TPTP Problem File: SLH0453^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Actuarial_Mathematics/0001_Interest/prob_00469_019854__12921954_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1337 ( 765 unt; 58 typ; 0 def)
% Number of atoms : 3004 (1780 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 9430 ( 305 ~; 82 |; 131 &;8074 @)
% ( 0 <=>; 838 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 152 ( 152 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 55 usr; 11 con; 0-3 aty)
% Number of variables : 2794 ( 65 ^;2644 !; 85 ?;2794 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:13:39.799
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
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 (55)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_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_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Interest_Oacc,type,
acc: real > nat > nat > 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__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_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_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_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
exp_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
powr_real: real > real > real ).
thf(sy_v_i,type,
i: real ).
thf(sy_v_m,type,
m: nat ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1271)
thf(fact_0_that_I2_J,axiom,
i != zero_zero_real ).
% that(2)
thf(fact_1_that_I1_J,axiom,
m != zero_zero_nat ).
% that(1)
thf(fact_2_interest__axioms,axiom,
interest @ i ).
% interest_axioms
thf(fact_3_s_H_H__s,axiom,
! [M: nat,N: nat] :
( ( M != zero_zero_nat )
=> ( ( acc_due @ i @ M @ N )
= ( times_times_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( acc @ i @ M @ N ) ) ) ) ).
% s''_s
thf(fact_4_s__calc,axiom,
! [M: nat,N: nat] :
( ( M != zero_zero_nat )
=> ( ( i != zero_zero_real )
=> ( ( acc @ i @ M @ N )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ N ) @ one_one_real ) @ ( i_nom @ i @ M ) ) ) ) ) ).
% s_calc
thf(fact_5_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_6_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_7_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_8_s_H_H__calc__i__0,axiom,
! [M: nat,N: nat] :
( ( M != zero_zero_nat )
=> ( ( i = zero_zero_real )
=> ( ( acc_due @ i @ M @ N )
= ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% s''_calc_i_0
thf(fact_9_s__calc__i__0,axiom,
! [M: nat,N: nat] :
( ( M != zero_zero_nat )
=> ( ( i = zero_zero_real )
=> ( ( acc @ i @ M @ N )
= ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% s_calc_i_0
thf(fact_10_i__nom__def,axiom,
( i_nom
= ( ^ [I: real,M2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( minus_minus_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ I ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) ) @ one_one_real ) ) ) ) ).
% i_nom_def
thf(fact_11_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_12_perp__def,axiom,
( perp
= ( ^ [I: real,M2: nat] : ( divide_divide_real @ one_one_real @ ( i_nom @ I @ M2 ) ) ) ) ).
% perp_def
thf(fact_13_powr__one__eq__one,axiom,
! [A: real] :
( ( powr_real @ one_one_real @ A )
= one_one_real ) ).
% powr_one_eq_one
thf(fact_14_real__divide__square__eq,axiom,
! [R: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A @ R ) ) ).
% real_divide_square_eq
thf(fact_15_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_16_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_17_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_18_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_19_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_20_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_21_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_22_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_23_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_24_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_25_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_26_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_27_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_28_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_29_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_30_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_31_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_32_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_33_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_34_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_35_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_36_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_37_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_38_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_39_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_40_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_41_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_42_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_43_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_44_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_45_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_46_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_47_powr__eq__0__iff,axiom,
! [W: real,Z: real] :
( ( ( powr_real @ W @ Z )
= zero_zero_real )
= ( W = zero_zero_real ) ) ).
% powr_eq_0_iff
thf(fact_48_powr__0,axiom,
! [Z: real] :
( ( powr_real @ zero_zero_real @ Z )
= zero_zero_real ) ).
% powr_0
thf(fact_49_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_50_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_51_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_52_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_53_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_54_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_55_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_56_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_57_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_58_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_59_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_60_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_61_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_62_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_63_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_64_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_65_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_66_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_67_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_68_powr__zero__eq__one,axiom,
! [X: real] :
( ( ( X = zero_zero_real )
=> ( ( powr_real @ X @ zero_zero_real )
= zero_zero_real ) )
& ( ( X != zero_zero_real )
=> ( ( powr_real @ X @ zero_zero_real )
= one_one_real ) ) ) ).
% powr_zero_eq_one
thf(fact_69_i__nom__1,axiom,
( ( i_nom @ i @ one_one_nat )
= i ) ).
% i_nom_1
thf(fact_70_a_H_H__calc__i__0,axiom,
! [M: nat,N: nat] :
( ( M != zero_zero_nat )
=> ( ( i = zero_zero_real )
=> ( ( ann_due @ i @ M @ N )
= ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% a''_calc_i_0
thf(fact_71_a__calc__i__0,axiom,
! [M: nat,N: nat] :
( ( M != zero_zero_nat )
=> ( ( i = zero_zero_real )
=> ( ( ann @ i @ M @ N )
= ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% a_calc_i_0
thf(fact_72_s__a,axiom,
! [M: nat,N: nat] :
( ( M != zero_zero_nat )
=> ( ( acc @ i @ M @ N )
= ( times_times_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ N ) @ ( ann @ i @ M @ N ) ) ) ) ).
% s_a
thf(fact_73_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_74_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_75_interest_Od__nom__0__iff__i__0,axiom,
! [I2: real,M: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( ( d_nom @ I2 @ M )
= zero_zero_real )
= ( I2 = zero_zero_real ) ) ) ) ).
% interest.d_nom_0_iff_i_0
thf(fact_76_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_77_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_78_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_79_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_80_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_81_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_82_power__mult,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_83_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_84_interest_Oi__nom__1,axiom,
! [I2: real] :
( ( interest @ I2 )
=> ( ( i_nom @ I2 @ one_one_nat )
= I2 ) ) ).
% interest.i_nom_1
thf(fact_85_interest_Os__calc__i__0,axiom,
! [I2: real,M: nat,N: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( I2 = zero_zero_real )
=> ( ( acc @ I2 @ M @ N )
= ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% interest.s_calc_i_0
thf(fact_86_interest_Oi__nom__0__iff__i__0,axiom,
! [I2: real,M: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( ( i_nom @ I2 @ M )
= zero_zero_real )
= ( I2 = zero_zero_real ) ) ) ) ).
