TPTP Problem File: SLH0103^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Actuarial_Mathematics/0000_Preliminaries/prob_00119_003593__12831996_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1350 ( 618 unt; 68 typ; 0 def)
% Number of atoms : 3488 (1401 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 11279 ( 325 ~; 94 |; 138 &;9364 @)
% ( 0 <=>;1358 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 428 ( 428 >; 0 *; 0 +; 0 <<)
% Number of symbols : 64 ( 61 usr; 11 con; 0-3 aty)
% Number of variables : 3524 ( 153 ^;3262 !; 109 ?;3524 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:09:11.210
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
filter_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__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 (61)
thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(sy_c_Derivative_Ofrechet__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
freche5035077971757218939l_real: ( real > real ) > filter_real > real > real ).
thf(sy_c_Derivative_Ovector__derivative_001t__Real__Oreal,type,
vector6775426145287551914e_real: ( real > real ) > filter_real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Int__Oint_J,type,
times_times_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
times_times_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_HOL_OUniq_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
uniq_real_real: ( ( real > real ) > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__Real__Oreal,type,
uniq_real: ( real > $o ) > $o ).
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_Inner__Product_Ogderiv_001t__Real__Oreal,type,
inner_gderiv_real: ( real > real ) > real > real > $o ).
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__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_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
top_top_real_o: real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_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_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Nat__Onat,type,
topolo8373849641844647293al_nat: filter_real > ( real > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
powr_real: real > real > real ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_a,type,
a: real ).
thf(sy_v_x,type,
x: real ).
% Relevant facts (1274)
thf(fact_0_f,axiom,
( has_fi5821293074295781190e_real
@ ^ [X: real] : X
@ one_one_real
@ ( topolo2177554685111907308n_real @ x @ top_top_set_real ) ) ).
% f
thf(fact_1__092_060open_062_I_I_O_094_J_Aa_Ahas__real__derivative_Aa_A_O_094_Ax_A_K_A1_A_K_Aln_Aa_J_A_Iat_Ax_J_092_060close_062,axiom,
has_fi5821293074295781190e_real @ ( powr_real @ a ) @ ( times_times_real @ ( times_times_real @ ( powr_real @ a @ x ) @ one_one_real ) @ ( ln_ln_real @ a ) ) @ ( topolo2177554685111907308n_real @ x @ top_top_set_real ) ).
% \<open>((.^) a has_real_derivative a .^ x * 1 * ln a) (at x)\<close>
thf(fact_2_DERIV__chain_H,axiom,
! [F: real > real,D: real,X2: real,S: set_real,G: real > real,E: real] :
( ( has_fi5821293074295781190e_real @ F @ D @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( ( has_fi5821293074295781190e_real @ G @ E @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( G @ ( F @ X ) )
@ ( times_times_real @ E @ D )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ) ).
% DERIV_chain'
thf(fact_3_DERIV__chain2,axiom,
! [F: real > real,Da: real,G: real > real,X2: real,Db: real,S: set_real] :
( ( has_fi5821293074295781190e_real @ F @ Da @ ( topolo2177554685111907308n_real @ ( G @ X2 ) @ top_top_set_real ) )
=> ( ( has_fi5821293074295781190e_real @ G @ Db @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( F @ ( G @ X ) )
@ ( times_times_real @ Da @ Db )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ) ).
% DERIV_chain2
thf(fact_4_DERIV__chain3,axiom,
! [G: real > real,G2: real > real,F: real > real,F2: real,X2: real] :
( ! [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( has_fi5821293074295781190e_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( G @ ( F @ X ) )
@ ( times_times_real @ F2 @ ( G2 @ ( F @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% DERIV_chain3
thf(fact_5_DERIV__chain__s,axiom,
! [S: set_real,G: real > real,G2: real > real,F: real > real,F2: real,X2: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
=> ( ( has_fi5821293074295781190e_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( member_real @ ( F @ X2 ) @ S )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( G @ ( F @ X ) )
@ ( times_times_real @ F2 @ ( G2 @ ( F @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% DERIV_chain_s
thf(fact_6_DERIV__cmult,axiom,
! [F: real > real,D: real,X2: real,S: set_real,C: real] :
( ( has_fi5821293074295781190e_real @ F @ D @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( times_times_real @ C @ ( F @ X ) )
@ ( times_times_real @ C @ D )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ).
% DERIV_cmult
thf(fact_7_DERIV__cmult__right,axiom,
! [F: real > real,D: real,X2: real,S: set_real,C: real] :
( ( has_fi5821293074295781190e_real @ F @ D @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ C )
@ ( times_times_real @ D @ C )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ).
% DERIV_cmult_right
thf(fact_8_DERIV__unique,axiom,
! [F: real > real,D: real,X2: real,E: real] :
( ( has_fi5821293074295781190e_real @ F @ D @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( has_fi5821293074295781190e_real @ F @ E @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( D = E ) ) ) ).
% DERIV_unique
thf(fact_9_has__field__derivative__at__within,axiom,
! [F: real > real,F2: real,X2: real,S: set_real] :
( ( has_fi5821293074295781190e_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ).
% has_field_derivative_at_within
thf(fact_10_DERIV__cmult__Id,axiom,
! [C: real,X2: real,S: set_real] : ( has_fi5821293074295781190e_real @ ( times_times_real @ C ) @ C @ ( topolo2177554685111907308n_real @ X2 @ S ) ) ).
% DERIV_cmult_Id
thf(fact_11_UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_I
thf(fact_12_iso__tuple__UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_13_top__set__def,axiom,
( top_top_set_real
= ( collect_real @ top_top_real_o ) ) ).
% top_set_def
thf(fact_14_DERIV__ident,axiom,
! [F3: filter_real] :
( has_fi5821293074295781190e_real
@ ^ [X: real] : X
@ one_one_real
@ F3 ) ).
% DERIV_ident
thf(fact_15_UNIV__witness,axiom,
? [X3: real] : ( member_real @ X3 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_16_UNIV__eq__I,axiom,
! [A: set_real] :
( ! [X3: real] : ( member_real @ X3 @ A )
=> ( top_top_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_17_DERIV__cong,axiom,
! [F: real > real,X4: real,F3: filter_real,Y: real] :
( ( has_fi5821293074295781190e_real @ F @ X4 @ F3 )
=> ( ( X4 = Y )
=> ( has_fi5821293074295781190e_real @ F @ Y @ F3 ) ) ) ).
% DERIV_cong
thf(fact_18_mult__commute__abs,axiom,
! [C: real] :
( ( ^ [X: real] : ( times_times_real @ X @ C ) )
= ( times_times_real @ C ) ) ).
% mult_commute_abs
thf(fact_19_mult__commute__abs,axiom,
! [C: nat] :
( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
= ( times_times_nat @ C ) ) ).
% mult_commute_abs
thf(fact_20_mult__commute__abs,axiom,
! [C: int] :
( ( ^ [X: int] : ( times_times_int @ X @ C ) )
= ( times_times_int @ C ) ) ).
% mult_commute_abs
thf(fact_21_UNIV__def,axiom,
( top_top_set_real
= ( collect_real
@ ^ [X: real] : $true ) ) ).
% UNIV_def
thf(fact_22_powr__one__eq__one,axiom,
! [A2: real] :
( ( powr_real @ one_one_real @ A2 )
= one_one_real ) ).
% powr_one_eq_one
thf(fact_23_mult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% mult_1
thf(fact_24_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_25_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_26_vector__space__over__itself_Oscale__one,axiom,
! [X2: real] :
( ( times_times_real @ one_one_real @ X2 )
= X2 ) ).
% vector_space_over_itself.scale_one
thf(fact_27_mult_Oright__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.right_neutral
thf(fact_28_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_29_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_30_set__times__intro,axiom,
! [A2: real,C2: set_real,B: real,D: set_real] :
( ( member_real @ A2 @ C2 )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( times_times_real @ A2 @ B ) @ ( times_times_set_real @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_31_set__times__intro,axiom,
! [A2: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_set_nat @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_32_set__times__intro,axiom,
! [A2: int,C2: set_int,B: int,D: set_int] :
( ( member_int @ A2 @ C2 )
=> ( ( member_int @ B @ D )
=> ( member_int @ ( times_times_int @ A2 @ B ) @ ( times_times_set_int @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_33_lambda__one,axiom,
( ( ^ [X: real] : X )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_34_lambda__one,axiom,
( ( ^ [X: nat] : X )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_35_lambda__one,axiom,
( ( ^ [X: int] : X )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_36_powr__powr,axiom,
! [X2: real,A2: real,B: real] :
( ( powr_real @ ( powr_real @ X2 @ A2 ) @ B )
= ( powr_real @ X2 @ ( times_times_real @ A2 @ B ) ) ) ).
% powr_powr
thf(fact_37_mult_Ocomm__neutral,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ one_one_real )
= A2 ) ).
