TPTP Problem File: SLH0545^1.p
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%------------------------------------------------------------------------------
% 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 : Actuarial_Mathematics/0001_Interest/prob_00588_023553__12948836_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1362 ( 688 unt; 78 typ; 0 def)
% Number of atoms : 3134 (1676 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 10152 ( 302 ~; 78 |; 155 &;8603 @)
% ( 0 <=>;1014 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 5 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 279 ( 279 >; 0 *; 0 +; 0 <<)
% Number of symbols : 73 ( 70 usr; 13 con; 0-3 aty)
% Number of variables : 3025 ( 155 ^;2828 !; 42 ?;3025 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:14:11.440
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Filter__Ofilter_It__Complex__Ocomplex_J,type,
filter_complex: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
filter_real: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Int__Oint_J,type,
filter_int: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (70)
thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
at_top_nat: filter_nat ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Complex__Ocomplex,type,
filter6923414461901439796omplex: ( nat > complex ) > filter_complex > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Int__Oint,type,
filterlim_nat_int: ( nat > int ) > filter_int > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__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__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__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_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
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_If_001t__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
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_Interest_Oacc,type,
acc: real > nat > nat > real ).
thf(sy_c_Interest_Oacc__cont,type,
acc_cont: real > real > real ).
thf(sy_c_Interest_Oacc__due,type,
acc_due: real > nat > nat > real ).
thf(sy_c_Interest_Oann,type,
ann: real > nat > nat > real ).
thf(sy_c_Interest_Oann__cont,type,
ann_cont: real > real > real ).
thf(sy_c_Interest_Oann__due,type,
ann_due: real > nat > nat > real ).
thf(sy_c_Interest_Od__nom,type,
d_nom: real > nat > real ).
thf(sy_c_Interest_Oi__force,type,
i_force: real > real ).
thf(sy_c_Interest_Oi__nom,type,
i_nom: real > nat > real ).
thf(sy_c_Interest_Ointerest,type,
interest: real > $o ).
thf(sy_c_Interest_Operp,type,
perp: real > nat > real ).
thf(sy_c_Interest_Operp__due,type,
perp_due: real > nat > real ).
thf(sy_c_Interest_Ov__pres,type,
v_pres: real > real ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
semiri8010041392384452111omplex: nat > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Complex__Ocomplex,type,
ord_less_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
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_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Complex__Ocomplex,type,
topolo2444363109189100025omplex: complex > filter_complex ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Int__Oint,type,
topolo8924058970096914807ds_int: int > filter_int ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Nat__Onat,type,
topolo8926549440605965083ds_nat: nat > filter_nat ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(sy_c_Transcendental_Opowr_001t__Complex__Ocomplex,type,
powr_complex: complex > complex > complex ).
thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
powr_real: real > real > real ).
thf(sy_v_i,type,
i: real ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1271)
thf(fact_0_False,axiom,
i != zero_zero_real ).
% False
thf(fact_1_delta__0__iff__i__0,axiom,
( ( ( i_force @ i )
= zero_zero_real )
= ( i = zero_zero_real ) ) ).
% delta_0_iff_i_0
thf(fact_2_interest__axioms,axiom,
interest @ i ).
% interest_axioms
thf(fact_3_v__futr__pos,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ i ) ).
% v_futr_pos
thf(fact_4_lim__d__nom,axiom,
filterlim_nat_real @ ( d_nom @ i ) @ ( topolo2815343760600316023s_real @ ( i_force @ i ) ) @ at_top_nat ).
% lim_d_nom
thf(fact_5_tendsto__zero__divide__iff,axiom,
! [C: complex,A: nat > complex] :
( ( C != zero_zero_complex )
=> ( ( filter6923414461901439796omplex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ ( A @ N ) @ C )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ at_top_nat )
= ( filter6923414461901439796omplex @ A @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ at_top_nat ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_6_tendsto__zero__divide__iff,axiom,
! [C: real,A: nat > real] :
( ( C != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ ( A @ N ) @ C )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
= ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_7_power__strict__decreasing__iff,axiom,
! [B: real,M: nat,N2: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_8_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_9_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_10_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_11_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_12_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_13_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_14_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_15_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_16_lim__i__nom,axiom,
filterlim_nat_real @ ( i_nom @ i ) @ ( topolo2815343760600316023s_real @ ( i_force @ i ) ) @ at_top_nat ).
% lim_i_nom
thf(fact_17_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_18_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_19_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_20_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
= zero_zero_complex ) ).
% power_0_Suc
thf(fact_21_power__0__Suc,axiom,
! [N2: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_22_power__inject__exp,axiom,
! [A: real,M: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N2 ) )
= ( M = N2 ) ) ) ).
% power_inject_exp
thf(fact_23_power__inject__exp,axiom,
! [A: nat,M: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N2 ) )
= ( M = N2 ) ) ) ).
% power_inject_exp
thf(fact_24_power__inject__exp,axiom,
! [A: int,M: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N2 ) )
= ( M = N2 ) ) ) ).
% power_inject_exp
thf(fact_25_i__nom__0__iff__i__0,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ( ( i_nom @ i @ M )
= zero_zero_real )
= ( i = zero_zero_real ) ) ) ).
% i_nom_0_iff_i_0
thf(fact_26_d__nom__0__iff__i__0,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ( ( d_nom @ i @ M )
= zero_zero_real )
= ( i = zero_zero_real ) ) ) ).
% d_nom_0_iff_i_0
thf(fact_27_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_28_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_29_power__one__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_30_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_31_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_32_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_33_nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_34_i__nom__pos__iff__i__pos,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ( ord_less_real @ zero_zero_real @ ( i_nom @ i @ M ) )
= ( ord_less_real @ zero_zero_real @ i ) ) ) ).
% i_nom_pos_iff_i_pos
thf(fact_35_d__nom__pos__iff__i__pos,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ( ord_less_real @ zero_zero_real @ ( d_nom @ i @ M ) )
= ( ord_less_real @ zero_zero_real @ i ) ) ) ).
% d_nom_pos_iff_i_pos
thf(fact_36_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_37_division__ring__divide__zero,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_38_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_39_divide__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_40_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_41_divide__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ C @ A )
= ( divide1717551699836669952omplex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_42_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_43_divide__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divide_eq_0_iff
thf(fact_44_power__one,axiom,
! [N2: nat] :
( ( power_power_real @ one_one_real @ N2 )
= one_one_real ) ).
% power_one
thf(fact_45_power__one,axiom,
! [N2: nat] :
( ( power_power_nat @ one_one_nat @ N2 )
= one_one_nat ) ).
% power_one
thf(fact_46_power__one,axiom,
! [N2: nat] :
( ( power_power_complex @ one_one_complex @ N2 )
= one_one_complex ) ).
% power_one
thf(fact_47_power__one,axiom,
! [N2: nat] :
( ( power_power_int @ one_one_int @ N2 )
= one_one_int ) ).
% power_one
thf(fact_48_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_49_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_50_power__Suc0__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_51_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_52_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_53_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_54_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_55_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_56_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_57_divide__self__if,axiom,
! [A: complex] :
( ( ( A = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= zero_zero_complex ) )
& ( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ) ).
% divide_self_if
thf(fact_58_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_59_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% divide_self
thf(fact_60_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_61_one__eq__divide__iff,axiom,
! [A: complex,B: complex] :
( ( one_one_complex
= ( divide1717551699836669952omplex @ A @ B ) )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_62_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_63_divide__eq__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_64_i__nom__1,axiom,
( ( i_nom @ i @ one_one_nat )
= i ) ).
% i_nom_1
thf(fact_65_power__eq__0__iff,axiom,
! [A: real,N2: nat] :
( ( ( power_power_real @ A @ N2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_66_power__eq__0__iff,axiom,
! [A: nat,N2: nat] :
( ( ( power_power_nat @ A @ N2 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_67_power__eq__0__iff,axiom,
! [A: complex,N2: nat] :
( ( ( power_power_complex @ A @ N2 )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_68_power__eq__0__iff,axiom,
! [A: int,N2: nat] :
( ( ( power_power_int @ A @ N2 )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% power_eq_0_iff
thf(fact_69_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_70_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_71_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_72_s_H_H__calc,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( i != zero_zero_real )
=> ( ( acc_due @ i @ M @ N2 )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ N2 ) @ one_one_real ) @ ( d_nom @ i @ M ) ) ) ) ) ).
% s''_calc
thf(fact_73_interest_Oi__nom__1,axiom,
! [I: real] :
( ( interest @ I )
=> ( ( i_nom @ I @ one_one_nat )
= I ) ) ).
% interest.i_nom_1
thf(fact_74_interest_Oi__nom__0__iff__i__0,axiom,
! [I: real,M: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( ( i_nom @ I @ M )
= zero_zero_real )
= ( I = zero_zero_real ) ) ) ) ).
% interest.i_nom_0_iff_i_0
thf(fact_75_power__gt__expt,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).
% power_gt_expt
thf(fact_76_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_77_interest_Oi__nom__pos__iff__i__pos,axiom,
! [I: real,M: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( ord_less_real @ zero_zero_real @ ( i_nom @ I @ M ) )
= ( ord_less_real @ zero_zero_real @ I ) ) ) ) ).
% interest.i_nom_pos_iff_i_pos
thf(fact_78_interest_Od__nom__0__iff__i__0,axiom,
! [I: real,M: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( ( d_nom @ I @ M )
= zero_zero_real )
= ( I = zero_zero_real ) ) ) ) ).
% interest.d_nom_0_iff_i_0
thf(fact_79_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= zero_zero_real ) ) ).
% zero_power
thf(fact_80_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_81_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_complex @ zero_zero_complex @ N2 )
= zero_zero_complex ) ) ).
% zero_power
thf(fact_82_zero__power,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ).
% zero_power
thf(fact_83_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_84_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_85_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_86_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_87_interest_Od__nom__pos__iff__i__pos,axiom,
! [I: real,M: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( ord_less_real @ zero_zero_real @ ( d_nom @ I @ M ) )
= ( ord_less_real @ zero_zero_real @ I ) ) ) ) ).
% interest.d_nom_pos_iff_i_pos
thf(fact_88_one__less__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_89_one__less__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_90_one__less__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).
% one_less_power
thf(fact_91_interest_Odelta__0__iff__i__0,axiom,
! [I: real] :
( ( interest @ I )
=> ( ( ( i_force @ I )
= zero_zero_real )
= ( I = zero_zero_real ) ) ) ).
% interest.delta_0_iff_i_0
thf(fact_92_interest_Olim__i__nom,axiom,
! [I: real] :
( ( interest @ I )
=> ( filterlim_nat_real @ ( i_nom @ I ) @ ( topolo2815343760600316023s_real @ ( i_force @ I ) ) @ at_top_nat ) ) ).
% interest.lim_i_nom
thf(fact_93_linordered__field__no__ub,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_94_linordered__field__no__lb,axiom,
! [X2: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X2 ) ).
% linordered_field_no_lb
thf(fact_95_tendsto__null__power,axiom,
! [F: nat > real,F2: filter_nat,N2: nat] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( power_power_real @ ( F @ X3 ) @ N2 )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ) ).