% interest.i_nom_0_iff_i_0
thf(fact_87_power__add,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% power_add
thf(fact_88_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_89_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_90_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_91_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_92_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_93_interest_Os_H_H__calc__i__0,axiom,
! [I2: real,M: nat,N: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( I2 = zero_zero_real )
=> ( ( acc_due @ I2 @ M @ N )
= ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% interest.s''_calc_i_0
thf(fact_94_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_95_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_96_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_97_interest_Od__nom__i__nom,axiom,
! [I2: real,M: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( minus_minus_real @ one_one_real @ ( divide_divide_real @ ( d_nom @ I2 @ M ) @ ( semiri5074537144036343181t_real @ M ) ) )
= ( divide_divide_real @ one_one_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ I2 @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ).
% interest.d_nom_i_nom
thf(fact_98_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_99_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_100_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_101_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_102_interest_Os__calc,axiom,
! [I2: real,M: nat,N: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( I2 != zero_zero_real )
=> ( ( acc @ I2 @ M @ N )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ I2 ) @ N ) @ one_one_real ) @ ( i_nom @ I2 @ M ) ) ) ) ) ) ).
% interest.s_calc
thf(fact_103_interest_Oi__nom__eff,axiom,
! [I2: real,M: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ I2 @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) @ M )
= ( plus_plus_real @ one_one_real @ I2 ) ) ) ) ).
% interest.i_nom_eff
thf(fact_104_interest_Oi__nom__i,axiom,
! [I2: real,M: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ I2 @ M ) @ ( semiri5074537144036343181t_real @ M ) ) )
= ( powr_real @ ( plus_plus_real @ one_one_real @ I2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ).
% interest.i_nom_i
thf(fact_105_interest_Os_H_H__s,axiom,
! [I2: real,M: nat,N: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( acc_due @ I2 @ M @ N )
= ( times_times_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ I2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( acc @ I2 @ M @ N ) ) ) ) ) ).
% interest.s''_s
thf(fact_106_power__commuting__commutes,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_107_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_108_power__commuting__commutes,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_109_power__mult__distrib,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_110_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_111_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_112_power__commutes,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_commutes
thf(fact_113_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_114_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_115_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_116_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_117_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_118_power__divide,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
= ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_divide
thf(fact_119_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_120_d__nom__def,axiom,
( d_nom
= ( ^ [I: real,M2: nat] : ( divide_divide_real @ ( i_nom @ I @ M2 ) @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ I @ M2 ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ) ) ).
% d_nom_def
thf(fact_121_left__right__inverse__power,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_122_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_123_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_124_power__one__over,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% power_one_over
thf(fact_125_powr__add,axiom,
! [X: real,A: real,B: real] :
( ( powr_real @ X @ ( plus_plus_real @ A @ B ) )
= ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ).
% powr_add
thf(fact_126_powr__diff,axiom,
! [W: real,Z1: real,Z2: real] :
( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z2 ) )
= ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z2 ) ) ) ).
% powr_diff
thf(fact_127_delta__0__iff__i__0,axiom,
( ( ( i_force @ i )
= zero_zero_real )
= ( i = zero_zero_real ) ) ).
% delta_0_iff_i_0
thf(fact_128_s__s_H_H,axiom,
! [M: nat,N: nat] :
( ( M != zero_zero_nat )
=> ( ( acc @ i @ M @ N )
= ( times_times_real @ ( powr_real @ ( v_pres @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( acc_due @ i @ M @ N ) ) ) ) ).
% s_s''
thf(fact_129_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_130_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_131_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_132_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_133_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_134_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_135_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_136_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_137_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_138_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_139_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_140_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_141_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_142_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_143_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_144_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_145_v__pos,axiom,
ord_less_real @ zero_zero_real @ ( v_pres @ i ) ).
% v_pos
thf(fact_146_v__1__iff__i__0,axiom,
( ( ( v_pres @ i )
= one_one_real )
= ( i = zero_zero_real ) ) ).
% v_1_iff_i_0
thf(fact_147_v__futr__pos,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ i ) ).
% v_futr_pos
thf(fact_148_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_149_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_150_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_151_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_152_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_153_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_154_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_155_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_156_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_157_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_158_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_159_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_160_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_161_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_162_a__calc,axiom,
! [M: nat,N: nat] :
( ( M != zero_zero_nat )
=> ( ( i != zero_zero_real )
=> ( ( ann @ i @ M @ N )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( v_pres @ i ) @ N ) ) @ ( i_nom @ i @ M ) ) ) ) ) ).
% a_calc
thf(fact_163_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_164_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_165_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_166_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_167_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_168_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_169_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_170_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_171_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_172_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_173_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_174_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_175_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_176_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_177_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_178_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_179_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_180_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_181_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_182_a_H_H__a,axiom,
! [M: nat,N: nat] :
( ( M != zero_zero_nat )
=> ( ( ann_due @ i @ M @ N )
= ( times_times_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( ann @ i @ M @ N ) ) ) ) ).
% a''_a
thf(fact_183_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_184_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_185_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_186_power__inject__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_187_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_188_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_189_a_H_H__calc,axiom,
! [M: nat,N: nat] :
( ( M != zero_zero_nat )
=> ( ( i != zero_zero_real )
=> ( ( ann_due @ i @ M @ N )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( v_pres @ i ) @ N ) ) @ ( d_nom @ i @ M ) ) ) ) ) ).
% a''_calc
thf(fact_190_a__a_H_H,axiom,
! [M: nat,N: nat] :
( ( M != zero_zero_nat )
=> ( ( ann @ i @ M @ N )
= ( times_times_real @ ( powr_real @ ( v_pres @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( ann_due @ i @ M @ N ) ) ) ) ).
% a_a''
thf(fact_191_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_192_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_193_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_194_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_195_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_196_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_197_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_198_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_199_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_200_power__strict__decreasing__iff,axiom,
! [B: real,M: nat,N: 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 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_201_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: 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 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_202_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: 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 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_203_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_204_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_205_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_214_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_215_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_216_linordered__field__no__ub,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_217_linordered__field__no__lb,axiom,
! [X2: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X2 ) ).
% linordered_field_no_lb
thf(fact_218_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_219_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_220_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_221_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_222_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_223_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_224_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_225_power__less__imp__less__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_226_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_227_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_228_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: real] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_229_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_230_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_231_interest_Ov__pos,axiom,
! [I2: real] :
( ( interest @ I2 )
=> ( ord_less_real @ zero_zero_real @ ( v_pres @ I2 ) ) ) ).