% mult.comm_neutral
thf(fact_38_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_39_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_40_comm__monoid__mult__class_Omult__1,axiom,
! [A2: real] :
( ( times_times_real @ one_one_real @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_41_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_42_comm__monoid__mult__class_Omult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_43_DERIV__compose__FDERIV,axiom,
! [F: real > real,F2: real,G: real > real,X2: real,G2: real > real,S: set_real] :
( ( has_fi5821293074295781190e_real @ F @ F2 @ ( topolo2177554685111907308n_real @ ( G @ X2 ) @ top_top_set_real ) )
=> ( ( has_de1759254742604945161l_real @ G @ G2 @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( F @ ( G @ X ) )
@ ^ [X: real] : ( times_times_real @ ( G2 @ X ) @ F2 )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ) ).
% DERIV_compose_FDERIV
thf(fact_44_DERIV__Uniq,axiom,
! [F: real > real,X2: real] :
( uniq_real
@ ^ [D2: real] : ( has_fi5821293074295781190e_real @ F @ D2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% DERIV_Uniq
thf(fact_45_has__derivative__eq__rhs,axiom,
! [F: real > real,F2: real > real,F3: filter_real,G2: real > real] :
( ( has_de1759254742604945161l_real @ F @ F2 @ F3 )
=> ( ( F2 = G2 )
=> ( has_de1759254742604945161l_real @ F @ G2 @ F3 ) ) ) ).
% has_derivative_eq_rhs
thf(fact_46_has__derivative__ident,axiom,
! [F3: filter_real] :
( has_de1759254742604945161l_real
@ ^ [X: real] : X
@ ^ [X: real] : X
@ F3 ) ).
% has_derivative_ident
thf(fact_47_has__derivative__transform,axiom,
! [X2: real,S: set_real,G: real > real,F: real > real,F2: real > real] :
( ( member_real @ X2 @ S )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ( G @ X3 )
= ( F @ X3 ) ) )
=> ( ( has_de1759254742604945161l_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( has_de1759254742604945161l_real @ G @ F2 @ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ) ) ).
% has_derivative_transform
thf(fact_48_has__derivative__Uniq,axiom,
! [F: real > real,X2: real] :
( uniq_real_real
@ ^ [F4: real > real] : ( has_de1759254742604945161l_real @ F @ F4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% has_derivative_Uniq
thf(fact_49_has__derivative__mult__left,axiom,
! [G: real > real,G2: real > real,F3: filter_real,Y2: real] :
( ( has_de1759254742604945161l_real @ G @ G2 @ F3 )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( times_times_real @ ( G @ X ) @ Y2 )
@ ^ [X: real] : ( times_times_real @ ( G2 @ X ) @ Y2 )
@ F3 ) ) ).
% has_derivative_mult_left
thf(fact_50_has__derivative__mult__right,axiom,
! [G: real > real,G2: real > real,F3: filter_real,X2: real] :
( ( has_de1759254742604945161l_real @ G @ G2 @ F3 )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( times_times_real @ X2 @ ( G @ X ) )
@ ^ [X: real] : ( times_times_real @ X2 @ ( G2 @ X ) )
@ F3 ) ) ).
% has_derivative_mult_right
thf(fact_51_has__derivative__unique,axiom,
! [F: real > real,F3: real > real,X2: real,F5: real > real] :
( ( has_de1759254742604945161l_real @ F @ F3 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( has_de1759254742604945161l_real @ F @ F5 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( F3 = F5 ) ) ) ).
% has_derivative_unique
thf(fact_52_has__field__derivative__def,axiom,
( has_fi5821293074295781190e_real
= ( ^ [F6: real > real,D2: real] : ( has_de1759254742604945161l_real @ F6 @ ( times_times_real @ D2 ) ) ) ) ).
% has_field_derivative_def
thf(fact_53_has__derivative__imp__has__field__derivative,axiom,
! [F: real > real,D: real > real,F3: filter_real,D3: real] :
( ( has_de1759254742604945161l_real @ F @ D @ F3 )
=> ( ! [X3: real] :
( ( times_times_real @ X3 @ D3 )
= ( D @ X3 ) )
=> ( has_fi5821293074295781190e_real @ F @ D3 @ F3 ) ) ) ).
% has_derivative_imp_has_field_derivative
thf(fact_54_has__field__derivative__imp__has__derivative,axiom,
! [F: real > real,D: real,F3: filter_real] :
( ( has_fi5821293074295781190e_real @ F @ D @ F3 )
=> ( has_de1759254742604945161l_real @ F @ ( times_times_real @ D ) @ F3 ) ) ).
% has_field_derivative_imp_has_derivative
thf(fact_55_has__derivative__compose,axiom,
! [F: real > real,F2: real > real,X2: real,S: set_real,G: real > real,G2: real > real] :
( ( has_de1759254742604945161l_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( ( has_de1759254742604945161l_real @ G @ G2 @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( G @ ( F @ X ) )
@ ^ [X: real] : ( G2 @ ( F2 @ X ) )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ) ).
% has_derivative_compose
thf(fact_56_mem__Collect__eq,axiom,
! [A2: real,P: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_58_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_59_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_60_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_61_mult_Oassoc,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A2 @ B ) @ C )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_62_mult_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_63_mult_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_64_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_65_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_66_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_67_mult_Oleft__commute,axiom,
! [B: real,A2: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A2 @ C ) )
= ( times_times_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_68_mult_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_69_mult_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A2 @ C ) )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_70_vector__space__over__itself_Oscale__scale,axiom,
! [A2: real,B: real,X2: real] :
( ( times_times_real @ A2 @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ ( times_times_real @ A2 @ B ) @ X2 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_71_vector__space__over__itself_Oscale__left__commute,axiom,
! [A2: real,B: real,X2: real] :
( ( times_times_real @ A2 @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ B @ ( times_times_real @ A2 @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_72_set__times__elim,axiom,
! [X2: real,A: set_real,B3: set_real] :
( ( member_real @ X2 @ ( times_times_set_real @ A @ B3 ) )
=> ~ ! [A4: real,B4: real] :
( ( X2
= ( times_times_real @ A4 @ B4 ) )
=> ( ( member_real @ A4 @ A )
=> ~ ( member_real @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_73_set__times__elim,axiom,
! [X2: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ X2 @ ( times_times_set_nat @ A @ B3 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X2
= ( times_times_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A )
=> ~ ( member_nat @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_74_set__times__elim,axiom,
! [X2: int,A: set_int,B3: set_int] :
( ( member_int @ X2 @ ( times_times_set_int @ A @ B3 ) )
=> ~ ! [A4: int,B4: int] :
( ( X2
= ( times_times_int @ A4 @ B4 ) )
=> ( ( member_int @ A4 @ A )
=> ~ ( member_int @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_75_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_76_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_77_powr__powr__swap,axiom,
! [X2: real,A2: real,B: real] :
( ( powr_real @ ( powr_real @ X2 @ A2 ) @ B )
= ( powr_real @ ( powr_real @ X2 @ B ) @ A2 ) ) ).
% powr_powr_swap
thf(fact_78_has__derivative__subst,axiom,
! [F: real > real,Df: real > real,X2: real,D4: real > real] :
( ( has_de1759254742604945161l_real @ F @ Df @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( Df = D4 )
=> ( has_de1759254742604945161l_real @ F @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% has_derivative_subst
thf(fact_79_has__derivative__at__withinI,axiom,
! [F: real > real,F2: real > real,X2: real,S: set_real] :
( ( has_de1759254742604945161l_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( has_de1759254742604945161l_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ).
% has_derivative_at_withinI
thf(fact_80_top__empty__eq,axiom,
( top_top_real_o
= ( ^ [X: real] : ( member_real @ X @ top_top_set_real ) ) ) ).
% top_empty_eq
thf(fact_81_DERIV__const__ratio__const,axiom,
! [A2: real,B: real,F: real > real,K: real] :
( ( A2 != B )
=> ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( minus_minus_real @ ( F @ B ) @ ( F @ A2 ) )
= ( times_times_real @ ( minus_minus_real @ B @ A2 ) @ K ) ) ) ) ).
% DERIV_const_ratio_const
thf(fact_82_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_83_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_84_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_85_diff__eq__diff__eq,axiom,
! [A2: real,B: real,C: real,D4: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D4 ) )
=> ( ( A2 = B )
= ( C = D4 ) ) ) ).
% diff_eq_diff_eq
thf(fact_86_diff__eq__diff__eq,axiom,
! [A2: int,B: int,C: int,D4: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D4 ) )
=> ( ( A2 = B )
= ( C = D4 ) ) ) ).
% diff_eq_diff_eq
thf(fact_87_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A2: real,X2: real,Y2: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ X2 @ Y2 ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ X2 ) @ ( times_times_real @ A2 @ Y2 ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_88_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A2: real,B: real,X2: real] :
( ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ X2 )
= ( minus_minus_real @ ( times_times_real @ A2 @ X2 ) @ ( times_times_real @ B @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_89_right__diff__distrib_H,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_90_right__diff__distrib_H,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_91_right__diff__distrib_H,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_92_left__diff__distrib_H,axiom,
! [B: real,C: real,A2: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A2 )
= ( minus_minus_real @ ( times_times_real @ B @ A2 ) @ ( times_times_real @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_93_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A2 )
= ( minus_minus_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_94_left__diff__distrib_H,axiom,
! [B: int,C: int,A2: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A2 )
= ( minus_minus_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_95_right__diff__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_96_right__diff__distrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_97_left__diff__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_98_left__diff__distrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_99_Deriv_Ofield__differentiable__diff,axiom,
! [F: real > real,F2: real,F3: filter_real,G: real > real,G2: real] :
( ( has_fi5821293074295781190e_real @ F @ F2 @ F3 )
=> ( ( has_fi5821293074295781190e_real @ G @ G2 @ F3 )
=> ( has_fi5821293074295781190e_real
@ ^ [Z: real] : ( minus_minus_real @ ( F @ Z ) @ ( G @ Z ) )
@ ( minus_minus_real @ F2 @ G2 )
@ F3 ) ) ) ).