% tendsto_null_power
thf(fact_96_tendsto__null__power,axiom,
! [F: nat > complex,F2: filter_nat,N2: nat] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( power_power_complex @ ( F @ X3 ) @ N2 )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 ) ) ) ).
% tendsto_null_power
thf(fact_97_interest__def,axiom,
( interest
= ( ^ [I2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ I2 ) ) ) ) ).
% interest_def
thf(fact_98_interest_Ov__futr__pos,axiom,
! [I: real] :
( ( interest @ I )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ I ) ) ) ).
% interest.v_futr_pos
thf(fact_99_interest_Ointro,axiom,
! [I: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ I ) )
=> ( interest @ I ) ) ).
% interest.intro
thf(fact_100_power__not__zero,axiom,
! [A: real,N2: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N2 )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_101_power__not__zero,axiom,
! [A: nat,N2: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N2 )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_102_power__not__zero,axiom,
! [A: complex,N2: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N2 )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_103_power__not__zero,axiom,
! [A: int,N2: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N2 )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_104_interest_Olim__d__nom,axiom,
! [I: real] :
( ( interest @ I )
=> ( filterlim_nat_real @ ( d_nom @ I ) @ ( topolo2815343760600316023s_real @ ( i_force @ I ) ) @ at_top_nat ) ) ).
% interest.lim_d_nom
thf(fact_105_add__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_106_add__divide__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_107_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_108_diff__divide__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
= ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_109_power__divide,axiom,
! [A: real,B: real,N2: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N2 )
= ( divide_divide_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% power_divide
thf(fact_110_power__divide,axiom,
! [A: complex,B: complex,N2: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N2 )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).
% power_divide
thf(fact_111_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_112_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_113_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_114_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_115_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_116_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_117_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_118_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_119_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_120_zero__less__power,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_121_zero__less__power,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_122_zero__less__power,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% zero_less_power
thf(fact_123_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_124_right__inverse__eq,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_125_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= one_one_real ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N2 )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_126_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= one_one_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_127_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N2 )
= one_one_complex ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N2 )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_128_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= one_one_int ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_129_power__strict__increasing,axiom,
! [N2: nat,N3: nat,A: real] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_130_power__strict__increasing,axiom,
! [N2: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_131_power__strict__increasing,axiom,
! [N2: nat,N3: nat,A: int] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_132_power__less__imp__less__exp,axiom,
! [A: real,M: nat,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_133_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_134_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_135_power__one__over,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N2 )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% power_one_over
thf(fact_136_power__one__over,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N2 )
= ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_one_over
thf(fact_137_tendsto__add__const__iff,axiom,
! [C: real,F: nat > real,D: real,F2: filter_nat] :
( ( filterlim_nat_real
@ ^ [X3: nat] : ( plus_plus_real @ C @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( plus_plus_real @ C @ D ) )
@ F2 )
= ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ D ) @ F2 ) ) ).
% tendsto_add_const_iff
thf(fact_138_tendsto__add__const__iff,axiom,
! [C: complex,F: nat > complex,D: complex,F2: filter_nat] :
( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( plus_plus_complex @ C @ ( F @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( plus_plus_complex @ C @ D ) )
@ F2 )
= ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ D ) @ F2 ) ) ).
% tendsto_add_const_iff
thf(fact_139_tendsto__add,axiom,
! [F: nat > nat,A: nat,F2: filter_nat,G: nat > nat,B: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ A ) @ F2 )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ B ) @ F2 )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( plus_plus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo8926549440605965083ds_nat @ ( plus_plus_nat @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_add
thf(fact_140_tendsto__add,axiom,
! [F: nat > real,A: real,F2: filter_nat,G: nat > real,B: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( plus_plus_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( plus_plus_real @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_add
thf(fact_141_tendsto__add,axiom,
! [F: nat > complex,A: complex,F2: filter_nat,G: nat > complex,B: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( ( filter6923414461901439796omplex @ G @ ( topolo2444363109189100025omplex @ B ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( plus_plus_complex @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( plus_plus_complex @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_add
thf(fact_142_tendsto__diff,axiom,
! [F: nat > real,A: real,F2: filter_nat,G: nat > real,B: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( minus_minus_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( minus_minus_real @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_diff
thf(fact_143_tendsto__diff,axiom,
! [F: nat > complex,A: complex,F2: filter_nat,G: nat > complex,B: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( ( filter6923414461901439796omplex @ G @ ( topolo2444363109189100025omplex @ B ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( minus_minus_complex @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( minus_minus_complex @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_diff
thf(fact_144_tendsto__power__strong,axiom,
! [F: nat > nat,A: nat,F2: filter_nat,G: nat > nat,B: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ A ) @ F2 )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ B ) @ F2 )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( power_power_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo8926549440605965083ds_nat @ ( power_power_nat @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_power_strong
thf(fact_145_tendsto__power__strong,axiom,
! [F: nat > int,A: int,F2: filter_nat,G: nat > nat,B: nat] :
( ( filterlim_nat_int @ F @ ( topolo8924058970096914807ds_int @ A ) @ F2 )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ B ) @ F2 )
=> ( filterlim_nat_int
@ ^ [X3: nat] : ( power_power_int @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo8924058970096914807ds_int @ ( power_power_int @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_power_strong
thf(fact_146_tendsto__power,axiom,
! [F: nat > real,A: real,F2: filter_nat,N2: nat] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( power_power_real @ ( F @ X3 ) @ N2 )
@ ( topolo2815343760600316023s_real @ ( power_power_real @ A @ N2 ) )
@ F2 ) ) ).
% tendsto_power
thf(fact_147_tendsto__power,axiom,
! [F: nat > complex,A: complex,F2: filter_nat,N2: nat] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( power_power_complex @ ( F @ X3 ) @ N2 )
@ ( topolo2444363109189100025omplex @ ( power_power_complex @ A @ N2 ) )
@ F2 ) ) ).
% tendsto_power
thf(fact_148_less__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_149_divide__less__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_150_power__strict__decreasing,axiom,
! [N2: nat,N3: nat,A: real] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_151_power__strict__decreasing,axiom,
! [N2: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_152_power__strict__decreasing,axiom,
! [N2: nat,N3: nat,A: int] :
( ( ord_less_nat @ N2 @ N3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_153_less__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_154_gt__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_155_power__gt1,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_156_power__gt1,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_157_power__gt1,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_158_tendsto__add__zero,axiom,
! [F: nat > nat,F2: filter_nat,G: nat > nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ zero_zero_nat ) @ F2 )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ zero_zero_nat ) @ F2 )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( plus_plus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo8926549440605965083ds_nat @ zero_zero_nat )
@ F2 ) ) ) ).
% tendsto_add_zero
thf(fact_159_tendsto__add__zero,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( plus_plus_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ) ).
% tendsto_add_zero
thf(fact_160_tendsto__add__zero,axiom,
! [F: nat > complex,F2: filter_nat,G: nat > complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 )
=> ( ( filter6923414461901439796omplex @ G @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( plus_plus_complex @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 ) ) ) ).
% tendsto_add_zero
thf(fact_161_Lim__transform__eq,axiom,
! [F: nat > real,G: nat > real,F2: filter_nat,A: real] :
( ( filterlim_nat_real
@ ^ [X3: nat] : ( minus_minus_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
=> ( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
= ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ A ) @ F2 ) ) ) ).
% Lim_transform_eq
thf(fact_162_Lim__transform__eq,axiom,
! [F: nat > complex,G: nat > complex,F2: filter_nat,A: complex] :
( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( minus_minus_complex @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 )
=> ( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
= ( filter6923414461901439796omplex @ G @ ( topolo2444363109189100025omplex @ A ) @ F2 ) ) ) ).
% Lim_transform_eq
thf(fact_163_LIM__zero__cancel,axiom,
! [F: nat > real,L: real,F2: filter_nat] :
( ( filterlim_nat_real
@ ^ [X3: nat] : ( minus_minus_real @ ( F @ X3 ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 ) ) ).
% LIM_zero_cancel
thf(fact_164_LIM__zero__cancel,axiom,
! [F: nat > complex,L: complex,F2: filter_nat] :
( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( minus_minus_complex @ ( F @ X3 ) @ L )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ F2 ) ) ).
% LIM_zero_cancel
thf(fact_165_Lim__transform2,axiom,
! [F: nat > real,A: real,F2: filter_nat,G: nat > real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_nat_real
@ ^ [X3: nat] : ( minus_minus_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
=> ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ A ) @ F2 ) ) ) ).
% Lim_transform2
thf(fact_166_Lim__transform2,axiom,
! [F: nat > complex,A: complex,F2: filter_nat,G: nat > complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( minus_minus_complex @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 )
=> ( filter6923414461901439796omplex @ G @ ( topolo2444363109189100025omplex @ A ) @ F2 ) ) ) ).
% Lim_transform2
thf(fact_167_Lim__transform,axiom,
! [G: nat > real,A: real,F2: filter_nat,F: nat > real] :
( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_nat_real
@ ^ [X3: nat] : ( minus_minus_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 ) ) ) ).
% Lim_transform
thf(fact_168_Lim__transform,axiom,
! [G: nat > complex,A: complex,F2: filter_nat,F: nat > complex] :
( ( filter6923414461901439796omplex @ G @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( minus_minus_complex @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 ) ) ) ).
% Lim_transform
thf(fact_169_LIM__zero__iff,axiom,
! [F: nat > real,L: real,F2: filter_nat] :
( ( filterlim_nat_real
@ ^ [X3: nat] : ( minus_minus_real @ ( F @ X3 ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
= ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 ) ) ).
% LIM_zero_iff
thf(fact_170_LIM__zero__iff,axiom,
! [F: nat > complex,L: complex,F2: filter_nat] :
( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( minus_minus_complex @ ( F @ X3 ) @ L )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 )
= ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ F2 ) ) ).
% LIM_zero_iff
thf(fact_171_LIM__zero,axiom,
! [F: nat > real,L: real,F2: filter_nat] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( minus_minus_real @ ( F @ X3 ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ).
% LIM_zero
thf(fact_172_LIM__zero,axiom,
! [F: nat > complex,L: complex,F2: filter_nat] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( minus_minus_complex @ ( F @ X3 ) @ L )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 ) ) ).
% LIM_zero
thf(fact_173_tendsto__divide__zero,axiom,
! [F: nat > real,F2: filter_nat,C: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( divide_divide_real @ ( F @ X3 ) @ C )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ).
% tendsto_divide_zero
thf(fact_174_tendsto__divide__zero,axiom,
! [F: nat > complex,F2: filter_nat,C: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( divide1717551699836669952omplex @ ( F @ X3 ) @ C )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 ) ) ).
% tendsto_divide_zero
thf(fact_175_tendsto__divide,axiom,
! [F: nat > real,A: real,F2: filter_nat,G: nat > real,B: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F2 )
=> ( ( B != zero_zero_real )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( divide_divide_real @ A @ B ) )
@ F2 ) ) ) ) ).