% interest.v_pos
thf(fact_232_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_233_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_234_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_235_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_236_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_237_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_238_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_239_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: real] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_250_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_251_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_252_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_253_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_254_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_255_interest_Ov__lt__1__iff__i__pos,axiom,
! [I2: real] :
( ( interest @ I2 )
=> ( ( ord_less_real @ ( v_pres @ I2 ) @ one_one_real )
= ( ord_less_real @ zero_zero_real @ I2 ) ) ) ).
% interest.v_lt_1_iff_i_pos
thf(fact_256_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_257_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_258_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_259_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_260_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_261_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_262_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_263_mult__imp__less__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_264_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_265_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_266_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_267_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_268_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_269_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_270_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_271_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_272_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_273_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_274_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_275_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_276_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_277_powr__non__neg,axiom,
! [A: real,X: real] :
~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% powr_non_neg
thf(fact_278_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_279_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_280_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_281_frac__less__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_282_sum__squares__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_283_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_284_power__gt1__lemma,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_285_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_286_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_287_power__less__power__Suc,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_288_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_289_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_290_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_291_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_292_interest_Oa__a_H_H,axiom,
! [I2: real,M: nat,N: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( ann @ I2 @ M @ N )
= ( times_times_real @ ( powr_real @ ( v_pres @ I2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( ann_due @ I2 @ M @ N ) ) ) ) ) ).
% interest.a_a''
thf(fact_293_v__pres__def,axiom,
( v_pres
= ( ^ [I: real] : ( divide_divide_real @ one_one_real @ ( plus_plus_real @ one_one_real @ I ) ) ) ) ).
% v_pres_def
thf(fact_294_interest_Odelta__0__iff__i__0,axiom,
! [I2: real] :
( ( interest @ I2 )
=> ( ( ( i_force @ I2 )
= zero_zero_real )
= ( I2 = zero_zero_real ) ) ) ).
% interest.delta_0_iff_i_0
thf(fact_295_interest_Ov__1__iff__i__0,axiom,
! [I2: real] :
( ( interest @ I2 )
=> ( ( ( v_pres @ I2 )
= one_one_real )
= ( I2 = zero_zero_real ) ) ) ).
% interest.v_1_iff_i_0
thf(fact_296_power__Suc__less,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_297_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_298_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_299_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N3: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_300_powr__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
= ( power_power_real @ X @ N ) ) ) ).
% powr_realpow
thf(fact_301_interest_Ointro,axiom,
! [I2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ I2 ) )
=> ( interest @ I2 ) ) ).
% interest.intro
thf(fact_302_interest_Ov__futr__pos,axiom,
! [I2: real] :
( ( interest @ I2 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ I2 ) ) ) ).
% interest.v_futr_pos
thf(fact_303_interest__def,axiom,
( interest
= ( ^ [I: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ I ) ) ) ) ).
% interest_def
thf(fact_304_interest_Oi__nom__pos__iff__i__pos,axiom,
! [I2: real,M: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( ord_less_real @ zero_zero_real @ ( i_nom @ I2 @ M ) )
= ( ord_less_real @ zero_zero_real @ I2 ) ) ) ) ).
% interest.i_nom_pos_iff_i_pos
thf(fact_305_interest_Od__nom__pos__iff__i__pos,axiom,
! [I2: real,M: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( ord_less_real @ zero_zero_real @ ( d_nom @ I2 @ M ) )
= ( ord_less_real @ zero_zero_real @ I2 ) ) ) ) ).
% interest.d_nom_pos_iff_i_pos
thf(fact_306_power__eq__if,axiom,
( power_power_real
= ( ^ [P: real,M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P @ ( power_power_real @ P @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_307_power__eq__if,axiom,
( power_power_nat
= ( ^ [P: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P @ ( power_power_nat @ P @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_308_power__eq__if,axiom,
( power_power_int
= ( ^ [P: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P @ ( power_power_int @ P @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_309_interest_Oa__calc,axiom,
! [I2: real,M: nat,N: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( I2 != zero_zero_real )
=> ( ( ann @ I2 @ M @ N )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( v_pres @ I2 ) @ N ) ) @ ( i_nom @ I2 @ M ) ) ) ) ) ) ).
% interest.a_calc
thf(fact_310_interest_Oa_H_H__calc,axiom,
! [I2: real,M: nat,N: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( I2 != zero_zero_real )
=> ( ( ann_due @ I2 @ M @ N )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( v_pres @ I2 ) @ N ) ) @ ( d_nom @ I2 @ M ) ) ) ) ) ) ).
% interest.a''_calc
thf(fact_311_times__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_312_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_313_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_314_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_315_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_316_interest_Oa_H_H__a,axiom,
! [I2: real,M: nat,N: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( ann_due @ I2 @ M @ N )
= ( times_times_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ I2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( ann @ I2 @ M @ N ) ) ) ) ) ).
% interest.a''_a
thf(fact_317_interest_Oa__calc__i__0,axiom,
! [I2: real,M: nat,N: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( I2 = zero_zero_real )
=> ( ( ann @ I2 @ M @ N )
= ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% interest.a_calc_i_0
thf(fact_318_interest_Oa_H_H__calc__i__0,axiom,
! [I2: real,M: nat,N: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( I2 = zero_zero_real )
=> ( ( ann_due @ I2 @ M @ N )
= ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% interest.a''_calc_i_0
thf(fact_319_interest_Ov__futr__m__pos,axiom,
! [I2: real,M: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ I2 @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ).
% interest.v_futr_m_pos
thf(fact_320_frac__eq__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_321_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_322_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_323_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_324_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_325_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_326_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_327_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_328_interest_Od__nom__v,axiom,
! [I2: real,M: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( d_nom @ I2 @ M )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( minus_minus_real @ one_one_real @ ( powr_real @ ( v_pres @ I2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ).
% interest.d_nom_v
thf(fact_329_interest_Od__nom__i__nom__v,axiom,
! [I2: real,M: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( d_nom @ I2 @ M )
= ( times_times_real @ ( i_nom @ I2 @ M ) @ ( powr_real @ ( v_pres @ I2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ).