% Deriv.field_differentiable_diff
thf(fact_100_has__derivative__diff,axiom,
! [F: real > real,F2: real > real,F3: filter_real,G: real > real,G2: real > real] :
( ( has_de1759254742604945161l_real @ F @ F2 @ F3 )
=> ( ( has_de1759254742604945161l_real @ G @ G2 @ F3 )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
@ ^ [X: real] : ( minus_minus_real @ ( F2 @ X ) @ ( G2 @ X ) )
@ F3 ) ) ) ).
% has_derivative_diff
thf(fact_101_DERIV__diff,axiom,
! [F: real > real,D: real,X2: real,S: set_real,G: real > real,E: real] :
( ( has_fi5821293074295781190e_real @ F @ D @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( ( has_fi5821293074295781190e_real @ G @ E @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
@ ( minus_minus_real @ D @ E )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ) ).
% DERIV_diff
thf(fact_102_has__real__derivative__iff,axiom,
! [F: real > real,F3: filter_real] :
( ( ? [C3: real] : ( has_fi5821293074295781190e_real @ F @ C3 @ F3 ) )
= ( ? [D2: real > real] : ( has_de1759254742604945161l_real @ F @ D2 @ F3 ) ) ) ).
% has_real_derivative_iff
thf(fact_103_has__real__derivative,axiom,
! [F: real > real,F2: real > real,F3: filter_real] :
( ( has_de1759254742604945161l_real @ F @ F2 @ F3 )
=> ~ ! [C4: real] :
~ ( has_fi5821293074295781190e_real @ F @ C4 @ F3 ) ) ).
% has_real_derivative
thf(fact_104_inf__period_I1_J,axiom,
! [P: real > $o,D: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D ) ) ) )
=> ! [X5: real,K3: real] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D ) ) )
& ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_105_inf__period_I1_J,axiom,
! [P: int > $o,D: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ! [X5: int,K3: int] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) )
& ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_106_inf__period_I2_J,axiom,
! [P: real > $o,D: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D ) ) ) )
=> ! [X5: real,K3: real] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D ) ) )
| ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_107_inf__period_I2_J,axiom,
! [P: int > $o,D: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ! [X5: int,K3: int] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) )
| ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_108_gderiv__deriv,axiom,
( inner_gderiv_real
= ( ^ [F6: real > real,X: real,D2: real] : ( has_fi5821293074295781190e_real @ F6 @ D2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% gderiv_deriv
thf(fact_109_frechet__derivative__ident,axiom,
! [A2: real] :
( ( freche5035077971757218939l_real
@ ^ [X: real] : X
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
= ( ^ [X: real] : X ) ) ).
% frechet_derivative_ident
thf(fact_110_has__vector__derivative__id__at,axiom,
! [A2: real] :
( ( vector6775426145287551914e_real
@ ^ [X: real] : X
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
= one_one_real ) ).
% has_vector_derivative_id_at
thf(fact_111_frechet__derivative__at,axiom,
! [F: real > real,F2: real > real,X2: real] :
( ( has_de1759254742604945161l_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( F2
= ( freche5035077971757218939l_real @ F @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% frechet_derivative_at
thf(fact_112_CARAT__DERIV,axiom,
! [F: real > real,L: real,X2: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
= ( ? [G3: real > real] :
( ! [Z: real] :
( ( minus_minus_real @ ( F @ Z ) @ ( F @ X2 ) )
= ( times_times_real @ ( G3 @ Z ) @ ( minus_minus_real @ Z @ X2 ) ) )
& ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ G3 )
& ( ( G3 @ X2 )
= L ) ) ) ) ).
% CARAT_DERIV
thf(fact_113_DERIV__const__ratio__const2,axiom,
! [A2: real,B: real,F: real > real,K: real] :
( ( A2 != B )
=> ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A2 ) ) @ ( minus_minus_real @ B @ A2 ) )
= K ) ) ) ).
% DERIV_const_ratio_const2
thf(fact_114_has__real__derivative__powr,axiom,
! [Z2: real,R: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( has_fi5821293074295781190e_real
@ ^ [Z: real] : ( powr_real @ Z @ R )
@ ( times_times_real @ R @ ( powr_real @ Z2 @ ( minus_minus_real @ R @ one_one_real ) ) )
@ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).
% has_real_derivative_powr
thf(fact_115_assms,axiom,
ord_less_real @ zero_zero_real @ a ).
% assms
thf(fact_116_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_117_mult__zero__left,axiom,
! [A2: real] :
( ( times_times_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_118_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_119_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_120_mult__zero__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_121_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_122_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_123_mult__eq__0__iff,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_124_mult__eq__0__iff,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_125_mult__eq__0__iff,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_126_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A2: real,X2: real] :
( ( ( times_times_real @ A2 @ X2 )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_127_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_128_vector__space__over__itself_Oscale__zero__right,axiom,
! [A2: real] :
( ( times_times_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_129_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ( times_times_real @ A2 @ X2 )
= ( times_times_real @ A2 @ Y2 ) )
= ( ( X2 = Y2 )
| ( A2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_130_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A2: real,X2: real,B: real] :
( ( ( times_times_real @ A2 @ X2 )
= ( times_times_real @ B @ X2 ) )
= ( ( A2 = B )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_131_mult__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( ( times_times_real @ C @ A2 )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_132_mult__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_133_mult__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_134_mult__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( ( times_times_real @ A2 @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_135_mult__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_136_mult__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_137_diff__self,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% diff_self
thf(fact_138_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_139_diff__0__right,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_0_right
thf(fact_140_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_141_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_142_diff__zero,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_zero
thf(fact_143_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_144_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_145_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_146_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_147_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_148_div__0,axiom,
! [A2: real] :
( ( divide_divide_real @ zero_zero_real @ A2 )
= zero_zero_real ) ).
% div_0
thf(fact_149_div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% div_0
thf(fact_150_div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% div_0
thf(fact_151_div__by__0,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_152_div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_153_div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_154_div__by__1,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ one_one_real )
= A2 ) ).
% div_by_1
thf(fact_155_div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% div_by_1
thf(fact_156_div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% div_by_1
thf(fact_157_powr__0,axiom,
! [Z2: real] :
( ( powr_real @ zero_zero_real @ Z2 )
= zero_zero_real ) ).
% powr_0
thf(fact_158_powr__eq__0__iff,axiom,
! [W: real,Z2: real] :
( ( ( powr_real @ W @ Z2 )
= zero_zero_real )
= ( W = zero_zero_real ) ) ).
% powr_eq_0_iff
thf(fact_159_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_160_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_161_mult__cancel__left2,axiom,
! [C: real,A2: real] :
( ( ( times_times_real @ C @ A2 )
= C )
= ( ( C = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_162_mult__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ( times_times_int @ C @ A2 )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_163_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_164_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_165_mult__cancel__right2,axiom,
! [A2: real,C: real] :
( ( ( times_times_real @ A2 @ C )
= C )
= ( ( C = zero_zero_real )
| ( A2 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_166_mult__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ( times_times_int @ A2 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_167_diff__gt__0__iff__gt,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_real @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_168_diff__gt__0__iff__gt,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_int @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_169_nonzero__mult__div__cancel__left,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_170_nonzero__mult__div__cancel__left,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_171_nonzero__mult__div__cancel__left,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B ) @ A2 )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_172_nonzero__mult__div__cancel__right,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_173_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_174_nonzero__mult__div__cancel__right,axiom,
! [B: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ B ) @ B )
= A2 ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_175_divide__mult__cancel,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( times_times_real @ ( divide_divide_real @ A2 @ B ) @ B )
= A2 ) ) ).
% divide_mult_cancel
thf(fact_176_div__self,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ).
% div_self
thf(fact_177_div__self,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( divide_divide_nat @ A2 @ A2 )
= one_one_nat ) ) ).
% div_self
thf(fact_178_div__self,axiom,
! [A2: int] :
( ( A2 != zero_zero_int )
=> ( ( divide_divide_int @ A2 @ A2 )
= one_one_int ) ) ).
% div_self
thf(fact_179_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_180_powr__zero__eq__one,axiom,
! [X2: real] :
( ( ( X2 = zero_zero_real )
=> ( ( powr_real @ X2 @ zero_zero_real )
= zero_zero_real ) )
& ( ( X2 != zero_zero_real )
=> ( ( powr_real @ X2 @ zero_zero_real )
= one_one_real ) ) ) ).
% powr_zero_eq_one
thf(fact_181_ln__less__cancel__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) )
= ( ord_less_real @ X2 @ Y2 ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_182_ln__inj__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ( ln_ln_real @ X2 )
= ( ln_ln_real @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% ln_inj_iff
thf(fact_183_powr__gt__zero,axiom,
! [X2: real,A2: real] :
( ( ord_less_real @ zero_zero_real @ ( powr_real @ X2 @ A2 ) )
= ( X2 != zero_zero_real ) ) ).