% tendsto_divide
thf(fact_176_tendsto__divide,axiom,
! [F: nat > complex,A: complex,F2: filter_nat,G: nat > complex,B: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( ( filter6923414461901439796omplex @ G @ ( topolo2444363109189100025omplex @ B ) @ F2 )
=> ( ( B != zero_zero_complex )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( divide1717551699836669952omplex @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( divide1717551699836669952omplex @ A @ B ) )
@ F2 ) ) ) ) ).
% tendsto_divide
thf(fact_177_power__Suc__less__one,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_178_power__Suc__less__one,axiom,
! [A: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_179_power__Suc__less__one,axiom,
! [A: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_180_s__calc,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( i != zero_zero_real )
=> ( ( acc @ i @ M @ N2 )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ N2 ) @ one_one_real ) @ ( i_nom @ i @ M ) ) ) ) ) ).
% s_calc
thf(fact_181_power__tendsto__0__iff,axiom,
! [N2: nat,F: nat > real,F2: filter_nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( filterlim_nat_real
@ ^ [X3: nat] : ( power_power_real @ ( F @ X3 ) @ N2 )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
= ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 ) ) ) ).
% power_tendsto_0_iff
thf(fact_182_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_183_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_184_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_185_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% div_self
thf(fact_186_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_187_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_188_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_189_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_190_diff__gt__0__iff__gt,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ zero_zero_complex @ ( minus_minus_complex @ A @ B ) )
= ( ord_less_complex @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_191_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_192_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_193_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_194_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_195_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_196_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_197_less__add__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ ( plus_plus_complex @ B @ A ) )
= ( ord_less_complex @ zero_zero_complex @ B ) ) ).
% less_add_same_cancel2
thf(fact_198_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_199_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_200_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_201_less__add__same__cancel1,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ ( plus_plus_complex @ A @ B ) )
= ( ord_less_complex @ zero_zero_complex @ B ) ) ).
% less_add_same_cancel1
thf(fact_202_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_203_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_204_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_205_add__less__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A @ B ) @ B )
= ( ord_less_complex @ A @ zero_zero_complex ) ) ).
% add_less_same_cancel2
thf(fact_206_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_207_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_208_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_209_add__less__same__cancel1,axiom,
! [B: complex,A: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ B @ A ) @ B )
= ( ord_less_complex @ A @ zero_zero_complex ) ) ).
% add_less_same_cancel1
thf(fact_210_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_211_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_212_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_213_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_214_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_215_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_216_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_217_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_218_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_219_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_220_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_221_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_222_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_223_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_224_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_225_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_226_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_227_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_228_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( ( plus_plus_complex @ B @ A )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_229_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_230_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_231_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_232_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_233_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_234_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_235_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_236_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ B @ A ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_237_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_238_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_239_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_240_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ A @ B ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_241_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_242_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_243_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_244_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_245_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_246_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_247_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_248_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_249_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_250_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_251_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_252_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_253_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_254_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_255_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_256_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_257_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_258_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_259_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_260_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_261_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_262_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_263_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_264_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_265_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_266_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_267_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_268_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_269_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_270_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_271_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_272_div__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% div_0
thf(fact_273_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_274_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_275_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_276_div__by__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_277_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_278_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_279_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_280_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_281_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_282_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_283_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_284_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_285_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_286_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_287_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_288_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_289_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_290_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_291_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_292_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_293_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_294_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_295_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_296_div__by__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ one_one_complex )
= A ) ).
% div_by_1
thf(fact_297_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_298_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_299_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_300_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_301_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_302_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_303_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_304_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_305_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_306_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_307_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_308_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_309_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_310_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_311_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_312_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_313_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_314_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_315_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_316_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_317_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_318_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_319_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_320_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_321_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_322_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_323_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_324_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_325_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_326_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_327_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_328_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_329_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_330_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_331_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_332_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_333_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_334_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_335_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_336_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_337_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_338_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_339_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_340_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_341_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_342_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_343_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_344_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_345_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_346_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_347_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_348_gr__implies__not__zero,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_349_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_350_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_351_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_352_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_353_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_354_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_355_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_356_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_357_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_358_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_359_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_360_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_361_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_362_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_363_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_364_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_365_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: complex,Z: complex] : ( Y3 = Z ) )
= ( ^ [A3: complex,B3: complex] :
( ( minus_minus_complex @ A3 @ B3 )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_366_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_367_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B3: real] :
( ( minus_minus_real @ A3 @ B3 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_368_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_369_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_370_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_371_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_372_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_373_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_374_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_375_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_376_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_377_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_378_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_379_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_380_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_381_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_382_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_383_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_384_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_385_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_386_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_387_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_388_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_389_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_390_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_391_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_392_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_393_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_394_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_395_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_396_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_397_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_398_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_399_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_400_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_401_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_402_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_403_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_404_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_405_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_406_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_407_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_408_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_409_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_410_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_411_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_412_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_413_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_414_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_415_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_416_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_417_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_418_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_419_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_420_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_421_diff__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_422_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_423_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_424_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_425_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_426_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_427_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_428_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_429_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_430_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_431_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_432_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_433_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_434_add__neg__neg,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ zero_zero_complex )
=> ( ( ord_less_complex @ B @ zero_zero_complex )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ B ) @ zero_zero_complex ) ) ) ).
% add_neg_neg
thf(fact_435_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_436_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_437_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_438_add__pos__pos,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ zero_zero_complex @ A )
=> ( ( ord_less_complex @ zero_zero_complex @ B )
=> ( ord_less_complex @ zero_zero_complex @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_439_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_440_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_441_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_442_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_443_pos__add__strict,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ zero_zero_complex @ A )
=> ( ( ord_less_complex @ B @ C )
=> ( ord_less_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_444_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_445_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_446_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_447_less__iff__diff__less__0,axiom,
( ord_less_complex
= ( ^ [A3: complex,B3: complex] : ( ord_less_complex @ ( minus_minus_complex @ A3 @ B3 ) @ zero_zero_complex ) ) ) ).
% less_iff_diff_less_0
thf(fact_448_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_449_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_450_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_451_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_452_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_453_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_454_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_455_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_456_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_457_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_458_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_459_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_460_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_461_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_462_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_463_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_464_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_465_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_466_interest_Os__calc,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( I != zero_zero_real )
=> ( ( acc @ I @ M @ N2 )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ I ) @ N2 ) @ one_one_real ) @ ( i_nom @ I @ M ) ) ) ) ) ) ).
% interest.s_calc
thf(fact_467_realpow__pos__nth2,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ ( suc @ N2 ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_468_realpow__pos__nth,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N2 )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_469_realpow__pos__nth__unique,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ( power_power_real @ X4 @ N2 )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N2 )
= A ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_470_interest_Os_H_H__calc,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( I != zero_zero_real )
=> ( ( acc_due @ I @ M @ N2 )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ I ) @ N2 ) @ one_one_real ) @ ( d_nom @ I @ M ) ) ) ) ) ) ).
% interest.s''_calc
thf(fact_471_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_472_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_473_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_474_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_475_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_476_reals__power__lt__ex,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ Y )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y ) @ K2 ) @ X ) ) ) ) ).
% reals_power_lt_ex
thf(fact_477_d__nom__i__nom,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ( minus_minus_real @ one_one_real @ ( divide_divide_real @ ( d_nom @ i @ M ) @ ( semiri5074537144036343181t_real @ M ) ) )
= ( divide_divide_real @ one_one_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ i @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ).
% d_nom_i_nom
thf(fact_478_v__futr__m__pos,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ i @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% v_futr_m_pos
thf(fact_479_perp__due__def,axiom,
( perp_due
= ( ^ [I2: real,M2: nat] : ( divide_divide_real @ one_one_real @ ( d_nom @ I2 @ M2 ) ) ) ) ).
% perp_due_def
thf(fact_480_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_481_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_482_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_483_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_484_add__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ M @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% add_Suc_right
thf(fact_485_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_486_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= ( semiri8010041392384452111omplex @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_487_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_488_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_489_s__calc__i__0,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( i = zero_zero_real )
=> ( ( acc @ i @ M @ N2 )
= ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% s_calc_i_0
thf(fact_490_s_H_H__calc__i__0,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( i = zero_zero_real )
=> ( ( acc_due @ i @ M @ N2 )
= ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% s''_calc_i_0
thf(fact_491_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_add
thf(fact_492_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_add
thf(fact_493_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_add
thf(fact_494_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% of_nat_add
thf(fact_495_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_496_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_497_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_498_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_499_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_500_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_501_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_502_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_503_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_504_Suc__diff__diff,axiom,
! [M: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_505_diff__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_506_i__nom__eff,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ i @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) @ M )
= ( plus_plus_real @ one_one_real @ i ) ) ) ).
% i_nom_eff
thf(fact_507_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_508_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_509_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_510_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= zero_zero_complex )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_511_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_512_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_513_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_514_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_515_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_516_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_517_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_518_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_519_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_520_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_521_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_522_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_523_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_524_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri5074537144036343181t_real @ N2 )
= one_one_real )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_525_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri8010041392384452111omplex @ N2 )
= one_one_complex )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_526_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_527_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_528_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_529_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_530_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_531_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_532_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_533_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_534_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_535_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_536_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_537_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_538_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_539_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri8010041392384452111omplex @ X )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_540_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_541_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_542_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_543_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
= ( semiri8010041392384452111omplex @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_544_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N2 ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_545_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N2 ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_546_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N2 ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_547_of__nat__power,axiom,
! [M: nat,N2: nat] :
( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N2 ) )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N2 ) ) ).
% of_nat_power
thf(fact_548_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_549_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_550_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% of_nat_Suc
thf(fact_551_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ M ) )
= ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% of_nat_Suc
thf(fact_552_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_553_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_554_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_555_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_556_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_557_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_558_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_559_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_560_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_561_of__nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_562_of__nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_563_of__nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_564_lim__m__s,axiom,
! [N2: nat] :
( filterlim_nat_real
@ ^ [M2: nat] : ( acc @ i @ M2 @ N2 )
@ ( topolo2815343760600316023s_real @ ( acc_cont @ i @ ( semiri5074537144036343181t_real @ N2 ) ) )
@ at_top_nat ) ).
% lim_m_s
thf(fact_565_diff__add__inverse2,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
= M ) ).
% diff_add_inverse2
thf(fact_566_diff__add__inverse,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
= M ) ).
% diff_add_inverse
thf(fact_567_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_568_diff__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% diff_cancel2
thf(fact_569_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_570_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_571_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_572_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_573_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_574_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_575_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_576_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
!= zero_zero_complex ) ).