% interest.d_nom_i_nom_v
thf(fact_330_interest_Os__s_H_H,axiom,
! [I2: real,M: nat,N: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( acc @ I2 @ M @ N )
= ( times_times_real @ ( powr_real @ ( v_pres @ I2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( acc_due @ I2 @ M @ N ) ) ) ) ) ).
% interest.s_s''
thf(fact_331_interest_Os__a,axiom,
! [I2: real,M: nat,N: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( acc @ I2 @ M @ N )
= ( times_times_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ I2 ) @ N ) @ ( ann @ I2 @ M @ N ) ) ) ) ) ).
% interest.s_a
thf(fact_332_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_333_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_334_add__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_335_add__frac__num,axiom,
! [Y: real,X: real,Z: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_336_add__num__frac,axiom,
! [Y: real,Z: real,X: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_337_add__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_338_divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_339_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_340_diff__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_341_diff__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_342_divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_343_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_344_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_345_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_346_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_347_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_348_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_349_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_350_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_351_divide__mult__cancel,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( times_times_real @ ( divide_divide_real @ A @ B ) @ B )
= A ) ) ).
% divide_mult_cancel
thf(fact_352_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_353_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_354_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_355_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_356_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_357_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_358_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_359_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_360_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_361_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_362_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_363_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_364_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_365_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_366_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_367_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_368_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_369_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_370_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_371_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_372_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_373_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_374_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_375_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_376_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_377_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_378_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_379_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_380_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_381_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_382_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_383_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_384_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_385_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_386_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_387_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_388_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_389_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_390_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_391_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_392_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_393_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_394_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_395_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_396_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_397_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_398_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_399_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_400_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_401_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_402_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_403_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_404_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_405_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_406_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_407_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_408_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_409_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_410_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_411_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_412_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_413_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_414_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_415_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_416_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_417_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_418_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_419_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_420_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_421_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_422_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_423_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_424_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_425_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_426_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_427_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_428_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_429_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_430_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_431_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_432_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_433_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_434_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_435_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_436_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_437_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_438_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_439_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_440_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_441_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_442_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_443_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_444_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_445_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_446_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_447_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_448_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_449_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_450_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_451_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_452_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_453_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_454_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_455_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_456_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_457_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_458_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_459_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_460_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_461_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_462_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_463_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_464_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_465_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_466_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_467_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_468_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_469_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_470_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_471_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_472_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_473_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_474_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_475_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_476_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_477_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_478_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_479_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_480_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_481_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_482_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_483_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_484_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_485_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_486_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_487_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_488_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_489_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_490_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_491_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_492_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_493_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_494_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_495_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_496_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_497_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_498_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_499_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_500_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_501_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_502_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( P2 @ M3 ) )
=> ( P2 @ N4 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_503_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P2 @ N4 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_504_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_505_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_506_zdiv__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_507_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_508_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_509_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_510_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_511_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_512_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_513_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P2 @ N4 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_514_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_515_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_516_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_517_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_518_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_519_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_520_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_521_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_522_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_523_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_524_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_525_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_526_less__mult__imp__div__less,axiom,
! [M: nat,I2: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I2 @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I2 ) ) ).
% less_mult_imp_div_less
thf(fact_527_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_528_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
=> ( P2 @ D ) ) ) ) ).
% nat_diff_split
thf(fact_529_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
& ~ ( P2 @ D ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_530_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_531_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_532_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_533_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_534_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_535_nat__power__less__imp__less,axiom,
! [I2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_536_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_537_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_538_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_539_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_540_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_541_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_542_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_543_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_544_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_545_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_546_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_547_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_548_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_549_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_550_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_551_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_552_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_553_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_554_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_555_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_556_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_557_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_558_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_559_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_560_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_561_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_562_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_563_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_564_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_565_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_566_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_567_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_568_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_569_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_570_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_571_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_572_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_573_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_574_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_575_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_576_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_577_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_578_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_579_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_580_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_581_group__cancel_Oadd1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_582_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_583_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_584_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I2 @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_585_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_586_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I2 @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_587_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_588_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_589_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_590_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_591_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_592_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_593_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_594_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_595_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_596_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_597_split__div,axiom,
! [P2: nat > $o,M: nat,N: nat] :
( ( P2 @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P2 @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I ) @ J2 ) ) )
=> ( P2 @ I ) ) ) ) ) ).
% split_div
thf(fact_598_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_599_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_600_power__minus__mult,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_real @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_601_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_602_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_603_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_604_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y3: real] :
( ( ( ord_less_real @ zero_zero_real @ Y3 )
& ( ( power_power_real @ Y3 @ N )
= A ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_605_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_606_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_607_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_608_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_609_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_610_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_611_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_612_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_613_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_614_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_615_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_616_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_617_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_618_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_619_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_620_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_621_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_622_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_623_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_624_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_625_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_626_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_627_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_628_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_629_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_630_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_631_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_632_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_633_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_634_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_635_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_636_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_637_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_638_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z3: real] : ( Y4 = Z3 ) )
= ( ^ [A2: real,B2: real] :
( ( minus_minus_real @ A2 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_639_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A2: int,B2: int] :
( ( minus_minus_int @ A2 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_640_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_641_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_642_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_643_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_644_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_645_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_646_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_647_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_648_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_649_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_650_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_651_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_652_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_653_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_654_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_655_add__strict__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_656_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_657_add__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_658_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_659_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_660_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_661_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( I2 = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_662_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_663_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_664_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_665_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_666_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_667_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_668_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_669_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_670_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_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_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_686_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_687_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_688_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_689_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_690_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_691_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_692_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_693_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_694_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_695_diff__strict__mono,axiom,
! [A: real,B: real,D2: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D2 @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_696_diff__strict__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_697_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_698_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_699_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_700_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_701_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_702_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_703_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_704_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_705_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_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_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_715_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_716_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_717_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_718_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_719_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_720_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_721_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_722_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_723_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_724_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_725_group__cancel_Osub1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_726_group__cancel_Osub1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_727_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_728_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_729_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_730_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_731_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_732_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_733_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_734_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_735_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_736_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_737_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_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_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_747_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_748_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_749_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_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_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_758_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_759_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_760_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_761_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_762_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_763_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_764_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_765_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_766_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_767_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_768_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_769_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_770_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_771_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_772_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_773_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_774_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_775_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_776_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_777_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_778_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_779_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_780_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_781_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_782_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_783_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_784_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_785_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_786_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_787_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_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_796_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_797_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_798_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_799_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_800_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_801_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_802_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_803_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_804_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_805_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_806_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_807_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_808_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_809_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_810_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_811_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_812_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_813_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_814_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_815_eq__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D2: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_816_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_817_eq__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D2: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_818_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_819_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_820_not__sum__squares__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_821_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_822_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_823_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_824_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_825_less__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
= ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_826_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_827_less__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D2: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_828_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_829_square__diff__one__factored,axiom,
! [X: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_830_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_831_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_832_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_833_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_834_perp__due__def,axiom,
( perp_due
= ( ^ [I: real,M2: nat] : ( divide_divide_real @ one_one_real @ ( d_nom @ I @ M2 ) ) ) ) ).