% powr_gt_zero
thf(fact_184_powr__less__cancel__iff,axiom,
! [X2: real,A2: real,B: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ( ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B ) )
= ( ord_less_real @ A2 @ B ) ) ) ).
% powr_less_cancel_iff
thf(fact_185_ln__less__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
= ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_186_ln__gt__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
= ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% ln_gt_zero_iff
thf(fact_187_ln__eq__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ( ln_ln_real @ X2 )
= zero_zero_real )
= ( X2 = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_188_powr__eq__one__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less_real @ one_one_real @ A2 )
=> ( ( ( powr_real @ A2 @ X2 )
= one_one_real )
= ( X2 = zero_zero_real ) ) ) ).
% powr_eq_one_iff
thf(fact_189_powr__eq__one__iff__gen,axiom,
! [A2: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( A2 != one_one_real )
=> ( ( ( powr_real @ A2 @ X2 )
= one_one_real )
= ( X2 = zero_zero_real ) ) ) ) ).
% powr_eq_one_iff_gen
thf(fact_190_vector__derivative__const__at,axiom,
! [C: real,A2: real] :
( ( vector6775426145287551914e_real
@ ^ [X: real] : C
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
= zero_zero_real ) ).
% vector_derivative_const_at
thf(fact_191_frechet__derivative__const,axiom,
! [C: real,A2: real] :
( ( freche5035077971757218939l_real
@ ^ [X: real] : C
@ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
= ( ^ [X: real] : zero_zero_real ) ) ).
% frechet_derivative_const
thf(fact_192_GDERIV__const,axiom,
! [K: real,X2: real] :
( inner_gderiv_real
@ ^ [X: real] : K
@ X2
@ zero_zero_real ) ).
% GDERIV_const
thf(fact_193_GDERIV__subst,axiom,
! [F: real > real,X2: real,Df: real,D4: real] :
( ( inner_gderiv_real @ F @ X2 @ Df )
=> ( ( Df = D4 )
=> ( inner_gderiv_real @ F @ X2 @ D4 ) ) ) ).
% GDERIV_subst
thf(fact_194_lt__ex,axiom,
! [X2: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X2 ) ).
% lt_ex
thf(fact_195_lt__ex,axiom,
! [X2: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X2 ) ).
% lt_ex
thf(fact_196_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_197_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_198_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_199_dense,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ? [Z3: real] :
( ( ord_less_real @ X2 @ Z3 )
& ( ord_less_real @ Z3 @ Y2 ) ) ) ).
% dense
thf(fact_200_less__imp__neq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_201_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_202_less__imp__neq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_203_order_Oasym,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order.asym
thf(fact_204_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_205_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_206_ord__eq__less__trans,axiom,
! [A2: real,B: real,C: real] :
( ( A2 = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_207_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_208_ord__eq__less__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_209_ord__less__eq__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_210_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_211_ord__less__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_212_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_213_antisym__conv3,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_real @ Y2 @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_214_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_215_antisym__conv3,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_int @ Y2 @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_216_linorder__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_217_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_218_linorder__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_219_dual__order_Oasym,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ~ ( ord_less_real @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_220_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_221_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_222_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_223_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_224_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_225_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_226_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_227_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_228_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_229_order_Ostrict__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_230_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_231_order_Ostrict__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_232_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_233_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_234_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_235_dual__order_Ostrict__trans,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_236_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_237_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_238_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_239_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_240_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_241_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_242_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_243_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_244_linorder__neqE,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_245_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_246_linorder__neqE,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_247_order__less__asym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_248_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_249_order__less__asym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_250_linorder__neq__iff,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
= ( ( ord_less_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_251_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_252_linorder__neq__iff,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
= ( ( ord_less_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_253_order__less__asym_H,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_254_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_255_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_256_order__less__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_257_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_258_order__less__trans,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_259_ord__eq__less__subst,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_260_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_261_ord__eq__less__subst,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_262_ord__eq__less__subst,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_263_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_264_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_265_ord__eq__less__subst,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_266_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_267_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_268_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_269_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_270_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_271_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_272_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_273_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_274_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_275_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_276_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_277_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_278_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_279_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_280_order__less__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_281_order__less__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_282_order__less__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_283_order__less__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_284_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_285_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_286_order__less__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_287_order__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_288_order__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_289_order__less__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_290_order__less__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_291_order__less__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_292_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_293_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_294_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_295_order__less__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_296_order__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_297_order__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_298_order__less__not__sym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_299_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_300_order__less__not__sym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_301_order__less__imp__triv,axiom,
! [X2: real,Y2: real,P: $o] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_302_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_303_order__less__imp__triv,axiom,
! [X2: int,Y2: int,P: $o] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_304_linorder__less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_305_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_306_linorder__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_307_order__less__imp__not__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_308_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_309_order__less__imp__not__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_310_order__less__imp__not__eq2,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_311_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_312_order__less__imp__not__eq2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_313_order__less__imp__not__less,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_314_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_315_order__less__imp__not__less,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_316_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_317_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_318_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_319_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_320_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_321_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_322_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_323_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_324_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_325_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_326_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_327_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_328_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ~ ( ord_less_real @ X5 @ T ) ) ).
% pinf(5)
thf(fact_329_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_330_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_331_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ord_less_real @ T @ X5 ) ) ).
% pinf(7)
thf(fact_332_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_333_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_334_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_335_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_336_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_337_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_338_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_339_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_340_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_341_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_342_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_343_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_344_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_345_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_346_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ord_less_real @ X5 @ T ) ) ).
% minf(5)
thf(fact_347_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_348_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_349_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ~ ( ord_less_real @ T @ X5 ) ) ).
% minf(7)
thf(fact_350_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_351_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_352_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_353_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_354_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_355_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_356_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_357_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_358_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_359_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_360_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_361_mult__neg__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_362_mult__neg__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_363_not__square__less__zero,axiom,
! [A2: real] :
~ ( ord_less_real @ ( times_times_real @ A2 @ A2 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_364_not__square__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_365_mult__less__0__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_366_mult__less__0__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_367_mult__neg__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_368_mult__neg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_369_mult__neg__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_370_mult__pos__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_371_mult__pos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_372_mult__pos__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_373_mult__pos__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_374_mult__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_375_mult__pos__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_376_mult__pos__neg2,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A2 ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_377_mult__pos__neg2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_378_mult__pos__neg2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_379_zero__less__mult__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_380_zero__less__mult__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_381_zero__less__mult__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_382_zero__less__mult__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_383_zero__less__mult__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_384_zero__less__mult__pos2,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_385_zero__less__mult__pos2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_386_zero__less__mult__pos2,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_387_mult__less__cancel__left__neg,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_388_mult__less__cancel__left__neg,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_389_mult__less__cancel__left__pos,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A2 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_390_mult__less__cancel__left__pos,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A2 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_391_mult__strict__left__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_392_mult__strict__left__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_393_mult__strict__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_394_mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_395_mult__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_396_mult__less__cancel__left__disj,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A2 @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_397_mult__less__cancel__left__disj,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_398_mult__strict__right__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_399_mult__strict__right__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_400_mult__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_401_mult__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_402_mult__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_403_mult__less__cancel__right__disj,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A2 @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_404_mult__less__cancel__right__disj,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_405_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_406_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_407_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_408_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_409_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_410_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_411_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_412_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_413_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_414_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_415_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_416_ln__less__self,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% ln_less_self
thf(fact_417_powr__less__mono2__neg,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( powr_real @ Y2 @ A2 ) @ ( powr_real @ X2 @ A2 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_418_powr__non__neg,axiom,
! [A2: real,X2: real] :
~ ( ord_less_real @ ( powr_real @ A2 @ X2 ) @ zero_zero_real ) ).
% powr_non_neg
thf(fact_419_powr__less__cancel2,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y2 @ A2 ) )
=> ( ord_less_real @ X2 @ Y2 ) ) ) ) ) ).
% powr_less_cancel2
thf(fact_420_ln__div,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ln_ln_real @ ( divide_divide_real @ X2 @ Y2 ) )
= ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) ) ) ) ) ).
% ln_div
thf(fact_421_ln__gt__zero__imp__gt__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_422_ln__less__zero,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_423_ln__gt__zero,axiom,
! [X2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% ln_gt_zero
thf(fact_424_gr__one__powr,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ one_one_real @ ( powr_real @ X2 @ Y2 ) ) ) ) ).
% gr_one_powr
thf(fact_425_powr__inj,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( A2 != one_one_real )
=> ( ( ( powr_real @ A2 @ X2 )
= ( powr_real @ A2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% powr_inj
thf(fact_426_top_Oextremum__strict,axiom,
! [A2: set_real] :
~ ( ord_less_set_real @ top_top_set_real @ A2 ) ).
% top.extremum_strict
thf(fact_427_top_Onot__eq__extremum,axiom,
! [A2: set_real] :
( ( A2 != top_top_set_real )
= ( ord_less_set_real @ A2 @ top_top_set_real ) ) ).
% top.not_eq_extremum
thf(fact_428_diff__strict__mono,axiom,
! [A2: real,B: real,D4: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ D4 @ C )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ D4 ) ) ) ) ).