% of_nat_neq_0
thf(fact_577_of__nat__less__imp__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_578_of__nat__less__imp__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_579_of__nat__less__imp__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_580_less__imp__of__nat__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_581_less__imp__of__nat__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_582_less__imp__of__nat__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_583_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M )
= zero_zero_nat )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_584_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= M )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_585_diff__add__0,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_586_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_587_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_588_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_589_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_590_add__Suc__shift,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N2 )
= ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_591_add__Suc,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N2 )
= ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% add_Suc
thf(fact_592_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_593_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_594_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_595_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_596_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_597_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_598_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_599_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( P @ M3 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_600_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_601_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_602_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_603_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_604_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_605_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_606_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_607_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_608_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_609_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_610_diff__less__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_611_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_612_add__diff__inverse__nat,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less_nat @ M @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_613_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_614_interest_Os__calc__i__0,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( I = zero_zero_real )
=> ( ( acc @ I @ M @ N2 )
= ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% interest.s_calc_i_0
thf(fact_615_interest_Os_H_H__calc__i__0,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( I = zero_zero_real )
=> ( ( acc_due @ I @ M @ N2 )
= ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% interest.s''_calc_i_0
thf(fact_616_lim__const__over__n,axiom,
! [A: real] :
( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% lim_const_over_n
thf(fact_617_lim__const__over__n,axiom,
! [A: complex] :
( filter6923414461901439796omplex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ N ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ at_top_nat ) ).
% lim_const_over_n
thf(fact_618_d__nom__def,axiom,
( d_nom
= ( ^ [I2: real,M2: nat] : ( divide_divide_real @ ( i_nom @ I2 @ M2 ) @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ I2 @ M2 ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ) ) ).
% d_nom_def
thf(fact_619_LIMSEQ__n__over__Suc__n,axiom,
( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_620_LIMSEQ__n__over__Suc__n,axiom,
( filter6923414461901439796omplex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) )
@ ( topolo2444363109189100025omplex @ one_one_complex )
@ at_top_nat ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_621_LIMSEQ__Suc__n__over__n,axiom,
( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri5074537144036343181t_real @ N ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_622_LIMSEQ__Suc__n__over__n,axiom,
( filter6923414461901439796omplex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri8010041392384452111omplex @ N ) )
@ ( topolo2444363109189100025omplex @ one_one_complex )
@ at_top_nat ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_623_lim__inverse__n_H,axiom,
( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% lim_inverse_n'
thf(fact_624_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_625_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_626_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_627_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_628_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_629_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X4: nat,Y2: nat] :
( ( P @ X4 @ Y2 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_630_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_631_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_632_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_633_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_634_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_635_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_636_add__is__1,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_637_one__is__add,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N2 ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_638_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_639_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_640_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_641_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_642_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_643_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_644_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_645_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_646_diff__less,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).
% diff_less
thf(fact_647_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_648_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_649_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_650_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_651_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_652_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_653_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_654_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_655_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N2 @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_656_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
=> ( P @ I2 ) ) )
= ( ( P @ N2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_657_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_658_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_659_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
& ( P @ I2 ) ) )
= ( ( P @ N2 )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_660_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_661_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_662_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_663_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_664_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_665_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_666_less__natE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ! [Q: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M @ Q ) ) ) ) ).
% less_natE
thf(fact_667_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_668_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_669_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N: nat] :
? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_670_less__imp__Suc__add,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_671_Suc__diff__Suc,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_672_diff__less__Suc,axiom,
! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_673_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_674_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_675_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_676_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ M @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_677_interest_Oi__nom__eff,axiom,
! [I: real,M: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ I @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) @ M )
= ( plus_plus_real @ one_one_real @ I ) ) ) ) ).
% interest.i_nom_eff
thf(fact_678_interest_Ov__futr__m__pos,axiom,
! [I: real,M: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ I @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ).
% interest.v_futr_m_pos
thf(fact_679_interest_Od__nom__i__nom,axiom,
! [I: real,M: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( minus_minus_real @ one_one_real @ ( divide_divide_real @ ( d_nom @ I @ M ) @ ( semiri5074537144036343181t_real @ M ) ) )
= ( divide_divide_real @ one_one_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ I @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ).
% interest.d_nom_i_nom
thf(fact_680_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N2: nat,M: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_681_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_682_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N2: nat,M: nat] :
( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_683_lift__Suc__mono__less,axiom,
! [F: nat > real,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ord_less_real @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_684_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_685_lift__Suc__mono__less,axiom,
! [F: nat > int,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ord_less_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_686_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_687_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M2: nat] :
( N2
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_688_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_689_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_690_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_691_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_692_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_693_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_694_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_695_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_696_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_697_a__calc__i__0,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( i = zero_zero_real )
=> ( ( ann @ i @ M @ N2 )
= ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% a_calc_i_0
thf(fact_698_a_H_H__calc__i__0,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( i = zero_zero_real )
=> ( ( ann_due @ i @ M @ N2 )
= ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% a''_calc_i_0
thf(fact_699_lim__1__over__n,axiom,
( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% lim_1_over_n
thf(fact_700_lim__1__over__n,axiom,
( filter6923414461901439796omplex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ one_one_complex @ ( semiri8010041392384452111omplex @ N ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ at_top_nat ) ).
% lim_1_over_n
thf(fact_701_perp__def,axiom,
( perp
= ( ^ [I2: real,M2: nat] : ( divide_divide_real @ one_one_real @ ( i_nom @ I2 @ M2 ) ) ) ) ).
% perp_def
thf(fact_702_tendsto__const,axiom,
! [K: nat,F2: filter_nat] :
( filterlim_nat_nat
@ ^ [X3: nat] : K
@ ( topolo8926549440605965083ds_nat @ K )
@ F2 ) ).
% tendsto_const
thf(fact_703_tendsto__const,axiom,
! [K: real,F2: filter_nat] :
( filterlim_nat_real
@ ^ [X3: nat] : K
@ ( topolo2815343760600316023s_real @ K )
@ F2 ) ).
% tendsto_const
thf(fact_704_tendsto__const,axiom,
! [K: complex,F2: filter_nat] :
( filter6923414461901439796omplex
@ ^ [X3: nat] : K
@ ( topolo2444363109189100025omplex @ K )
@ F2 ) ).
% tendsto_const
thf(fact_705_nat__approx__posE,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ~ ! [N4: nat] :
~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ E ) ) ).
% nat_approx_posE
thf(fact_706_tendsto__inverse__real,axiom,
! [U: nat > real,L: real,F2: filter_nat] :
( ( filterlim_nat_real @ U @ ( topolo2815343760600316023s_real @ L ) @ F2 )
=> ( ( L != zero_zero_real )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( divide_divide_real @ one_one_real @ ( U @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( divide_divide_real @ one_one_real @ L ) )
@ F2 ) ) ) ).
% tendsto_inverse_real
thf(fact_707_lim__m__a,axiom,
! [N2: nat] :
( filterlim_nat_real
@ ^ [M2: nat] : ( ann @ i @ M2 @ N2 )
@ ( topolo2815343760600316023s_real @ ( ann_cont @ i @ ( semiri5074537144036343181t_real @ N2 ) ) )
@ at_top_nat ) ).
% lim_m_a
thf(fact_708_lim__m__a_H_H,axiom,
! [N2: nat] :
( filterlim_nat_real
@ ^ [M2: nat] : ( ann_due @ i @ M2 @ N2 )
@ ( topolo2815343760600316023s_real @ ( ann_cont @ i @ ( semiri5074537144036343181t_real @ N2 ) ) )
@ at_top_nat ) ).
% lim_m_a''
thf(fact_709_reals__Archimedean2,axiom,
! [X: real] :
? [N4: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% reals_Archimedean2
thf(fact_710_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N4: nat] :
( ~ ( P @ N4 )
& ( P @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_711_interest_Oa_H_H__calc__i__0,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( I = zero_zero_real )
=> ( ( ann_due @ I @ M @ N2 )
= ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% interest.a''_calc_i_0
thf(fact_712_interest_Oa__calc__i__0,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( I = zero_zero_real )
=> ( ( ann @ I @ M @ N2 )
= ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% interest.a_calc_i_0
thf(fact_713_tendsto__eq__rhs,axiom,
! [F: nat > nat,X: nat,F2: filter_nat,Y: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ X ) @ F2 )
=> ( ( X = Y )
=> ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ Y ) @ F2 ) ) ) ).
% tendsto_eq_rhs
thf(fact_714_tendsto__eq__rhs,axiom,
! [F: nat > real,X: real,F2: filter_nat,Y: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ X ) @ F2 )
=> ( ( X = Y )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ Y ) @ F2 ) ) ) ).
% tendsto_eq_rhs
thf(fact_715_tendsto__eq__rhs,axiom,
! [F: nat > complex,X: complex,F2: filter_nat,Y: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ X ) @ F2 )
=> ( ( X = Y )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ Y ) @ F2 ) ) ) ).
% tendsto_eq_rhs
thf(fact_716_tendsto__cong__limit,axiom,
! [F: nat > nat,L: nat,F2: filter_nat,K: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ L ) @ F2 )
=> ( ( K = L )
=> ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ K ) @ F2 ) ) ) ).
% tendsto_cong_limit
thf(fact_717_tendsto__cong__limit,axiom,
! [F: nat > real,L: real,F2: filter_nat,K: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 )
=> ( ( K = L )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ K ) @ F2 ) ) ) ).
% tendsto_cong_limit
thf(fact_718_tendsto__cong__limit,axiom,
! [F: nat > complex,L: complex,F2: filter_nat,K: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ F2 )
=> ( ( K = L )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ K ) @ F2 ) ) ) ).
% tendsto_cong_limit
thf(fact_719_LIMSEQ__unique,axiom,
! [X5: nat > nat,A: nat,B: nat] :
( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ A ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ B ) @ at_top_nat )
=> ( A = B ) ) ) ).
% LIMSEQ_unique
thf(fact_720_LIMSEQ__unique,axiom,
! [X5: nat > real,A: real,B: real] :
( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ A ) @ at_top_nat )
=> ( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ B ) @ at_top_nat )
=> ( A = B ) ) ) ).
% LIMSEQ_unique
thf(fact_721_LIMSEQ__unique,axiom,
! [X5: nat > complex,A: complex,B: complex] :
( ( filter6923414461901439796omplex @ X5 @ ( topolo2444363109189100025omplex @ A ) @ at_top_nat )
=> ( ( filter6923414461901439796omplex @ X5 @ ( topolo2444363109189100025omplex @ B ) @ at_top_nat )
=> ( A = B ) ) ) ).
% LIMSEQ_unique
thf(fact_722_LIMSEQ__offset,axiom,
! [F: nat > nat,K: nat,A: nat] :
( ( filterlim_nat_nat
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) )
@ ( topolo8926549440605965083ds_nat @ A )
@ at_top_nat )
=> ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ A ) @ at_top_nat ) ) ).
% LIMSEQ_offset
thf(fact_723_LIMSEQ__offset,axiom,
! [F: nat > real,K: nat,A: real] :
( ( filterlim_nat_real
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) )
@ ( topolo2815343760600316023s_real @ A )
@ at_top_nat )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ at_top_nat ) ) ).
% LIMSEQ_offset
thf(fact_724_LIMSEQ__offset,axiom,
! [F: nat > complex,K: nat,A: complex] :
( ( filter6923414461901439796omplex
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) )
@ ( topolo2444363109189100025omplex @ A )
@ at_top_nat )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ at_top_nat ) ) ).