% perp_due_def
thf(fact_835_one__less__of__natD,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ one_one_nat @ N ) ) ).
% one_less_of_natD
thf(fact_836_one__less__of__natD,axiom,
! [N: nat] :
( ( ord_less_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ one_one_nat @ N ) ) ).
% one_less_of_natD
thf(fact_837_one__less__of__natD,axiom,
! [N: nat] :
( ( ord_less_int @ one_one_int @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ one_one_nat @ N ) ) ).
% one_less_of_natD
thf(fact_838_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_839_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_840_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_841_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_842_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_843_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_844_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_845_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_846_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_847_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_848_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_849_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_850_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_851_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_852_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_853_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_854_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_855_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_856_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_857_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_858_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_859_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_860_vector__space__over__itself_Oscale__one,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% vector_space_over_itself.scale_one
thf(fact_861_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_862_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_863_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( times_times_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_864_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_865_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X: real,B: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_866_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X: real,Y: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_867_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_868_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_869_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_870_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_871_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_872_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_873_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_874_zmult__zless__mono2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_875_int__less__induct,axiom,
! [I2: int,K: int,P2: int > $o] :
( ( ord_less_int @ I2 @ K )
=> ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_876_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_877_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_878_int__gr__induct,axiom,
! [K: int,I2: int,P2: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_879_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_880_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_881_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_882_vector__space__over__itself_Oscale__scale,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_883_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_884_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N4: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% int_diff_cases
thf(fact_885_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A: real,B: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_886_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_887_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( times_times_real @ A @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ A @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_888_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ B @ X ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_889_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( times_times_real @ A @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ A @ Y ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_890_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ X )
= ( minus_minus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ B @ X ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_891_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_892_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N4: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% pos_int_cases
thf(fact_893_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( K
= ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_894_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K: nat] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_895_i__v__powr,axiom,
! [A: real] :
( ( powr_real @ ( plus_plus_real @ one_one_real @ i ) @ A )
= ( powr_real @ ( v_pres @ i ) @ ( uminus_uminus_real @ A ) ) ) ).
% i_v_powr
thf(fact_896_i__v,axiom,
( ( plus_plus_real @ one_one_real @ i )
= ( powr_real @ ( v_pres @ i ) @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% i_v
thf(fact_897_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_898_square__bound__lemma,axiom,
! [X: real] : ( ord_less_real @ X @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X ) @ ( plus_plus_real @ one_one_real @ X ) ) ) ).
% square_bound_lemma
thf(fact_899_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_900_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_901_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_902_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_903_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_904_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_905_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_906_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_907_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_908_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_909_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_910_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_911_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_912_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_913_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_914_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_915_vector__space__over__itself_Oscale__minus__left,axiom,
! [A: real,X: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ X )
= ( uminus_uminus_real @ ( times_times_real @ A @ X ) ) ) ).
% vector_space_over_itself.scale_minus_left
thf(fact_916_vector__space__over__itself_Oscale__minus__right,axiom,
! [A: real,X: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ X ) ) ) ).
% vector_space_over_itself.scale_minus_right
thf(fact_917_mult__minus__right,axiom,
! [A: real,B: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_918_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_919_minus__mult__minus,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A @ B ) ) ).
% minus_mult_minus
thf(fact_920_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_921_mult__minus__left,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_922_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_923_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_924_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_925_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_926_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_927_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_928_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_929_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_930_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_931_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_932_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_933_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_934_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_935_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_936_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_937_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_938_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_939_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_940_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_941_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_942_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_943_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_944_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_945_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_946_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_947_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_948_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_949_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_950_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_951_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_952_diff__minus__eq__add,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_953_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_954_uminus__add__conv__diff,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
= ( minus_minus_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_955_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_956_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_957_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_958_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_959_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_960_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_961_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_962_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_963_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_964_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_965_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_966_left__minus__one__mult__self,axiom,
! [N: nat,A: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_967_left__minus__one__mult__self,axiom,
! [N: nat,A: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_968_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_969_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_970_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_971_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_972_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_973_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_974_int__distrib_I1_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(1)
thf(fact_975_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(2)
thf(fact_976_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_977_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_978_int__distrib_I3_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(3)
thf(fact_979_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(4)
thf(fact_980_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_981_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_982_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_983_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_984_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_985_minus__divide__left,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_986_minus__divide__divide,axiom,
! [A: real,B: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ).