% diff_strict_mono
thf(fact_429_diff__strict__mono,axiom,
! [A2: int,B: int,D4: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ D4 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D4 ) ) ) ) ).
% diff_strict_mono
thf(fact_430_diff__eq__diff__less,axiom,
! [A2: real,B: real,C: real,D4: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D4 ) )
=> ( ( ord_less_real @ A2 @ B )
= ( ord_less_real @ C @ D4 ) ) ) ).
% diff_eq_diff_less
thf(fact_431_diff__eq__diff__less,axiom,
! [A2: int,B: int,C: int,D4: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D4 ) )
=> ( ( ord_less_int @ A2 @ B )
= ( ord_less_int @ C @ D4 ) ) ) ).
% diff_eq_diff_less
thf(fact_432_diff__strict__left__mono,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_433_diff__strict__left__mono,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_434_diff__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_435_diff__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_436_continuous__within__ln,axiom,
! [X2: real,S: set_real,F: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S ) @ F )
=> ( ( ( F @ X2 )
!= zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S )
@ ^ [X: real] : ( ln_ln_real @ ( F @ X ) ) ) ) ) ).
% continuous_within_ln
thf(fact_437_continuous__at__within__powr,axiom,
! [A2: real,S: set_real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ G )
=> ( ( ( F @ A2 )
!= zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S )
@ ^ [X: real] : ( powr_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_at_within_powr
thf(fact_438_mult__not__zero,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
!= zero_zero_real )
=> ( ( A2 != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_439_mult__not__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_440_mult__not__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
!= zero_zero_int )
=> ( ( A2 != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_441_divisors__zero,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ B )
= zero_zero_real )
=> ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_442_divisors__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_443_divisors__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
=> ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_444_no__zero__divisors,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A2 @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_445_no__zero__divisors,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_446_no__zero__divisors,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A2 @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_447_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A2: real,X2: real,Y2: real] :
( ( A2 != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ X2 )
= ( times_times_real @ A2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_448_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: real,A2: real,B: real] :
( ( X2 != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ X2 )
= ( times_times_real @ B @ X2 ) )
=> ( A2 = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_449_mult__left__cancel,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A2 )
= ( times_times_real @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_450_mult__left__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_451_mult__left__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_452_mult__right__cancel,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C )
= ( times_times_real @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_453_mult__right__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_454_mult__right__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_455_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_456_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_457_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_458_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: real,Z5: real] : ( Y5 = Z5 ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_459_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z5: int] : ( Y5 = Z5 ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_460_isCont__ln,axiom,
! [X2: real] :
( ( X2 != zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ ln_ln_real ) ) ).
% isCont_ln
thf(fact_461_ln__eq__minus__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ( ln_ln_real @ X2 )
= ( minus_minus_real @ X2 @ one_one_real ) )
=> ( X2 = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_462_has__real__derivative__neg__dec__left,axiom,
! [F: real > real,L: real,X2: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H: real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( member_real @ ( minus_minus_real @ X2 @ H ) @ S2 )
=> ( ( ord_less_real @ H @ D5 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_463_has__real__derivative__pos__inc__left,axiom,
! [F: real > real,L: real,X2: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H: real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( member_real @ ( minus_minus_real @ X2 @ H ) @ S2 )
=> ( ( ord_less_real @ H @ D5 )
=> ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_464_lambda__zero,axiom,
( ( ^ [H2: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_465_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_466_lambda__zero,axiom,
( ( ^ [H2: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_467_isCont__ln_H,axiom,
! [X2: real,F: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F )
=> ( ( ( F @ X2 )
!= zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real )
@ ^ [X: real] : ( ln_ln_real @ ( F @ X ) ) ) ) ) ).
% isCont_ln'
thf(fact_468_isCont__powr,axiom,
! [A2: real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ G )
=> ( ( ( F @ A2 )
!= zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X: real] : ( powr_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ) ).
% isCont_powr
thf(fact_469_DERIV__ln__divide,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% DERIV_ln_divide
thf(fact_470_DERIV__const,axiom,
! [K: real,F3: filter_real] :
( has_fi5821293074295781190e_real
@ ^ [X: real] : K
@ zero_zero_real
@ F3 ) ).
% DERIV_const
thf(fact_471_has__derivative__const,axiom,
! [C: real,F3: filter_real] :
( has_de1759254742604945161l_real
@ ^ [X: real] : C
@ ^ [X: real] : zero_zero_real
@ F3 ) ).
% has_derivative_const
thf(fact_472_DERIV__continuous,axiom,
! [F: real > real,D: real,X2: real,S: set_real] :
( ( has_fi5821293074295781190e_real @ F @ D @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S ) @ F ) ) ).
% DERIV_continuous
thf(fact_473_has__derivative__continuous,axiom,
! [F: real > real,F2: real > real,X2: real,S: set_real] :
( ( has_de1759254742604945161l_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S ) @ F ) ) ).
% has_derivative_continuous
thf(fact_474_less__1__mult,axiom,
! [M2: real,N: real] :
( ( ord_less_real @ one_one_real @ M2 )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_475_less__1__mult,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_476_less__1__mult,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ one_one_int @ M2 )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_477_powr__diff,axiom,
! [W: real,Z1: real,Z22: real] :
( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
= ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% powr_diff
thf(fact_478_DERIV__neg__dec__left,axiom,
! [F: real > real,L: real,X2: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H: real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( ord_less_real @ H @ D5 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_479_DERIV__pos__inc__left,axiom,
! [F: real > real,L: real,X2: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H: real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( ord_less_real @ H @ D5 )
=> ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_480_DERIV__divide,axiom,
! [F: real > real,D: real,X2: real,S: set_real,G: real > real,E: real] :
( ( has_fi5821293074295781190e_real @ F @ D @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( ( has_fi5821293074295781190e_real @ G @ E @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( ( ( G @ X2 )
!= zero_zero_real )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
@ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ D @ ( G @ X2 ) ) @ ( times_times_real @ ( F @ X2 ) @ E ) ) @ ( times_times_real @ ( G @ X2 ) @ ( G @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ) ) ).
% DERIV_divide
thf(fact_481_has__derivative__divide_H,axiom,
! [F: real > real,F2: real > real,X2: real,S2: set_real,G: real > real,G2: real > real] :
( ( has_de1759254742604945161l_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( has_de1759254742604945161l_real @ G @ G2 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( ( G @ X2 )
!= zero_zero_real )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
@ ^ [H2: real] : ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( F2 @ H2 ) @ ( G @ X2 ) ) @ ( times_times_real @ ( F @ X2 ) @ ( G2 @ H2 ) ) ) @ ( times_times_real @ ( G @ X2 ) @ ( G @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_482_powr__less__mono,axiom,
! [A2: real,B: real,X2: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ one_one_real @ X2 )
=> ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B ) ) ) ) ).
% powr_less_mono
thf(fact_483_powr__less__cancel,axiom,
! [X2: real,A2: real,B: real] :
( ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B ) )
=> ( ( ord_less_real @ one_one_real @ X2 )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% powr_less_cancel
thf(fact_484_DERIV__cdivide,axiom,
! [F: real > real,D: real,X2: real,S: set_real,C: real] :
( ( has_fi5821293074295781190e_real @ F @ D @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ C )
@ ( divide_divide_real @ D @ C )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ).
% DERIV_cdivide
thf(fact_485_DERIV__isCont,axiom,
! [F: real > real,D: real,X2: real] :
( ( has_fi5821293074295781190e_real @ F @ D @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F ) ) ).
% DERIV_isCont
thf(fact_486_GDERIV__diff,axiom,
! [F: real > real,X2: real,Df: real,G: real > real,Dg: real] :
( ( inner_gderiv_real @ F @ X2 @ Df )
=> ( ( inner_gderiv_real @ G @ X2 @ Dg )
=> ( inner_gderiv_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
@ X2
@ ( minus_minus_real @ Df @ Dg ) ) ) ) ).
% GDERIV_diff
thf(fact_487_DERIV__fun__powr2,axiom,
! [A2: real,F: real > real,R: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( powr_real @ A2 @ ( F @ X ) )
@ ( times_times_real @ ( times_times_real @ ( powr_real @ A2 @ ( F @ X2 ) ) @ R ) @ ( ln_ln_real @ A2 ) )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% DERIV_fun_powr2
thf(fact_488_DERIV__isconst__all,axiom,
! [F: real > real,X2: real,Y2: real] :
( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( F @ X2 )
= ( F @ Y2 ) ) ) ).
% DERIV_isconst_all
thf(fact_489_ln__powr,axiom,
! [X2: real,Y2: real] :
( ( X2 != zero_zero_real )
=> ( ( ln_ln_real @ ( powr_real @ X2 @ Y2 ) )
= ( times_times_real @ Y2 @ ( ln_ln_real @ X2 ) ) ) ) ).
% ln_powr
thf(fact_490_DERIV__caratheodory__within,axiom,
! [F: real > real,L: real,X2: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
= ( ? [G3: real > real] :
( ! [Z: real] :
( ( minus_minus_real @ ( F @ Z ) @ ( F @ X2 ) )
= ( times_times_real @ ( G3 @ Z ) @ ( minus_minus_real @ Z @ X2 ) ) )
& ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S2 ) @ G3 )
& ( ( G3 @ X2 )
= L ) ) ) ) ).