% LIMSEQ_offset
thf(fact_725_LIMSEQ__ignore__initial__segment,axiom,
! [F: nat > nat,A: nat,K: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ A ) @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) )
@ ( topolo8926549440605965083ds_nat @ A )
@ at_top_nat ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_726_LIMSEQ__ignore__initial__segment,axiom,
! [F: nat > real,A: real,K: nat] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) )
@ ( topolo2815343760600316023s_real @ A )
@ at_top_nat ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_727_LIMSEQ__ignore__initial__segment,axiom,
! [F: nat > complex,A: complex,K: nat] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ at_top_nat )
=> ( filter6923414461901439796omplex
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) )
@ ( topolo2444363109189100025omplex @ A )
@ at_top_nat ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_728_LIMSEQ__const__iff,axiom,
! [K: nat,L: nat] :
( ( filterlim_nat_nat
@ ^ [N: nat] : K
@ ( topolo8926549440605965083ds_nat @ L )
@ at_top_nat )
= ( K = L ) ) ).
% LIMSEQ_const_iff
thf(fact_729_LIMSEQ__const__iff,axiom,
! [K: real,L: real] :
( ( filterlim_nat_real
@ ^ [N: nat] : K
@ ( topolo2815343760600316023s_real @ L )
@ at_top_nat )
= ( K = L ) ) ).
% LIMSEQ_const_iff
thf(fact_730_LIMSEQ__const__iff,axiom,
! [K: complex,L: complex] :
( ( filter6923414461901439796omplex
@ ^ [N: nat] : K
@ ( topolo2444363109189100025omplex @ L )
@ at_top_nat )
= ( K = L ) ) ).
% LIMSEQ_const_iff
thf(fact_731_interest_Olim__m__s,axiom,
! [I: real,N2: nat] :
( ( interest @ I )
=> ( filterlim_nat_real
@ ^ [M2: nat] : ( acc @ I @ M2 @ N2 )
@ ( topolo2815343760600316023s_real @ ( acc_cont @ I @ ( semiri5074537144036343181t_real @ N2 ) ) )
@ at_top_nat ) ) ).
% interest.lim_m_s
thf(fact_732_approx__from__above__dense__linorder,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [U2: nat > real] :
( ! [N6: nat] : ( ord_less_real @ X @ ( U2 @ N6 ) )
& ( filterlim_nat_real @ U2 @ ( topolo2815343760600316023s_real @ X ) @ at_top_nat ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_733_approx__from__below__dense__linorder,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ Y @ X )
=> ? [U2: nat > real] :
( ! [N6: nat] : ( ord_less_real @ ( U2 @ N6 ) @ X )
& ( filterlim_nat_real @ U2 @ ( topolo2815343760600316023s_real @ X ) @ at_top_nat ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_734_LIMSEQ__Suc,axiom,
! [F: nat > nat,L: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ L ) @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F @ ( suc @ N ) )
@ ( topolo8926549440605965083ds_nat @ L )
@ at_top_nat ) ) ).
% LIMSEQ_Suc
thf(fact_735_LIMSEQ__Suc,axiom,
! [F: nat > real,L: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F @ ( suc @ N ) )
@ ( topolo2815343760600316023s_real @ L )
@ at_top_nat ) ) ).
% LIMSEQ_Suc
thf(fact_736_LIMSEQ__Suc,axiom,
! [F: nat > complex,L: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ at_top_nat )
=> ( filter6923414461901439796omplex
@ ^ [N: nat] : ( F @ ( suc @ N ) )
@ ( topolo2444363109189100025omplex @ L )
@ at_top_nat ) ) ).
% LIMSEQ_Suc
thf(fact_737_LIMSEQ__imp__Suc,axiom,
! [F: nat > nat,L: nat] :
( ( filterlim_nat_nat
@ ^ [N: nat] : ( F @ ( suc @ N ) )
@ ( topolo8926549440605965083ds_nat @ L )
@ at_top_nat )
=> ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ L ) @ at_top_nat ) ) ).
% LIMSEQ_imp_Suc
thf(fact_738_LIMSEQ__imp__Suc,axiom,
! [F: nat > real,L: real] :
( ( filterlim_nat_real
@ ^ [N: nat] : ( F @ ( suc @ N ) )
@ ( topolo2815343760600316023s_real @ L )
@ at_top_nat )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ).
% LIMSEQ_imp_Suc
thf(fact_739_LIMSEQ__imp__Suc,axiom,
! [F: nat > complex,L: complex] :
( ( filter6923414461901439796omplex
@ ^ [N: nat] : ( F @ ( suc @ N ) )
@ ( topolo2444363109189100025omplex @ L )
@ at_top_nat )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ at_top_nat ) ) ).
% LIMSEQ_imp_Suc
thf(fact_740_seq__offset__neg,axiom,
! [F: nat > nat,L: nat,K: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ L ) @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [I2: nat] : ( F @ ( minus_minus_nat @ I2 @ K ) )
@ ( topolo8926549440605965083ds_nat @ L )
@ at_top_nat ) ) ).
% seq_offset_neg
thf(fact_741_seq__offset__neg,axiom,
! [F: nat > real,L: real,K: nat] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [I2: nat] : ( F @ ( minus_minus_nat @ I2 @ K ) )
@ ( topolo2815343760600316023s_real @ L )
@ at_top_nat ) ) ).
% seq_offset_neg
thf(fact_742_seq__offset__neg,axiom,
! [F: nat > complex,L: complex,K: nat] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ at_top_nat )
=> ( filter6923414461901439796omplex
@ ^ [I2: nat] : ( F @ ( minus_minus_nat @ I2 @ K ) )
@ ( topolo2444363109189100025omplex @ L )
@ at_top_nat ) ) ).
% seq_offset_neg
thf(fact_743_a_H_H__calc,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( i != zero_zero_real )
=> ( ( ann_due @ i @ M @ N2 )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( v_pres @ i ) @ N2 ) ) @ ( d_nom @ i @ M ) ) ) ) ) ).
% a''_calc
thf(fact_744_a__calc,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( i != zero_zero_real )
=> ( ( ann @ i @ M @ N2 )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( v_pres @ i ) @ N2 ) ) @ ( i_nom @ i @ M ) ) ) ) ) ).
% a_calc
thf(fact_745_s_H_H__a_H_H,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( acc_due @ i @ M @ N2 )
= ( times_times_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ N2 ) @ ( ann_due @ i @ M @ N2 ) ) ) ) ).
% s''_a''
thf(fact_746_s__a,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( acc @ i @ M @ N2 )
= ( times_times_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ N2 ) @ ( ann @ i @ M @ N2 ) ) ) ) ).
% s_a
thf(fact_747_i__nom__i,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ i @ M ) @ ( semiri5074537144036343181t_real @ M ) ) )
= ( powr_real @ ( plus_plus_real @ one_one_real @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% i_nom_i
thf(fact_748_v__pos,axiom,
ord_less_real @ zero_zero_real @ ( v_pres @ i ) ).
% v_pos
thf(fact_749_v__1__iff__i__0,axiom,
( ( ( v_pres @ i )
= one_one_real )
= ( i = zero_zero_real ) ) ).
% v_1_iff_i_0
thf(fact_750_v__lt__1__iff__i__pos,axiom,
( ( ord_less_real @ ( v_pres @ i ) @ one_one_real )
= ( ord_less_real @ zero_zero_real @ i ) ) ).
% v_lt_1_iff_i_pos
thf(fact_751_mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_752_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_753_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_754_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_755_mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_756_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_757_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_758_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_759_mult__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_760_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_761_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_762_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_763_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_764_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_765_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_766_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_767_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_768_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_769_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_770_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_771_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_772_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_773_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_774_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_775_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_776_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_777_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_778_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_779_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_780_times__divide__eq__right,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_781_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_782_divide__divide__eq__right,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_783_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_784_divide__divide__eq__left,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
= ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_785_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_786_times__divide__eq__left,axiom,
! [B: complex,C: complex,A: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
= ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_787_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_788_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mult
thf(fact_789_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_mult
thf(fact_790_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N2 ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% of_nat_mult
thf(fact_791_real__divide__square__eq,axiom,
! [R2: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A @ R2 ) ) ).
% real_divide_square_eq
thf(fact_792_d__nom__v,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ( d_nom @ i @ M )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( minus_minus_real @ one_one_real @ ( powr_real @ ( v_pres @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ).
% d_nom_v
thf(fact_793_d__nom__i__nom__v,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ( d_nom @ i @ M )
= ( times_times_real @ ( i_nom @ i @ M ) @ ( powr_real @ ( v_pres @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ).
% d_nom_i_nom_v
thf(fact_794_a_H_H__a,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( ann_due @ i @ M @ N2 )
= ( times_times_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( ann @ i @ M @ N2 ) ) ) ) ).
% a''_a
thf(fact_795_s_H_H__s,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( acc_due @ i @ M @ N2 )
= ( times_times_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( acc @ i @ M @ N2 ) ) ) ) ).
% s''_s
thf(fact_796_mult__cancel__right2,axiom,
! [A: complex,C: complex] :
( ( ( times_times_complex @ A @ C )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_797_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_798_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_799_mult__cancel__right1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_800_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_801_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_802_mult__cancel__left2,axiom,
! [C: complex,A: complex] :
( ( ( times_times_complex @ C @ A )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_803_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_804_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_805_mult__cancel__left1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_806_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_807_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_808_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_809_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_810_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_811_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_812_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_813_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_814_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_815_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_816_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_817_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_818_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_819_mult__divide__mult__cancel__left__if,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( C = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= zero_zero_complex ) )
& ( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_820_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_821_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_822_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_823_nonzero__mult__div__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_824_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_825_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_826_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_827_nonzero__mult__div__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_828_a__a_H_H,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( ann @ i @ M @ N2 )
= ( times_times_real @ ( powr_real @ ( v_pres @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( ann_due @ i @ M @ N2 ) ) ) ) ).
% a_a''
thf(fact_829_s__s_H_H,axiom,
! [M: nat,N2: nat] :
( ( M != zero_zero_nat )
=> ( ( acc @ i @ M @ N2 )
= ( times_times_real @ ( powr_real @ ( v_pres @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( acc_due @ i @ M @ N2 ) ) ) ) ).
% s_s''
thf(fact_830_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_831_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_832_nonzero__divide__mult__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_833_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_834_nonzero__divide__mult__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_835_tendsto__mult__right__iff,axiom,
! [C: real,F: nat > real,L: real,F2: filter_nat] :
( ( C != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [X3: nat] : ( times_times_real @ ( F @ X3 ) @ C )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ L @ C ) )
@ F2 )
= ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 ) ) ) ).
% tendsto_mult_right_iff
thf(fact_836_tendsto__mult__right__iff,axiom,
! [C: complex,F: nat > complex,L: complex,F2: filter_nat] :
( ( C != zero_zero_complex )
=> ( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( times_times_complex @ ( F @ X3 ) @ C )
@ ( topolo2444363109189100025omplex @ ( times_times_complex @ L @ C ) )
@ F2 )
= ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ F2 ) ) ) ).