% minus_divide_divide
thf(fact_987_minus__divide__right,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_divide_right
thf(fact_988_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_989_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_990_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_991_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_992_group__cancel_Oneg1,axiom,
! [A3: real,K: real,A: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( uminus_uminus_real @ A3 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_993_group__cancel_Oneg1,axiom,
! [A3: int,K: int,A: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_994_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_995_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_996_minus__mult__commute,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_mult_commute
thf(fact_997_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_998_square__eq__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ A )
= ( times_times_real @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% square_eq_iff
thf(fact_999_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_1000_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_1001_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_1002_add__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_1003_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_1004_ab__group__add__class_Oab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1005_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1006_add_Oinverse__unique,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_1007_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_1008_eq__neg__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1009_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1010_neg__eq__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1011_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1012_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_1013_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_1014_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_1015_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_1016_square__eq__1__iff,axiom,
! [X: real] :
( ( ( times_times_real @ X @ X )
= one_one_real )
= ( ( X = one_one_real )
| ( X
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_1017_square__eq__1__iff,axiom,
! [X: int] :
( ( ( times_times_int @ X @ X )
= one_one_int )
= ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_1018_nonzero__minus__divide__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_1019_nonzero__minus__divide__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_1020_group__cancel_Osub2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( minus_minus_real @ A @ B3 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_1021_group__cancel_Osub2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_1022_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A2: real,B2: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1023_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1024_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A2: real,B2: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1025_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1026_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_1027_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_1028_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_1029_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_1030_eq__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= ( uminus_uminus_real @ B ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1031_minus__divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
= A )
= ( ( ( C != zero_zero_real )
=> ( ( uminus_uminus_real @ B )
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1032_nonzero__neg__divide__eq__eq,axiom,
! [B: real,A: real,C: real] :
( ( B != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= C )
= ( ( uminus_uminus_real @ A )
= ( times_times_real @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1033_nonzero__neg__divide__eq__eq2,axiom,
! [B: real,C: real,A: real] :
( ( B != zero_zero_real )
=> ( ( C
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
= ( ( times_times_real @ C @ B )
= ( uminus_uminus_real @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1034_divide__eq__minus__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B != zero_zero_real )
& ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_1035_power__minus,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% power_minus
thf(fact_1036_power__minus,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% power_minus
thf(fact_1037_powr__minus__divide,axiom,
! [X: real,A: real] :
( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
= ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).
% powr_minus_divide
thf(fact_1038_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1039_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1040_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1041_divide__powr__uminus,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
= ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% divide_powr_uminus
thf(fact_1042_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A2 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1043_int__if,axiom,
! [P2: $o,A: nat,B: nat] :
( ( P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P2
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P2 @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1044_less__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_1045_minus__divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_1046_neg__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_1047_neg__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_1048_pos__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_1049_pos__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_1050_add__divide__eq__if__simps_I3_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1051_minus__divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1052_add__divide__eq__if__simps_I6_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_1053_add__divide__eq__if__simps_I5_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_1054_minus__divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_1055_powr__neg__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( divide_divide_real @ one_one_real @ X ) ) ) ).
% powr_neg_one
thf(fact_1056_Preliminaries_Oinverse__powr,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( powr_real @ ( divide_divide_real @ one_one_real @ A ) @ B )
= ( powr_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% Preliminaries.inverse_powr
thf(fact_1057_interest_Oi__v,axiom,
! [I2: real] :
( ( interest @ I2 )
=> ( ( plus_plus_real @ one_one_real @ I2 )
= ( powr_real @ ( v_pres @ I2 ) @ ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% interest.i_v
thf(fact_1058_interest_Oi__v__powr,axiom,
! [I2: real,A: real] :
( ( interest @ I2 )
=> ( ( powr_real @ ( plus_plus_real @ one_one_real @ I2 ) @ A )
= ( powr_real @ ( v_pres @ I2 ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% interest.i_v_powr
thf(fact_1059_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_1060_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1061_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1062_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1063_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1064_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1065_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1066_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1067_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1068_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_1069_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_1070_plusinfinity,axiom,
! [D2: int,P3: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P3 @ X3 )
= ( P3 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [X_12: int] : ( P3 @ X_12 )
=> ? [X_1: int] : ( P2 @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1071_minusinfinity,axiom,
! [D2: int,P1: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P2 @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1072_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1073_int__cases2,axiom,
! [Z: int] :
( ! [N4: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% int_cases2
thf(fact_1074_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1075_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1076_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1077_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1078_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1079_minf_I7_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ~ ( ord_less_real @ T @ X2 ) ) ).
% minf(7)
thf(fact_1080_minf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ~ ( ord_less_nat @ T @ X2 ) ) ).
% minf(7)
thf(fact_1081_minf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ~ ( ord_less_int @ T @ X2 ) ) ).
% minf(7)
thf(fact_1082_minf_I5_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( ord_less_real @ X2 @ T ) ) ).
% minf(5)
thf(fact_1083_minf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ord_less_nat @ X2 @ T ) ) ).
% minf(5)
thf(fact_1084_minf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( ord_less_int @ X2 @ T ) ) ).
% minf(5)
thf(fact_1085_minf_I4_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_1086_minf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_1087_minf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_1088_minf_I3_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_1089_minf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_1090_minf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_1091_minf_I2_J,axiom,
! [P2: real > $o,P3: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( ( ( P2 @ X2 )
| ( Q2 @ X2 ) )
= ( ( P3 @ X2 )
| ( Q3 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1092_minf_I2_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( ( P2 @ X2 )
| ( Q2 @ X2 ) )
= ( ( P3 @ X2 )
| ( Q3 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1093_minf_I2_J,axiom,
! [P2: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( ( ( P2 @ X2 )
| ( Q2 @ X2 ) )
= ( ( P3 @ X2 )
| ( Q3 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1094_minf_I1_J,axiom,
! [P2: real > $o,P3: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z5 )
=> ( ( ( P2 @ X2 )
& ( Q2 @ X2 ) )
= ( ( P3 @ X2 )
& ( Q3 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1095_minf_I1_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( ( P2 @ X2 )
& ( Q2 @ X2 ) )
= ( ( P3 @ X2 )
& ( Q3 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1096_minf_I1_J,axiom,
! [P2: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z5 )
=> ( ( ( P2 @ X2 )
& ( Q2 @ X2 ) )
= ( ( P3 @ X2 )
& ( Q3 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1097_pinf_I7_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( ord_less_real @ T @ X2 ) ) ).
% pinf(7)
thf(fact_1098_pinf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ord_less_nat @ T @ X2 ) ) ).
% pinf(7)
thf(fact_1099_pinf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( ord_less_int @ T @ X2 ) ) ).
% pinf(7)
thf(fact_1100_pinf_I5_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ~ ( ord_less_real @ X2 @ T ) ) ).
% pinf(5)
thf(fact_1101_pinf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ~ ( ord_less_nat @ X2 @ T ) ) ).
% pinf(5)
thf(fact_1102_pinf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ~ ( ord_less_int @ X2 @ T ) ) ).