% DERIV_caratheodory_within
thf(fact_491_Deriv_Ohas__derivative__zero__unique,axiom,
! [F3: real > real,X2: real] :
( ( has_de1759254742604945161l_real
@ ^ [X: real] : zero_zero_real
@ F3
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( F3
= ( ^ [H2: real] : zero_zero_real ) ) ) ).
% Deriv.has_derivative_zero_unique
thf(fact_492_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A2 @ B ) )
= ( divide_divide_real @ one_one_real @ A2 ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_493_nonzero__divide__mult__cancel__left,axiom,
! [A2: real,B: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ ( times_times_real @ A2 @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_494_zero__less__divide__1__iff,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% zero_less_divide_1_iff
thf(fact_495_less__divide__eq__1__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ord_less_real @ A2 @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_496_less__divide__eq__1__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ord_less_real @ B @ A2 ) ) ) ).
% less_divide_eq_1_neg
thf(fact_497_divide__less__eq__1__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ord_less_real @ B @ A2 ) ) ) ).
% divide_less_eq_1_pos
thf(fact_498_divide__less__eq__1__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ord_less_real @ A2 @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_499_divide__less__0__1__iff,axiom,
! [A2: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_500_not__real__square__gt__zero,axiom,
! [X2: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
= ( X2 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_501_zero__eq__1__divide__iff,axiom,
! [A2: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_502_divide__eq__0__iff,axiom,
! [A2: real,B: real] :
( ( ( divide_divide_real @ A2 @ B )
= zero_zero_real )
= ( ( A2 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_503_divide__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( ( divide_divide_real @ C @ A2 )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% divide_cancel_left
thf(fact_504_divide__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( ( divide_divide_real @ A2 @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A2 = B ) ) ) ).
% divide_cancel_right
thf(fact_505_division__ring__divide__zero,axiom,
! [A2: real] :
( ( divide_divide_real @ A2 @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_506_times__divide__eq__right,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A2 @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_507_divide__divide__eq__right,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ A2 @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_508_divide__divide__eq__left,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A2 @ B ) @ C )
= ( divide_divide_real @ A2 @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_509_times__divide__eq__left,axiom,
! [B: real,C: real,A2: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A2 )
= ( divide_divide_real @ ( times_times_real @ B @ A2 ) @ C ) ) ).
% times_divide_eq_left
thf(fact_510_real__divide__square__eq,axiom,
! [R: real,A2: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A2 ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A2 @ R ) ) ).
% real_divide_square_eq
thf(fact_511_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A2: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_512_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_513_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_514_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_515_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_516_divide__eq__1__iff,axiom,
! [A2: real,B: real] :
( ( ( divide_divide_real @ A2 @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A2 = B ) ) ) ).
% divide_eq_1_iff
thf(fact_517_one__eq__divide__iff,axiom,
! [A2: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A2 @ B ) )
= ( ( B != zero_zero_real )
& ( A2 = B ) ) ) ).
% one_eq_divide_iff
thf(fact_518_divide__self,axiom,
! [A2: real] :
( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ).
% divide_self
thf(fact_519_divide__self__if,axiom,
! [A2: real] :
( ( ( A2 = zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= zero_zero_real ) )
& ( ( A2 != zero_zero_real )
=> ( ( divide_divide_real @ A2 @ A2 )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_520_divide__eq__eq__1,axiom,
! [B: real,A2: real] :
( ( ( divide_divide_real @ B @ A2 )
= one_one_real )
= ( ( A2 != zero_zero_real )
& ( A2 = B ) ) ) ).
% divide_eq_eq_1
thf(fact_521_eq__divide__eq__1,axiom,
! [B: real,A2: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A2 ) )
= ( ( A2 != zero_zero_real )
& ( A2 = B ) ) ) ).
% eq_divide_eq_1
thf(fact_522_one__divide__eq__0__iff,axiom,
! [A2: real] :
( ( ( divide_divide_real @ one_one_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_523_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X5 ) ).
% linordered_field_no_lb
thf(fact_524_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_525_divide__divide__eq__left_H,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A2 @ B ) @ C )
= ( divide_divide_real @ A2 @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_526_divide__divide__times__eq,axiom,
! [X2: real,Y2: real,Z2: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ W ) @ ( times_times_real @ Y2 @ Z2 ) ) ) ).
% divide_divide_times_eq
thf(fact_527_times__divide__times__eq,axiom,
! [X2: real,Y2: real,Z2: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ Z2 @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ Z2 ) @ ( times_times_real @ Y2 @ W ) ) ) ).
% times_divide_times_eq
thf(fact_528_diff__divide__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_529_divide__neg__neg,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ zero_zero_real )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% divide_neg_neg
thf(fact_530_divide__neg__pos,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_531_divide__pos__neg,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ Y2 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_532_divide__pos__pos,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% divide_pos_pos
thf(fact_533_divide__less__0__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A2 @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_534_divide__less__cancel,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A2 @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A2 ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_535_zero__less__divide__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_536_divide__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_537_divide__strict__right__mono__neg,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_538_frac__eq__eq,axiom,
! [Y2: real,Z2: real,X2: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ( divide_divide_real @ X2 @ Y2 )
= ( divide_divide_real @ W @ Z2 ) )
= ( ( times_times_real @ X2 @ Z2 )
= ( times_times_real @ W @ Y2 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_539_divide__eq__eq,axiom,
! [B: real,C: real,A2: real] :
( ( ( divide_divide_real @ B @ C )
= A2 )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A2 @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_540_eq__divide__eq,axiom,
! [A2: real,B: real,C: real] :
( ( A2
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A2 @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A2 = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_541_divide__eq__imp,axiom,
! [C: real,B: real,A2: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A2 @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A2 ) ) ) ).
% divide_eq_imp
thf(fact_542_eq__divide__imp,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A2 @ C )
= B )
=> ( A2
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_543_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A2: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A2 )
= ( B
= ( times_times_real @ A2 @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_544_nonzero__eq__divide__eq,axiom,
! [C: real,A2: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A2
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A2 @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_545_right__inverse__eq,axiom,
! [B: real,A2: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A2 @ B )
= one_one_real )
= ( A2 = B ) ) ) ).
% right_inverse_eq
thf(fact_546_divide__less__eq,axiom,
! [B: real,C: real,A2: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A2 )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A2 @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_547_less__divide__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A2 @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_548_neg__divide__less__eq,axiom,
! [C: real,B: real,A2: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A2 )
= ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_549_neg__less__divide__eq,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A2 @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A2 @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_550_pos__divide__less__eq,axiom,
! [C: real,B: real,A2: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A2 )
= ( ord_less_real @ B @ ( times_times_real @ A2 @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_551_pos__less__divide__eq,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A2 @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_552_mult__imp__div__pos__less,axiom,
! [Y2: real,X2: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ X2 @ ( times_times_real @ Z2 @ Y2 ) )
=> ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ Z2 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_553_mult__imp__less__div__pos,axiom,
! [Y2: real,Z2: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( times_times_real @ Z2 @ Y2 ) @ X2 )
=> ( ord_less_real @ Z2 @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_554_divide__strict__left__mono,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_555_divide__strict__left__mono__neg,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_556_divide__less__eq__1,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ B @ A2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ A2 @ B ) )
| ( A2 = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_557_less__divide__eq__1,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_real @ A2 @ B ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_real @ B @ A2 ) ) ) ) ).
% less_divide_eq_1
thf(fact_558_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: real,A2: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B @ Z2 ) )
= A2 ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A2 @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_559_diff__frac__eq,axiom,
! [Y2: real,Z2: real,X2: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ W @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z2 ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_560_diff__divide__eq__iff,axiom,
! [Z2: real,X2: real,Y2: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ X2 @ ( divide_divide_real @ Y2 @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z2 ) @ Y2 ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_561_divide__diff__eq__iff,axiom,
! [Z2: real,X2: real,Y2: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Z2 ) @ Y2 )
= ( divide_divide_real @ ( minus_minus_real @ X2 @ ( times_times_real @ Y2 @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_562_frac__less__eq,axiom,
! [Y2: real,Z2: real,X2: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ W @ Z2 ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z2 ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_563_div__mult__mult1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ).
% div_mult_mult1
thf(fact_564_div__mult__mult1,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A2 @ B ) ) ) ).
% div_mult_mult1
thf(fact_565_div__mult__mult2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ).
% div_mult_mult2
thf(fact_566_div__mult__mult2,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A2 @ B ) ) ) ).
% div_mult_mult2
thf(fact_567_div__mult__mult1__if,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_568_div__mult__mult1__if,axiom,
! [C: int,A2: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_569_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_570_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_571_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_572_isCont__divide,axiom,
! [A2: real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ G )
=> ( ( ( G @ A2 )
!= zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ) ).
% isCont_divide
thf(fact_573_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_574_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_575_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_576_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_577_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_578_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_579_isCont__arcosh,axiom,
! [X2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcosh_real ) ) ).