% tendsto_mult_right_iff
thf(fact_837_tendsto__mult__left__iff,axiom,
! [C: real,F: nat > real,L: real,F2: filter_nat] :
( ( C != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [X3: nat] : ( times_times_real @ C @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ C @ L ) )
@ F2 )
= ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 ) ) ) ).
% tendsto_mult_left_iff
thf(fact_838_tendsto__mult__left__iff,axiom,
! [C: complex,F: nat > complex,L: complex,F2: filter_nat] :
( ( C != zero_zero_complex )
=> ( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( times_times_complex @ C @ ( F @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( times_times_complex @ C @ L ) )
@ F2 )
= ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ F2 ) ) ) ).
% tendsto_mult_left_iff
thf(fact_839_tendsto__zero__mult__right__iff,axiom,
! [C: real,A: nat > real] :
( ( C != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ C )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
= ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_840_tendsto__zero__mult__right__iff,axiom,
! [C: complex,A: nat > complex] :
( ( C != zero_zero_complex )
=> ( ( filter6923414461901439796omplex
@ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ C )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ at_top_nat )
= ( filter6923414461901439796omplex @ A @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ at_top_nat ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_841_tendsto__zero__mult__left__iff,axiom,
! [C: real,A: nat > real] :
( ( C != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [N: nat] : ( times_times_real @ C @ ( A @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
= ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_842_tendsto__zero__mult__left__iff,axiom,
! [C: complex,A: nat > complex] :
( ( C != zero_zero_complex )
=> ( ( filter6923414461901439796omplex
@ ^ [N: nat] : ( times_times_complex @ C @ ( A @ N ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ at_top_nat )
= ( filter6923414461901439796omplex @ A @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ at_top_nat ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_843_mult__not__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
!= zero_zero_complex )
=> ( ( A != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_844_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_845_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_846_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_847_divisors__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_848_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_849_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_850_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_851_no__zero__divisors,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( times_times_complex @ A @ B )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_852_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_853_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_854_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_855_mult__left__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_856_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_857_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_858_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_859_mult__right__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_860_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_861_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_862_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_863_mult__delta__right,axiom,
! [B: $o,X: complex,Y: complex] :
( ( B
=> ( ( times_times_complex @ X @ ( if_complex @ B @ Y @ zero_zero_complex ) )
= ( times_times_complex @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_complex @ X @ ( if_complex @ B @ Y @ zero_zero_complex ) )
= zero_zero_complex ) ) ) ).
% mult_delta_right
thf(fact_864_mult__delta__right,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_865_mult__delta__right,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_866_mult__delta__right,axiom,
! [B: $o,X: int,Y: int] :
( ( B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y @ zero_zero_int ) )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_867_mult__delta__left,axiom,
! [B: $o,X: complex,Y: complex] :
( ( B
=> ( ( times_times_complex @ ( if_complex @ B @ X @ zero_zero_complex ) @ Y )
= ( times_times_complex @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_complex @ ( if_complex @ B @ X @ zero_zero_complex ) @ Y )
= zero_zero_complex ) ) ) ).
% mult_delta_left
thf(fact_868_mult__delta__left,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_869_mult__delta__left,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_870_mult__delta__left,axiom,
! [B: $o,X: int,Y: int] :
( ( B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_871_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_872_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_873_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_874_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_875_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_876_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_877_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_878_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_879_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_880_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_881_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_882_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_883_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_884_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_885_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_886_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_887_comm__monoid__mult__class_Omult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_888_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_889_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_890_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_891_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_892_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_893_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_894_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_895_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_896_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_897_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_898_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_899_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_900_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_901_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_902_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_903_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_904_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_905_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_906_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_907_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_908_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_909_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_910_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_911_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_912_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_913_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_914_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_915_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_916_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_917_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_918_divide__divide__eq__left_H,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
= ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_919_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z2: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z2 ) ) ) ).
% divide_divide_times_eq
thf(fact_920_divide__divide__times__eq,axiom,
! [X: complex,Y: complex,Z2: complex,W: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z2 @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z2 ) ) ) ).
% divide_divide_times_eq
thf(fact_921_times__divide__times__eq,axiom,
! [X: real,Y: real,Z2: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_922_times__divide__times__eq,axiom,
! [X: complex,Y: complex,Z2: complex,W: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z2 @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_923_power__commuting__commutes,axiom,
! [X: complex,Y: complex,N2: nat] :
( ( ( times_times_complex @ X @ Y )
= ( times_times_complex @ Y @ X ) )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ Y )
= ( times_times_complex @ Y @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_924_power__commuting__commutes,axiom,
! [X: real,Y: real,N2: nat] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_925_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_926_power__commuting__commutes,axiom,
! [X: int,Y: int,N2: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_927_power__mult__distrib,axiom,
! [A: complex,B: complex,N2: nat] :
( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N2 )
= ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_928_power__mult__distrib,axiom,
! [A: real,B: real,N2: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N2 )
= ( times_times_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_929_power__mult__distrib,axiom,
! [A: nat,B: nat,N2: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N2 )
= ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_930_power__mult__distrib,axiom,
! [A: int,B: int,N2: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N2 )
= ( times_times_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_931_power__commutes,axiom,
! [A: complex,N2: nat] :
( ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_932_power__commutes,axiom,
! [A: real,N2: nat] :
( ( times_times_real @ ( power_power_real @ A @ N2 ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_933_power__commutes,axiom,
! [A: nat,N2: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_934_power__commutes,axiom,
! [A: int,N2: nat] :
( ( times_times_int @ ( power_power_int @ A @ N2 ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% power_commutes
thf(fact_935_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_936_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_937_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_938_mult__of__nat__commute,axiom,
! [X: nat,Y: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
= ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_939_lambda__zero,axiom,
( ( ^ [H: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_940_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_941_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_942_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_943_lambda__one,axiom,
( ( ^ [X3: complex] : X3 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_944_lambda__one,axiom,
( ( ^ [X3: real] : X3 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_945_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_946_lambda__one,axiom,
( ( ^ [X3: int] : X3 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_947_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_948_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_949_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_950_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_951_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_952_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_953_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_954_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_955_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_956_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_957_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_958_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_959_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_960_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_961_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_962_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_963_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_964_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_965_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_966_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_967_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_968_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_969_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_970_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_971_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_972_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_973_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_974_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_975_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_976_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_977_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_978_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_979_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_980_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_981_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_982_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_983_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_984_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_985_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_986_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_987_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_988_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_989_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_990_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_991_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_992_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_993_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_994_less__1__mult,axiom,
! [M: real,N2: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N2 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_995_less__1__mult,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_996_less__1__mult,axiom,
! [M: int,N2: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N2 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_997_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_998_nonzero__eq__divide__eq,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( A
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( times_times_complex @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_999_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1000_nonzero__divide__eq__eq,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ B @ C )
= A )
= ( B
= ( times_times_complex @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1001_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_1002_eq__divide__imp,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= B )
=> ( A
= ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_1003_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_1004_divide__eq__imp,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( B
= ( times_times_complex @ A @ C ) )
=> ( ( divide1717551699836669952omplex @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_1005_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_1006_eq__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ A @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq
thf(fact_1007_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_1008_divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ C )
= A )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq
thf(fact_1009_frac__eq__eq,axiom,
! [Y: real,Z2: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z2 ) )
= ( ( times_times_real @ X @ Z2 )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1010_frac__eq__eq,axiom,
! [Y: complex,Z2: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z2 != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X @ Y )
= ( divide1717551699836669952omplex @ W @ Z2 ) )
= ( ( times_times_complex @ X @ Z2 )
= ( times_times_complex @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1011_square__diff__square__factored,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_1012_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_1013_eq__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1014_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1015_eq__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1016_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1017_left__right__inverse__power,axiom,
! [X: complex,Y: complex,N2: nat] :
( ( ( times_times_complex @ X @ Y )
= one_one_complex )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
= one_one_complex ) ) ).
% left_right_inverse_power
thf(fact_1018_left__right__inverse__power,axiom,
! [X: real,Y: real,N2: nat] :
( ( ( times_times_real @ X @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_1019_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_1020_left__right__inverse__power,axiom,
! [X: int,Y: int,N2: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_1021_power__Suc2,axiom,
! [A: complex,N2: nat] :
( ( power_power_complex @ A @ ( suc @ N2 ) )
= ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_1022_power__Suc2,axiom,
! [A: real,N2: nat] :
( ( power_power_real @ A @ ( suc @ N2 ) )
= ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_1023_power__Suc2,axiom,
! [A: nat,N2: nat] :
( ( power_power_nat @ A @ ( suc @ N2 ) )
= ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_1024_power__Suc2,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ A @ ( suc @ N2 ) )
= ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).
% power_Suc2
thf(fact_1025_power__Suc,axiom,
! [A: int,N2: nat] :
( ( power_power_int @ A @ ( suc @ N2 ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% power_Suc
thf(fact_1026_interest_Od__nom__v,axiom,
! [I: real,M: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( d_nom @ I @ M )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( minus_minus_real @ one_one_real @ ( powr_real @ ( v_pres @ I ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ).
% interest.d_nom_v
thf(fact_1027_interest_Od__nom__i__nom__v,axiom,
! [I: real,M: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( d_nom @ I @ M )
= ( times_times_real @ ( i_nom @ I @ M ) @ ( powr_real @ ( v_pres @ I ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ).
% interest.d_nom_i_nom_v
thf(fact_1028_interest_Oa__a_H_H,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( ann @ I @ M @ N2 )
= ( times_times_real @ ( powr_real @ ( v_pres @ I ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( ann_due @ I @ M @ N2 ) ) ) ) ) ).
% interest.a_a''
thf(fact_1029_interest_Os__s_H_H,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( acc @ I @ M @ N2 )
= ( times_times_real @ ( powr_real @ ( v_pres @ I ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( acc_due @ I @ M @ N2 ) ) ) ) ) ).
% interest.s_s''
thf(fact_1030_i__nom__def,axiom,
( i_nom
= ( ^ [I2: real,M2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( minus_minus_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ I2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) ) @ one_one_real ) ) ) ) ).
% i_nom_def
thf(fact_1031_v__pres__def,axiom,
( v_pres
= ( ^ [I2: real] : ( divide_divide_real @ one_one_real @ ( plus_plus_real @ one_one_real @ I2 ) ) ) ) ).
% v_pres_def
thf(fact_1032_interest_Ov__pos,axiom,
! [I: real] :
( ( interest @ I )
=> ( ord_less_real @ zero_zero_real @ ( v_pres @ I ) ) ) ).
% interest.v_pos
thf(fact_1033_interest_Ov__1__iff__i__0,axiom,
! [I: real] :
( ( interest @ I )
=> ( ( ( v_pres @ I )
= one_one_real )
= ( I = zero_zero_real ) ) ) ).
% interest.v_1_iff_i_0
thf(fact_1034_interest_Oa_H_H__a,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( ann_due @ I @ M @ N2 )
= ( times_times_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ I ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( ann @ I @ M @ N2 ) ) ) ) ) ).