% pinf(5)
thf(fact_1103_pinf_I4_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_1104_pinf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_1105_pinf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_1106_pinf_I3_J,axiom,
! [T: real] :
? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_1107_pinf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_1108_pinf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_1109_pinf_I2_J,axiom,
! [P2: real > $o,P3: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( ( ( P2 @ X2 )
| ( Q2 @ X2 ) )
= ( ( P3 @ X2 )
| ( Q3 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1110_pinf_I2_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( ( P2 @ X2 )
| ( Q2 @ X2 ) )
= ( ( P3 @ X2 )
| ( Q3 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1111_pinf_I2_J,axiom,
! [P2: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( ( ( P2 @ X2 )
| ( Q2 @ X2 ) )
= ( ( P3 @ X2 )
| ( Q3 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1112_pinf_I1_J,axiom,
! [P2: real > $o,P3: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: real] :
! [X2: real] :
( ( ord_less_real @ Z5 @ X2 )
=> ( ( ( P2 @ X2 )
& ( Q2 @ X2 ) )
= ( ( P3 @ X2 )
& ( Q3 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1113_pinf_I1_J,axiom,
! [P2: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( ( P2 @ X2 )
& ( Q2 @ X2 ) )
= ( ( P3 @ X2 )
& ( Q3 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1114_pinf_I1_J,axiom,
! [P2: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: int] :
! [X2: int] :
( ( ord_less_int @ Z5 @ X2 )
=> ( ( ( P2 @ X2 )
& ( Q2 @ X2 ) )
= ( ( P3 @ X2 )
& ( Q3 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1115_int__cases4,axiom,
! [M: int] :
( ! [N4: nat] :
( M
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% int_cases4
thf(fact_1116_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N4: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
=> ~ ! [N4: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% int_cases3
thf(fact_1117_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_1118_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N4: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% neg_int_cases
thf(fact_1119_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_1120_inf__period_I1_J,axiom,
! [P2: real > $o,D3: real,Q2: real > $o] :
( ! [X3: real,K2: real] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D3 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D3 ) ) ) )
=> ! [X2: real,K3: real] :
( ( ( P2 @ X2 )
& ( Q2 @ X2 ) )
= ( ( P2 @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D3 ) ) )
& ( Q2 @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1121_inf__period_I1_J,axiom,
! [P2: int > $o,D3: int,Q2: int > $o] :
( ! [X3: int,K2: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ! [X2: int,K3: int] :
( ( ( P2 @ X2 )
& ( Q2 @ X2 ) )
= ( ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) )
& ( Q2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1122_inf__period_I2_J,axiom,
! [P2: real > $o,D3: real,Q2: real > $o] :
( ! [X3: real,K2: real] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D3 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D3 ) ) ) )
=> ! [X2: real,K3: real] :
( ( ( P2 @ X2 )
| ( Q2 @ X2 ) )
= ( ( P2 @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D3 ) ) )
| ( Q2 @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1123_inf__period_I2_J,axiom,
! [P2: int > $o,D3: int,Q2: int > $o] :
( ! [X3: int,K2: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ! [X2: int,K3: int] :
( ( ( P2 @ X2 )
| ( Q2 @ X2 ) )
= ( ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) )
| ( Q2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1124_add__diff__add,axiom,
! [A: real,C: real,B: real,D2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D2 ) ) ) ).
% add_diff_add
thf(fact_1125_add__diff__add,axiom,
! [A: int,C: int,B: int,D2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D2 ) ) ) ).
% add_diff_add
thf(fact_1126_minus__diff__minus,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_1127_minus__diff__minus,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_1128_mult__diff__mult,axiom,
! [X: real,Y: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_1129_mult__diff__mult,axiom,
! [X: int,Y: int,A: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_1130_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_1131_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X4: real,Y5: real] : ( plus_plus_real @ X4 @ ( uminus_uminus_real @ Y5 ) ) ) ) ).
% minus_real_def
thf(fact_1132_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_1133_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y3: real] :
? [N4: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1134_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1135_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1136_v__i__nom,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ( v_pres @ i )
= ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ i @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( ring_1_of_int_real @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ) ).
% v_i_nom
thf(fact_1137_ex__less__of__nat__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N4: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1138_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_real @ W )
= ( ring_1_of_int_real @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_1139_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
= ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% of_int_power
thf(fact_1140_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
= ( ring_1_of_int_real @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_1141_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int_real @ X )
= ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_1142_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_1143_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_1144_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1145_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1146_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1147_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= zero_zero_real )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1148_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_1149_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= one_one_real )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_1150_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_1151_of__int__1,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_1
thf(fact_1152_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_1153_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_1154_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
= ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_mult
thf(fact_1155_of__int__mult,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
= ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_mult
thf(fact_1156_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_add
thf(fact_1157_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_add
thf(fact_1158_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_minus
thf(fact_1159_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_minus
thf(fact_1160_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% of_int_of_nat_eq
thf(fact_1161_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_1162_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
= ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_diff
thf(fact_1163_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_diff
thf(fact_1164_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_1165_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_1166_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_1167_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_1168_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_1169_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_1170_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_1171_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_1172_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_1173_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_1174_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_1175_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_1176_ex__less__of__int,axiom,
! [X: real] :
? [Z5: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z5 ) ) ).
% ex_less_of_int
thf(fact_1177_ex__of__int__less,axiom,
! [X: real] :
? [Z5: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z5 ) @ X ) ).
% ex_of_int_less
thf(fact_1178_mult__of__int__commute,axiom,
! [X: int,Y: real] :
( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
= ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).
% mult_of_int_commute
thf(fact_1179_mult__of__int__commute,axiom,
! [X: int,Y: int] :
( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
= ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% mult_of_int_commute
thf(fact_1180_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_pos
thf(fact_1181_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_1182_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_1183_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_1184_reals__Archimedean2,axiom,
! [X: real] :
? [N4: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% reals_Archimedean2
thf(fact_1185_interest_Ov__i__nom,axiom,
! [I2: real,M: nat] :
( ( interest @ I2 )
=> ( ( M != zero_zero_nat )
=> ( ( v_pres @ I2 )
= ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ I2 @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) @ ( ring_1_of_int_real @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ) ) ).
% interest.v_i_nom
thf(fact_1186_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_1187_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_1188_v__delta,axiom,
( ( ln_ln_real @ ( v_pres @ i ) )
= ( uminus_uminus_real @ ( i_force @ i ) ) ) ).
% v_delta
thf(fact_1189_arsinh__minus__real,axiom,
! [X: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arsinh_real @ X ) ) ) ).