% isCont_arcosh
thf(fact_580_continuous__mult,axiom,
! [F3: filter_real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ F3 @ F )
=> ( ( topolo4422821103128117721l_real @ F3 @ G )
=> ( topolo4422821103128117721l_real @ F3
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% continuous_mult
thf(fact_581_continuous__mult_H,axiom,
! [F3: filter_real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ F3 @ F )
=> ( ( topolo4422821103128117721l_real @ F3 @ G )
=> ( topolo4422821103128117721l_real @ F3
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% continuous_mult'
thf(fact_582_continuous__mult__left,axiom,
! [F3: filter_real,F: real > real,C: real] :
( ( topolo4422821103128117721l_real @ F3 @ F )
=> ( topolo4422821103128117721l_real @ F3
@ ^ [X: real] : ( times_times_real @ C @ ( F @ X ) ) ) ) ).
% continuous_mult_left
thf(fact_583_continuous__mult__right,axiom,
! [F3: filter_real,F: real > real,C: real] :
( ( topolo4422821103128117721l_real @ F3 @ F )
=> ( topolo4422821103128117721l_real @ F3
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ C ) ) ) ).
% continuous_mult_right
thf(fact_584_continuous__diff,axiom,
! [F3: filter_real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ F3 @ F )
=> ( ( topolo4422821103128117721l_real @ F3 @ G )
=> ( topolo4422821103128117721l_real @ F3
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% continuous_diff
thf(fact_585_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_586_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_587_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_588_isCont__mult,axiom,
! [A2: real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ G )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% isCont_mult
thf(fact_589_continuous__at__within__divide,axiom,
! [A2: real,S: set_real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ G )
=> ( ( ( G @ A2 )
!= zero_zero_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S )
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_590_isCont__diff,axiom,
! [A2: real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ G )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% isCont_diff
thf(fact_591_bits__div__by__1,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ one_one_nat )
= A2 ) ).
% bits_div_by_1
thf(fact_592_bits__div__by__1,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ one_one_int )
= A2 ) ).
% bits_div_by_1
thf(fact_593_bits__div__by__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_594_bits__div__by__0,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_595_bits__div__0,axiom,
! [A2: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_596_bits__div__0,axiom,
! [A2: int] :
( ( divide_divide_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% bits_div_0
thf(fact_597_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_598_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_599_isCont__o2,axiom,
! [A2: real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ A2 ) @ top_top_set_real ) @ G )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X: real] : ( G @ ( F @ X ) ) ) ) ) ).
% isCont_o2
thf(fact_600_continuous__arsinh,axiom,
! [F3: filter_real,F: real > real] :
( ( topolo4422821103128117721l_real @ F3 @ F )
=> ( topolo4422821103128117721l_real @ F3
@ ^ [X: real] : ( arsinh_real @ ( F @ X ) ) ) ) ).
% continuous_arsinh
thf(fact_601_continuous__const,axiom,
! [F3: filter_real,C: real] :
( topolo4422821103128117721l_real @ F3
@ ^ [X: real] : C ) ).
% continuous_const
thf(fact_602_isCont__arsinh,axiom,
! [X2: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arsinh_real ) ).
% isCont_arsinh
thf(fact_603_continuous__ident,axiom,
! [X2: real,S2: set_real] :
( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S2 )
@ ^ [X: real] : X ) ).
% continuous_ident
thf(fact_604_continuous__at__imp__continuous__at__within,axiom,
! [X2: real,F: real > real,S: set_real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S ) @ F ) ) ).
% continuous_at_imp_continuous_at_within
thf(fact_605_continuous__within__compose3,axiom,
! [F: real > real,X2: real,G: real > real,S: set_real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S ) @ F )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ S )
@ ^ [X: real] : ( G @ ( F @ X ) ) ) ) ) ).
% continuous_within_compose3
thf(fact_606_DERIV__powr,axiom,
! [G: real > real,M2: real,X2: real,F: real > real,R: real] :
( ( has_fi5821293074295781190e_real @ G @ M2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
=> ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
@ ( times_times_real @ ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X2 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M2 @ ( F @ X2 ) ) @ ( G @ X2 ) ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% DERIV_powr
thf(fact_607_mult__if__delta,axiom,
! [P: $o,Q3: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q3 )
= Q3 ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q3 )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_608_mult__if__delta,axiom,
! [P: $o,Q3: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q3 )
= Q3 ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q3 )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_609_mult__if__delta,axiom,
! [P: $o,Q3: int] :
( ( P
=> ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q3 )
= Q3 ) )
& ( ~ P
=> ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q3 )
= zero_zero_int ) ) ) ).
% mult_if_delta
thf(fact_610_mult__less__iff1,axiom,
! [Z2: real,X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X2 @ Z2 ) @ ( times_times_real @ Y2 @ Z2 ) )
= ( ord_less_real @ X2 @ Y2 ) ) ) ).
% mult_less_iff1
thf(fact_611_mult__less__iff1,axiom,
! [Z2: int,X2: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X2 @ Z2 ) @ ( times_times_int @ Y2 @ Z2 ) )
= ( ord_less_int @ X2 @ Y2 ) ) ) ).
% mult_less_iff1
thf(fact_612_has__derivative__powr,axiom,
! [G: real > real,G2: real > real,X2: real,X4: set_real,F: real > real,F2: real > real] :
( ( has_de1759254742604945161l_real @ G @ G2 @ ( topolo2177554685111907308n_real @ X2 @ X4 ) )
=> ( ( has_de1759254742604945161l_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ X4 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
=> ( ( member_real @ X2 @ X4 )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
@ ^ [H2: real] : ( times_times_real @ ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) ) @ ( plus_plus_real @ ( times_times_real @ ( F2 @ H2 ) @ ( ln_ln_real @ ( G @ X2 ) ) ) @ ( divide_divide_real @ ( times_times_real @ ( G2 @ H2 ) @ ( F @ X2 ) ) @ ( G @ X2 ) ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ X4 ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_613_DERIV__log,axiom,
! [X2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X2 ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% DERIV_log
thf(fact_614_add__left__cancel,axiom,
! [A2: real,B: real,C: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_615_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_616_add__right__cancel,axiom,
! [B: real,A2: real,C: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_617_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_618_set__plus__intro,axiom,
! [A2: real,C2: set_real,B: real,D: set_real] :
( ( member_real @ A2 @ C2 )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( plus_plus_real @ A2 @ B ) @ ( plus_plus_set_real @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_619_set__plus__intro,axiom,
! [A2: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( plus_plus_nat @ A2 @ B ) @ ( plus_plus_set_nat @ C2 @ D ) ) ) ) ).
% set_plus_intro
thf(fact_620_add_Oright__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% add.right_neutral
thf(fact_621_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_622_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_623_double__zero__sym,axiom,
! [A2: real] :
( ( zero_zero_real
= ( plus_plus_real @ A2 @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_624_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_625_add__cancel__left__left,axiom,
! [B: real,A2: real] :
( ( ( plus_plus_real @ B @ A2 )
= A2 )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_626_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_627_add__cancel__left__left,axiom,
! [B: int,A2: int] :
( ( ( plus_plus_int @ B @ A2 )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_628_add__cancel__left__right,axiom,
! [A2: real,B: real] :
( ( ( plus_plus_real @ A2 @ B )
= A2 )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_629_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_630_add__cancel__left__right,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_631_add__cancel__right__left,axiom,
! [A2: real,B: real] :
( ( A2
= ( plus_plus_real @ B @ A2 ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_632_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_633_add__cancel__right__left,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ B @ A2 ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_634_add__cancel__right__right,axiom,
! [A2: real,B: real] :
( ( A2
= ( plus_plus_real @ A2 @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_635_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_636_add__cancel__right__right,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ A2 @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_637_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_638_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y2 ) )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_639_add__0,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% add_0
thf(fact_640_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_641_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_642_add__less__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_643_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_644_add__less__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_645_add__less__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_646_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_647_add__less__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_648_add__diff__cancel,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_649_add__diff__cancel,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_650_diff__add__cancel,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_651_diff__add__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_652_add__diff__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_653_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_654_add__diff__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_655_add__diff__cancel__left_H,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_656_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_657_add__diff__cancel__left_H,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_658_add__diff__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_659_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_660_add__diff__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_661_add__diff__cancel__right_H,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_662_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_663_add__diff__cancel__right_H,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_664_add__less__same__cancel1,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A2 ) @ B )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_665_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_666_add__less__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_667_add__less__same__cancel2,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ B ) @ B )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_668_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_669_add__less__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_670_less__add__same__cancel1,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_671_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_672_less__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_673_less__add__same__cancel2,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ ( plus_plus_real @ B @ A2 ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_674_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_675_less__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_676_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_677_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_678_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_679_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_680_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_681_Transcendental_Olog__one,axiom,
! [A2: real] :
( ( log @ A2 @ one_one_real )
= zero_zero_real ) ).
% Transcendental.log_one
thf(fact_682_div__mult__self4,axiom,
! [B: nat,C: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A2 ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self4
thf(fact_683_div__mult__self4,axiom,
! [B: int,C: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A2 ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_self4
thf(fact_684_div__mult__self3,axiom,
! [B: nat,C: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A2 ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self3
thf(fact_685_div__mult__self3,axiom,
! [B: int,C: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A2 ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_self3
thf(fact_686_div__mult__self2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self2
thf(fact_687_div__mult__self2,axiom,
! [B: int,A2: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_self2
thf(fact_688_div__mult__self1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self1
thf(fact_689_div__mult__self1,axiom,
! [B: int,A2: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B ) ) ) ) ).
