) ).
% binomial_unroll
thf(fact_1243_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_1244_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_1245_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_1246_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1247_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_1248_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P2 @ I5 ) )
& ( P2 @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1249_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_1250_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1251_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1252_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1253_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I2 @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1254_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1255_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_1256_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_1257_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_1258_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_1259_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1260_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1261_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_1262_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1263_int__of__nat__induct,axiom,
! [P2: int > $o,Z: int] :
( ! [N3: nat] : ( P2 @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P2 @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_1264_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_1265_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_1266_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1267_nat__power__less__imp__less,axiom,
! [I2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1268_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( power_power_real @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ n2 @ I ) ) @ j )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ n2 @ n ) ) )
= ( minus_minus_real
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( power_power_real @ ( semiri5074537144036343181t_real @ I ) @ j )
@ ( set_ord_atMost_nat @ n2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( power_power_real @ ( semiri5074537144036343181t_real @ I ) @ j )
@ ( set_ord_lessThan_nat @ n ) ) ) ) ).
%------------------------------------------------------------------------------