% arsinh_minus_real
thf(fact_1190_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_1191_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_1192_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1193_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1194_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_1195_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1196_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_1197_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_1198_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1199_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1200_ln__powr,axiom,
! [X: real,Y: real] :
( ( X != zero_zero_real )
=> ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
= ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).
% ln_powr
thf(fact_1201_i__force__def,axiom,
( i_force
= ( ^ [I: real] : ( ln_ln_real @ ( plus_plus_real @ one_one_real @ I ) ) ) ) ).
% i_force_def
thf(fact_1202_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= ( minus_minus_real @ X @ one_one_real ) )
=> ( X = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_1203_ln__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_1204_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_1205_interest_Ov__delta,axiom,
! [I2: real] :
( ( interest @ I2 )
=> ( ( ln_ln_real @ ( v_pres @ I2 ) )
= ( uminus_uminus_real @ ( i_force @ I2 ) ) ) ) ).
% interest.v_delta
thf(fact_1206_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% ln_realpow
thf(fact_1207_e__delta,axiom,
( ( exp_real @ ( i_force @ i ) )
= ( plus_plus_real @ one_one_real @ i ) ) ).
% e_delta
thf(fact_1208_exp__inj__iff,axiom,
! [X: real,Y: real] :
( ( ( exp_real @ X )
= ( exp_real @ Y ) )
= ( X = Y ) ) ).
% exp_inj_iff
thf(fact_1209_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% exp_less_mono
thf(fact_1210_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_1211_ln__exp,axiom,
! [X: real] :
( ( ln_ln_real @ ( exp_real @ X ) )
= X ) ).
% ln_exp
thf(fact_1212_exp__zero,axiom,
( ( exp_real @ zero_zero_real )
= one_one_real ) ).
% exp_zero
thf(fact_1213_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp_real @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ).
% exp_eq_one_iff
thf(fact_1214_one__less__exp__iff,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% one_less_exp_iff
thf(fact_1215_exp__less__one__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% exp_less_one_iff
thf(fact_1216_exp__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( exp_real @ ( ln_ln_real @ X ) )
= X ) ) ).
% exp_ln
thf(fact_1217_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp_real @ ( ln_ln_real @ X ) )
= X )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% exp_ln_iff
thf(fact_1218_ln__unique,axiom,
! [Y: real,X: real] :
( ( ( exp_real @ Y )
= X )
=> ( ( ln_ln_real @ X )
= Y ) ) ).
% ln_unique
thf(fact_1219_exp__add__commuting,axiom,
! [X: real,Y: real] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
= ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).
% exp_add_commuting
thf(fact_1220_mult__exp__exp,axiom,
! [X: real,Y: real] :
( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
= ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% mult_exp_exp
thf(fact_1221_exp__diff,axiom,
! [X: real,Y: real] :
( ( exp_real @ ( minus_minus_real @ X @ Y ) )
= ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% exp_diff
thf(fact_1222_exp__not__eq__zero,axiom,
! [X: real] :
( ( exp_real @ X )
!= zero_zero_real ) ).
% exp_not_eq_zero
thf(fact_1223_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% exp_less_cancel
thf(fact_1224_exp__times__arg__commute,axiom,
! [A3: real] :
( ( times_times_real @ ( exp_real @ A3 ) @ A3 )
= ( times_times_real @ A3 @ ( exp_real @ A3 ) ) ) ).
% exp_times_arg_commute
thf(fact_1225_exp__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X3: real] :
( ( exp_real @ X3 )
= Y ) ) ).
% exp_total
thf(fact_1226_exp__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% exp_gt_zero
thf(fact_1227_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% not_exp_less_zero
thf(fact_1228_exp__minus__inverse,axiom,
! [X: real] :
( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
= one_one_real ) ).
% exp_minus_inverse
thf(fact_1229_exp__of__nat__mult,axiom,
! [N: nat,X: real] :
( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) )
= ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% exp_of_nat_mult
thf(fact_1230_exp__of__nat2__mult,axiom,
! [X: real,N: nat] :
( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N ) ) )
= ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% exp_of_nat2_mult
thf(fact_1231_exp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% exp_gt_one
thf(fact_1232_powr__def,axiom,
( powr_real
= ( ^ [X4: real,A2: real] : ( if_real @ ( X4 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A2 @ ( ln_ln_real @ X4 ) ) ) ) ) ) ).
% powr_def
thf(fact_1233_interest_Oe__delta,axiom,
! [I2: real] :
( ( interest @ I2 )
=> ( ( exp_real @ ( i_force @ I2 ) )
= ( plus_plus_real @ one_one_real @ I2 ) ) ) ).
% interest.e_delta
thf(fact_1234_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_1235_exp__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% exp_le_cancel_iff
thf(fact_1236_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_1237_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1238_powr__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ one_one_real )
= X ) ) ).
% powr_one
thf(fact_1239_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_1240_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_1241_one__le__exp__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% one_le_exp_iff
thf(fact_1242_exp__le__one__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% exp_le_one_iff
thf(fact_1243_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_1244_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_1245_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1246_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).
% powr_ge_pzero
thf(fact_1247_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_1248_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_1249_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% exp_ge_zero
thf(fact_1250_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% not_exp_le_zero
thf(fact_1251_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_real @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_1252_real__eq__0__iff__le__ge__0,axiom,
! [X: real] :
( ( X = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_1253_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_1254_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_1255_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1256_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1257_ln__bound,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_bound
thf(fact_1258_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_1259_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_1260_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_1261_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_ge_zero
thf(fact_1262_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_1263_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_1264_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% exp_ge_add_one_self
thf(fact_1265_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1266_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% real_of_int_div4
thf(fact_1267_power__le__one__iff,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real )
= ( ( N = zero_zero_nat )
| ( ord_less_eq_real @ A @ one_one_real ) ) ) ) ).
% power_le_one_iff
thf(fact_1268_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1269_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N3: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_1270_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self
% Helper facts (7)
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_3_1_If_001t__Real__Oreal_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [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 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( acc_due @ i @ m @ n )
= ( divide_divide_real @ ( times_times_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ m ) ) ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ i ) @ n ) @ one_one_real ) ) @ ( i_nom @ i @ m ) ) ) ).
%------------------------------------------------------------------------------