% interest.a''_a
thf(fact_1035_interest_Os_H_H__s,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( acc_due @ I @ M @ N2 )
= ( times_times_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ I ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( acc @ I @ M @ N2 ) ) ) ) ) ).
% interest.s''_s
thf(fact_1036_interest_Ov__lt__1__iff__i__pos,axiom,
! [I: real] :
( ( interest @ I )
=> ( ( ord_less_real @ ( v_pres @ I ) @ one_one_real )
= ( ord_less_real @ zero_zero_real @ I ) ) ) ).
% interest.v_lt_1_iff_i_pos
thf(fact_1037_interest_Oi__nom__i,axiom,
! [I: real,M: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ I @ M ) @ ( semiri5074537144036343181t_real @ M ) ) )
= ( powr_real @ ( plus_plus_real @ one_one_real @ I ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ).
% interest.i_nom_i
thf(fact_1038_interest_Os__a,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( acc @ I @ M @ N2 )
= ( times_times_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ I ) @ N2 ) @ ( ann @ I @ M @ N2 ) ) ) ) ) ).
% interest.s_a
thf(fact_1039_interest_Os_H_H__a_H_H,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( acc_due @ I @ M @ N2 )
= ( times_times_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ I ) @ N2 ) @ ( ann_due @ I @ M @ N2 ) ) ) ) ) ).
% interest.s''_a''
thf(fact_1040_interest_Oa__calc,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( I != zero_zero_real )
=> ( ( ann @ I @ M @ N2 )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( v_pres @ I ) @ N2 ) ) @ ( i_nom @ I @ M ) ) ) ) ) ) ).
% interest.a_calc
thf(fact_1041_interest_Oa_H_H__calc,axiom,
! [I: real,M: nat,N2: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( I != zero_zero_real )
=> ( ( ann_due @ I @ M @ N2 )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( v_pres @ I ) @ N2 ) ) @ ( d_nom @ I @ M ) ) ) ) ) ) ).
% interest.a''_calc
thf(fact_1042_interest_Olim__m__a_H_H,axiom,
! [I: real,N2: nat] :
( ( interest @ I )
=> ( filterlim_nat_real
@ ^ [M2: nat] : ( ann_due @ I @ M2 @ N2 )
@ ( topolo2815343760600316023s_real @ ( ann_cont @ I @ ( semiri5074537144036343181t_real @ N2 ) ) )
@ at_top_nat ) ) ).
% interest.lim_m_a''
thf(fact_1043_interest_Olim__m__a,axiom,
! [I: real,N2: nat] :
( ( interest @ I )
=> ( filterlim_nat_real
@ ^ [M2: nat] : ( ann @ I @ M2 @ N2 )
@ ( topolo2815343760600316023s_real @ ( ann_cont @ I @ ( semiri5074537144036343181t_real @ N2 ) ) )
@ at_top_nat ) ) ).
% interest.lim_m_a
thf(fact_1044_powr__eq__one__iff__gen,axiom,
! [A: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ( powr_real @ A @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ) ) ).
% powr_eq_one_iff_gen
thf(fact_1045_powr__eq__one__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( powr_real @ A @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ) ).
% powr_eq_one_iff
thf(fact_1046_lim__1__over__real__power,axiom,
! [S: real] :
( ( ord_less_real @ zero_zero_real @ S )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ one_one_real @ ( powr_real @ ( semiri5074537144036343181t_real @ N ) @ S ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% lim_1_over_real_power
thf(fact_1047_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1048_div__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( divide_divide_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1049_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1050_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1051_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1052_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1053_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N2 ) )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1054_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1055_one__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N2 ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1056_mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1057_div__mult__self1__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1058_div__mult__self__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1059_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1060_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1061_powr__gt__zero,axiom,
! [X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
= ( X != zero_zero_real ) ) ).
% powr_gt_zero
thf(fact_1062_mult__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( times_times_nat @ M @ ( suc @ N2 ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_1063_powr__less__cancel__iff,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% powr_less_cancel_iff
thf(fact_1064_div__mult2__eq,axiom,
! [M: nat,N2: nat,Q2: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N2 @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N2 ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_1065_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N2: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1066_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1067_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( M = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_1068_add__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1069_add__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1070_diff__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1071_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1072_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1073_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1074_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N2 )
= ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1075_split__div,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N2 != zero_zero_nat )
=> ! [I2: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N2 )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J3 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_1076_dividend__less__div__times,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1077_dividend__less__times__div,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1078_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1079_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1080_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1081_mult__Suc,axiom,
! [M: nat,N2: nat] :
( ( times_times_nat @ ( suc @ M ) @ N2 )
= ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).
% mult_Suc
thf(fact_1082_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times_nat @ M @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1083_n__less__n__mult__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1084_n__less__m__mult__n,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1085_one__less__mult,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1086_exists__complex__root,axiom,
! [N2: nat,Z2: complex] :
( ( N2 != zero_zero_nat )
=> ~ ! [W2: complex] :
( Z2
!= ( power_power_complex @ W2 @ N2 ) ) ) ).
% exists_complex_root
thf(fact_1087_exists__complex__root__nonzero,axiom,
! [Z2: complex,N2: nat] :
( ( Z2 != zero_zero_complex )
=> ( ( N2 != zero_zero_nat )
=> ~ ! [W2: complex] :
( ( W2 != zero_zero_complex )
=> ( Z2
!= ( power_power_complex @ W2 @ N2 ) ) ) ) ) ).
% exists_complex_root_nonzero
thf(fact_1088_powr__powr__swap,axiom,
! [X: real,A: real,B: real] :
( ( powr_real @ ( powr_real @ X @ A ) @ B )
= ( powr_real @ ( powr_real @ X @ B ) @ A ) ) ).
% powr_powr_swap
thf(fact_1089_mult__nat__right__at__top,axiom,
! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
@ at_top_nat
@ at_top_nat ) ) ).
% mult_nat_right_at_top
thf(fact_1090_mult__nat__left__at__top,axiom,
! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% mult_nat_left_at_top
thf(fact_1091_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ) ) ) ).
% mult_eq_if
thf(fact_1092_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N2: nat] :
( ( ( divide_divide_nat @ M @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1093_powr__powr,axiom,
! [X: real,A: real,B: real] :
( ( powr_real @ ( powr_real @ X @ A ) @ B )
= ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% powr_powr
thf(fact_1094_div__eq__dividend__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N2 )
= M )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1095_div__less__dividend,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1096_powr__non__neg,axiom,
! [A: real,X: real] :
~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% powr_non_neg
thf(fact_1097_powr__less__cancel2,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) )
=> ( ord_less_real @ X @ Y ) ) ) ) ) ).
% powr_less_cancel2
thf(fact_1098_powr__less__mono2__neg,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_1099_powr__less__mono,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% powr_less_mono
thf(fact_1100_powr__less__cancel,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ A @ B ) ) ) ).
% powr_less_cancel
thf(fact_1101_int__power__div__base,axiom,
! [M: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1102_div__if,axiom,
( divide_divide_nat
= ( ^ [M2: nat,N: nat] :
( if_nat
@ ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% div_if
thf(fact_1103_powr__inj,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ( powr_real @ A @ X )
= ( powr_real @ A @ Y ) )
= ( X = Y ) ) ) ) ).
% powr_inj
thf(fact_1104_gr__one__powr,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_1105_powr__realpow,axiom,
! [X: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
= ( power_power_real @ X @ N2 ) ) ) ).
% powr_realpow
thf(fact_1106_a_H__calc,axiom,
! [N2: real] :
( ( i != zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ N2 )
=> ( ( ann_cont @ i @ N2 )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( powr_real @ ( v_pres @ i ) @ N2 ) ) @ ( i_force @ i ) ) ) ) ) ).
% a'_calc
thf(fact_1107_s_H__calc,axiom,
! [N2: real] :
( ( i != zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ N2 )
=> ( ( acc_cont @ i @ N2 )
= ( divide_divide_real @ ( minus_minus_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ i ) @ N2 ) @ one_one_real ) @ ( i_force @ i ) ) ) ) ) ).
% s'_calc
thf(fact_1108_s_H__a_H,axiom,
! [N2: real] :
( ( ord_less_eq_real @ zero_zero_real @ N2 )
=> ( ( acc_cont @ i @ N2 )
= ( times_times_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ i ) @ N2 ) @ ( ann_cont @ i @ N2 ) ) ) ) ).
% s'_a'
thf(fact_1109_a_H__calc__i__0,axiom,
! [N2: real] :
( ( i = zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ N2 )
=> ( ( ann_cont @ i @ N2 )
= N2 ) ) ) ).
% a'_calc_i_0
thf(fact_1110_s_H__calc__i__0,axiom,
! [N2: real] :
( ( i = zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ N2 )
=> ( ( acc_cont @ i @ N2 )
= N2 ) ) ) ).
% s'_calc_i_0
thf(fact_1111_powr__complexpow,axiom,
! [X: complex,N2: nat] :
( ( X != zero_zero_complex )
=> ( ( powr_complex @ X @ ( semiri8010041392384452111omplex @ N2 ) )
= ( power_power_complex @ X @ N2 ) ) ) ).
% powr_complexpow
thf(fact_1112_powr__nonneg__iff,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% powr_nonneg_iff
thf(fact_1113_powr__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ one_one_real )
= X ) ) ).
% powr_one
thf(fact_1114_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr_real @ X @ one_one_real )
= X )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% powr_one_gt_zero_iff
thf(fact_1115_powr__le__cancel__iff,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% powr_le_cancel_iff
thf(fact_1116_powr__nat,axiom,
! [Z2: complex,N2: nat] :
( ( ( Z2 = zero_zero_complex )
=> ( ( powr_complex @ Z2 @ ( semiri8010041392384452111omplex @ N2 ) )
= zero_zero_complex ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( powr_complex @ Z2 @ ( semiri8010041392384452111omplex @ N2 ) )
= ( power_power_complex @ Z2 @ N2 ) ) ) ) ).
% powr_nat
thf(fact_1117_powr__mono2,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_mono2
thf(fact_1118_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).
% powr_ge_pzero
thf(fact_1119_powr__mono,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% powr_mono
thf(fact_1120_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A4: real,B4: real,C2: real] :
( ( P @ A4 @ B4 )
=> ( ( P @ B4 @ C2 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C2 )
=> ( P @ A4 @ C2 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A4: real,B4: real] :
( ( ( ord_less_eq_real @ A4 @ X4 )
& ( ord_less_eq_real @ X4 @ B4 )
& ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D3 ) )
=> ( P @ A4 @ B4 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1121_powr__mono2_H,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% powr_mono2'
thf(fact_1122_powr__less__mono2,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_less_mono2
thf(fact_1123_powr__le1,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% powr_le1
thf(fact_1124_powr__mono__both,axiom,
! [A: real,B: real,X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_1125_ge__one__powr__ge__zero,axiom,
! [X: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_1126_powr__mult,axiom,
! [X: real,Y: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
= ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_mult
thf(fact_1127_powr__divide,axiom,
! [X: real,Y: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
= ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_divide
thf(fact_1128_interest_Oa_H__calc__i__0,axiom,
! [I: real,N2: real] :
( ( interest @ I )
=> ( ( I = zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ N2 )
=> ( ( ann_cont @ I @ N2 )
= N2 ) ) ) ) ).
% interest.a'_calc_i_0
thf(fact_1129_interest_Os_H__calc__i__0,axiom,
! [I: real,N2: real] :
( ( interest @ I )
=> ( ( I = zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ N2 )
=> ( ( acc_cont @ I @ N2 )
= N2 ) ) ) ) ).
% interest.s'_calc_i_0
thf(fact_1130_powr__mult__base,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
= ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).
% powr_mult_base
thf(fact_1131_linear__plus__1__le__power,axiom,
! [X: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N2 ) ) ) ).
% linear_plus_1_le_power
thf(fact_1132_nested__sequence__unique,axiom,
! [F: nat > real,G: nat > real] :
( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ! [N4: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N4 ) ) @ ( G @ N4 ) )
=> ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
=> ( ( filterlim_nat_real
@ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
=> ? [L2: real] :
( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L2 )
& ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat )
& ! [N6: nat] : ( ord_less_eq_real @ L2 @ ( G @ N6 ) )
& ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_1133_increasing__LIMSEQ,axiom,
! [F: nat > real,L: real] :
( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ L )
=> ( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [N6: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N6 ) @ E2 ) ) )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_1134_LIMSEQ__realpow__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_1135_interest_Os_H__a_H,axiom,
! [I: real,N2: real] :
( ( interest @ I )
=> ( ( ord_less_eq_real @ zero_zero_real @ N2 )
=> ( ( acc_cont @ I @ N2 )
= ( times_times_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ I ) @ N2 ) @ ( ann_cont @ I @ N2 ) ) ) ) ) ).
% interest.s'_a'
thf(fact_1136_interest_Os_H__calc,axiom,
! [I: real,N2: real] :
( ( interest @ I )
=> ( ( I != zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ N2 )
=> ( ( acc_cont @ I @ N2 )
= ( divide_divide_real @ ( minus_minus_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ I ) @ N2 ) @ one_one_real ) @ ( i_force @ I ) ) ) ) ) ) ).
% interest.s'_calc
thf(fact_1137_interest_Oa_H__calc,axiom,
! [I: real,N2: real] :
( ( interest @ I )
=> ( ( I != zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ N2 )
=> ( ( ann_cont @ I @ N2 )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( powr_real @ ( v_pres @ I ) @ N2 ) ) @ ( i_force @ I ) ) ) ) ) ) ).
% interest.a'_calc
thf(fact_1138_Youngs__inequality,axiom,
! [P2: real,Q2: real,A: real,B: real] :
( ( ord_less_real @ one_one_real @ P2 )
=> ( ( ord_less_real @ one_one_real @ Q2 )
=> ( ( ( plus_plus_real @ ( divide_divide_real @ one_one_real @ P2 ) @ ( divide_divide_real @ one_one_real @ Q2 ) )
= one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ ( plus_plus_real @ ( divide_divide_real @ ( powr_real @ A @ P2 ) @ P2 ) @ ( divide_divide_real @ ( powr_real @ B @ Q2 ) @ Q2 ) ) ) ) ) ) ) ) ).
% Youngs_inequality
thf(fact_1139_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1140_Youngs__inequality__0,axiom,
! [Alpha: real,Beta: real,A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ Alpha )
=> ( ( ord_less_eq_real @ zero_zero_real @ Beta )
=> ( ( ( plus_plus_real @ Alpha @ Beta )
= one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ ( powr_real @ A @ Alpha ) @ ( powr_real @ B @ Beta ) ) @ ( plus_plus_real @ ( times_times_real @ Alpha @ A ) @ ( times_times_real @ Beta @ B ) ) ) ) ) ) ) ) ).
% Youngs_inequality_0
thf(fact_1141_powr__to__1,axiom,
! [Z2: complex] :
( ( powr_complex @ Z2 @ one_one_complex )
= Z2 ) ).
% powr_to_1
thf(fact_1142_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1143_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1144_Suc__le__mono,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N2 @ M ) ) ).
% Suc_le_mono
thf(fact_1145_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1146_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1147_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1148_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1149_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1150_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1151_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1152_one__le__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_1153_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1154_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1155_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1156_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1157_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1158_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1159_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1160_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1161_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1162_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1163_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1164_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1165_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1166_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1167_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( M2 != N ) ) ) ) ).
% nat_less_le
thf(fact_1168_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1169_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1170_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1171_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1172_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1173_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1174_add__leD2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1175_add__leD1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% add_leD1
thf(fact_1176_le__add2,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% le_add2
thf(fact_1177_le__add1,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% le_add1
thf(fact_1178_add__leE,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1179_eq__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1180_le__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1181_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1182_diff__le__mono,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1183_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_1184_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1185_diff__le__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1186_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1187_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_1188_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_1189_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_1190_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1191_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1192_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_1193_le__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( M
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1194_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1195_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1196_le__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M @ N2 )
| ( M
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1197_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1198_not__less__eq__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1199_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N4 )
=> ( P @ M3 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1200_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ M )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1201_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ! [X4: nat] : ( R3 @ X4 @ X4 )
=> ( ! [X4: nat,Y2: nat,Z3: nat] :
( ( R3 @ X4 @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X4 @ Z3 ) ) )
=> ( ! [N4: nat] : ( R3 @ N4 @ ( suc @ N4 ) )
=> ( R3 @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1202_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1203_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1204_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1205_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1206_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1207_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1208_le__imp__less__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1209_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1210_less__Suc__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1211_le__less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1212_Suc__le__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1213_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1214_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1215_Suc__le__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_eq
thf(fact_1216_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_1217_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1218_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1219_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1220_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1221_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1222_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1223_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1224_Suc__diff__le,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
= ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1225_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1226_less__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1227_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1228_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1229_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1230_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1231_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1232_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1233_Suc__div__le__mono,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_div_le_mono
thf(fact_1234_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1235_div__times__less__eq__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1236_times__div__less__eq__dividend,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1237_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1238_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1239_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1240_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1241_nat__one__le__power,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N2 ) ) ) ).
% nat_one_le_power
thf(fact_1242_div__greater__zero__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1243_div__le__mono2,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1244_split__zdiv,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( divide_divide_int @ N2 @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I2: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
=> ( P @ I2 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I2: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_zdiv
thf(fact_1245_int__div__neg__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A @ B )
= Q2 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1246_int__div__pos__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A @ B )
= Q2 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1247_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q2 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1248_div__nat__eqI,axiom,
! [N2: nat,Q2: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q2 ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q2 ) ) )
=> ( ( divide_divide_nat @ M @ N2 )
= Q2 ) ) ) ).
% div_nat_eqI
thf(fact_1249_le__div__geq,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( ( divide_divide_nat @ M @ N2 )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% le_div_geq
thf(fact_1250_split__div_H,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q3 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q3 ) ) )
& ( P @ Q3 ) ) ) ) ).
% split_div'
thf(fact_1251_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1252_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1253_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1254_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1255_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1256_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1257_real__of__nat__div3,axiom,
! [N2: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_1258_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_1259_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N4: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N4 ) ) ) ).
% real_arch_pow
thf(fact_1260_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X @ N4 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1261_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N4: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1262_real__of__nat__div4,axiom,
! [N2: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1263_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_1264_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1265_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1266_real__of__nat__div2,axiom,
! [N2: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_1267_segment__bound__lemma,axiom,
! [B2: real,X: real,Y: real,U: real] :
( ( ord_less_eq_real @ B2 @ X )
=> ( ( ord_less_eq_real @ B2 @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ U @ one_one_real )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U ) @ X ) @ ( times_times_real @ U @ Y ) ) ) ) ) ) ) ).
% segment_bound_lemma
thf(fact_1268_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q4: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I3: nat] :
( ( Q4 @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I3 ) )
& ( ord_less_eq_real @ ( X4 @ I3 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X2: nat > real,I4: nat] : ( ord_less_eq_nat @ ( L2 @ X2 @ I4 ) @ one_one_nat )
& ! [X2: nat > real,I4: nat] :
( ( ( P @ X2 )
& ( Q4 @ I4 )
& ( ( X2 @ I4 )
= zero_zero_real ) )
=> ( ( L2 @ X2 @ I4 )
= zero_zero_nat ) )
& ! [X2: nat > real,I4: nat] :
( ( ( P @ X2 )
& ( Q4 @ I4 )
& ( ( X2 @ I4 )
= one_one_real ) )
=> ( ( L2 @ X2 @ I4 )
= one_one_nat ) )
& ! [X2: nat > real,I4: nat] :
( ( ( P @ X2 )
& ( Q4 @ I4 )
& ( ( L2 @ X2 @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X2 @ I4 ) @ ( F @ X2 @ I4 ) ) )
& ! [X2: nat > real,I4: nat] :
( ( ( P @ X2 )
& ( Q4 @ I4 )
& ( ( L2 @ X2 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X2 @ I4 ) @ ( X2 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1269_sum__le__prod1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% sum_le_prod1
thf(fact_1270_kuhn__lemma,axiom,
! [P2: nat,N2: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P2 )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P2 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ( Label @ X4 @ I3 )
= zero_zero_nat )
| ( ( Label @ X4 @ I3 )
= one_one_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P2 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ( X4 @ I3 )
= zero_zero_nat )
=> ( ( Label @ X4 @ I3 )
= zero_zero_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P2 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ( X4 @ I3 )
= P2 )
=> ( ( Label @ X4 @ I3 )
= one_one_nat ) ) ) )
=> ~ ! [Q: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_nat @ ( Q @ I4 ) @ P2 ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ? [R: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q @ J4 ) @ ( R @ J4 ) )
& ( ord_less_eq_nat @ ( R @ J4 ) @ ( plus_plus_nat @ ( Q @ J4 ) @ one_one_nat ) ) ) )
& ? [S2: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q @ J4 ) @ ( S2 @ J4 ) )
& ( ord_less_eq_nat @ ( S2 @ J4 ) @ ( plus_plus_nat @ ( Q @ J4 ) @ one_one_nat ) ) ) )
& ( ( Label @ R @ I4 )
!= ( Label @ S2 @ I4 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Complex__Ocomplex_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $true @ X @ Y )
= X ) ).
% Conjectures (4)
thf(conj_0,hypothesis,
$true ).
thf(conj_1,hypothesis,
$true ).
thf(conj_2,hypothesis,
( ( i_force @ i )
!= zero_zero_real ) ).
thf(conj_3,conjecture,
( filterlim_nat_real
@ ^ [M2: nat] : ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ n ) @ one_one_real ) @ ( d_nom @ i @ ( suc @ M2 ) ) )
@ ( topolo2815343760600316023s_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ n ) @ one_one_real ) @ ( i_force @ i ) ) )
@ at_top_nat ) ).
%------------------------------------------------------------------------------