% div_mult_self1
thf(fact_690_zero__less__log__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less_real @ one_one_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ ( log @ A2 @ X2 ) )
= ( ord_less_real @ one_one_real @ X2 ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_691_log__less__zero__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less_real @ one_one_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ ( log @ A2 @ X2 ) @ zero_zero_real )
= ( ord_less_real @ X2 @ one_one_real ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_692_one__less__log__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less_real @ one_one_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ one_one_real @ ( log @ A2 @ X2 ) )
= ( ord_less_real @ A2 @ X2 ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_693_log__less__one__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less_real @ one_one_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ ( log @ A2 @ X2 ) @ one_one_real )
= ( ord_less_real @ X2 @ A2 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_694_log__less__cancel__iff,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y2 ) )
= ( ord_less_real @ X2 @ Y2 ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_695_log__eq__one,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( A2 != one_one_real )
=> ( ( log @ A2 @ A2 )
= one_one_real ) ) ) ).
% log_eq_one
thf(fact_696_powr__log__cancel,axiom,
! [A2: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( A2 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( powr_real @ A2 @ ( log @ A2 @ X2 ) )
= X2 ) ) ) ) ).
% powr_log_cancel
thf(fact_697_log__powr__cancel,axiom,
! [A2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( A2 != one_one_real )
=> ( ( log @ A2 @ ( powr_real @ A2 @ Y2 ) )
= Y2 ) ) ) ).
% log_powr_cancel
thf(fact_698_GDERIV__add,axiom,
! [F: real > real,X2: real,Df: real,G: real > real,Dg: real] :
( ( inner_gderiv_real @ F @ X2 @ Df )
=> ( ( inner_gderiv_real @ G @ X2 @ Dg )
=> ( inner_gderiv_real
@ ^ [X: real] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) )
@ X2
@ ( plus_plus_real @ Df @ Dg ) ) ) ) ).
% GDERIV_add
thf(fact_699_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_700_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_701_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_702_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_703_group__cancel_Oadd1,axiom,
! [A: real,K: real,A2: real,B: real] :
( ( A
= ( plus_plus_real @ K @ A2 ) )
=> ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_704_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_705_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A2: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A2 @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_706_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A2: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_707_add_Oassoc,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_708_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_709_add_Oleft__cancel,axiom,
! [A2: real,B: real,C: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_710_add_Oright__cancel,axiom,
! [B: real,A2: real,C: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C @ A2 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_711_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_712_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_713_add_Oleft__commute,axiom,
! [B: real,A2: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A2 @ C ) )
= ( plus_plus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_714_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_715_add__left__imp__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_716_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_717_add__right__imp__eq,axiom,
! [B: real,A2: real,C: real] :
( ( ( plus_plus_real @ B @ A2 )
= ( plus_plus_real @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_718_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_719_set__plus__elim,axiom,
! [X2: real,A: set_real,B3: set_real] :
( ( member_real @ X2 @ ( plus_plus_set_real @ A @ B3 ) )
=> ~ ! [A4: real,B4: real] :
( ( X2
= ( plus_plus_real @ A4 @ B4 ) )
=> ( ( member_real @ A4 @ A )
=> ~ ( member_real @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_720_set__plus__elim,axiom,
! [X2: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ X2 @ ( plus_plus_set_nat @ A @ B3 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X2
= ( plus_plus_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A )
=> ~ ( member_nat @ B4 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_721_comm__monoid__add__class_Oadd__0,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_722_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_723_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_724_add_Ocomm__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% add.comm_neutral
thf(fact_725_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_726_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_727_add_Ogroup__left__neutral,axiom,
! [A2: real] :
( ( plus_plus_real @ zero_zero_real @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_728_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_729_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_730_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_731_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_732_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_733_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_734_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_735_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_736_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_737_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_738_add__strict__mono,axiom,
! [A2: real,B: real,C: real,D4: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ C @ D4 )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ D4 ) ) ) ) ).
% add_strict_mono
thf(fact_739_add__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D4: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D4 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).
% add_strict_mono
thf(fact_740_add__strict__mono,axiom,
! [A2: int,B: int,C: int,D4: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D4 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D4 ) ) ) ) ).
% add_strict_mono
thf(fact_741_add__strict__left__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_742_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_743_add__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_744_add__strict__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_745_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_746_add__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_747_add__less__imp__less__left,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_748_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_749_add__less__imp__less__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_750_add__less__imp__less__right,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_751_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_752_add__less__imp__less__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_753_ring__class_Oring__distribs_I2_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_754_ring__class_Oring__distribs_I2_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_755_ring__class_Oring__distribs_I1_J,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_756_ring__class_Oring__distribs_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_757_comm__semiring__class_Odistrib,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_758_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_759_comm__semiring__class_Odistrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_760_distrib__left,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ B ) @ ( times_times_real @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_761_distrib__left,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_762_distrib__left,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_763_distrib__right,axiom,
! [A2: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_764_distrib__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_765_distrib__right,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_766_combine__common__factor,axiom,
! [A2: real,E2: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_767_combine__common__factor,axiom,
! [A2: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_768_combine__common__factor,axiom,
! [A2: int,E2: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_769_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A2: real,B: real,X2: real] :
( ( times_times_real @ ( plus_plus_real @ A2 @ B ) @ X2 )
= ( plus_plus_real @ ( times_times_real @ A2 @ X2 ) @ ( times_times_real @ B @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_770_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A2: real,X2: real,Y2: real] :
( ( times_times_real @ A2 @ ( plus_plus_real @ X2 @ Y2 ) )
= ( plus_plus_real @ ( times_times_real @ A2 @ X2 ) @ ( times_times_real @ A2 @ Y2 ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_771_group__cancel_Osub1,axiom,
! [A: real,K: real,A2: real,B: real] :
( ( A
= ( plus_plus_real @ K @ A2 ) )
=> ( ( minus_minus_real @ A @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_772_group__cancel_Osub1,axiom,
! [A: int,K: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( minus_minus_int @ A @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_773_diff__eq__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ( minus_minus_real @ A2 @ B )
= C )
= ( A2
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_774_diff__eq__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( minus_minus_int @ A2 @ B )
= C )
= ( A2
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_775_eq__diff__eq,axiom,
! [A2: real,C: real,B: real] :
( ( A2
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_776_eq__diff__eq,axiom,
! [A2: int,C: int,B: int] :
( ( A2
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_777_add__diff__eq,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_778_add__diff__eq,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_779_diff__diff__eq2,axiom,
! [A2: real,B: real,C: real] :
( ( minus_minus_real @ A2 @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_780_diff__diff__eq2,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_781_diff__add__eq,axiom,
! [A2: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_782_diff__add__eq,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_783_diff__add__eq__diff__diff__swap,axiom,
! [A2: real,B: real,C: real] :
( ( minus_minus_real @ A2 @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_784_diff__add__eq__diff__diff__swap,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_785_add__implies__diff,axiom,
! [C: real,B: real,A2: real] :
( ( ( plus_plus_real @ C @ B )
= A2 )
=> ( C
= ( minus_minus_real @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_786_add__implies__diff,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_787_add__implies__diff,axiom,
! [C: int,B: int,A2: int] :
( ( ( plus_plus_int @ C @ B )
= A2 )
=> ( C
= ( minus_minus_int @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_788_diff__diff__eq,axiom,
! [A2: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( minus_minus_real @ A2 @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_789_diff__diff__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_790_diff__diff__eq,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_791_add__divide__distrib,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A2 @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_792_continuous__add,axiom,
! [F3: filter_real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ F3 @ F )
=> ( ( topolo4422821103128117721l_real @ F3 @ G )
=> ( topolo4422821103128117721l_real @ F3
@ ^ [X: real] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% continuous_add
thf(fact_793_Deriv_Ofield__differentiable__add,axiom,
! [F: real > real,F2: real,F3: filter_real,G: real > real,G2: real] :
( ( has_fi5821293074295781190e_real @ F @ F2 @ F3 )
=> ( ( has_fi5821293074295781190e_real @ G @ G2 @ F3 )
=> ( has_fi5821293074295781190e_real
@ ^ [Z: real] : ( plus_plus_real @ ( F @ Z ) @ ( G @ Z ) )
@ ( plus_plus_real @ F2 @ G2 )
@ F3 ) ) ) ).
% Deriv.field_differentiable_add
thf(fact_794_has__derivative__add,axiom,
! [F: real > real,F2: real > real,F3: filter_real,G: real > real,G2: real > real] :
( ( has_de1759254742604945161l_real @ F @ F2 @ F3 )
=> ( ( has_de1759254742604945161l_real @ G @ G2 @ F3 )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) )
@ ^ [X: real] : ( plus_plus_real @ ( F2 @ X ) @ ( G2 @ X ) )
@ F3 ) ) ) ).
% has_derivative_add
thf(fact_795_has__derivative__add__const,axiom,
! [F: real > real,F2: real > real,Net: filter_real,C: real] :
( ( has_de1759254742604945161l_real @ F @ F2 @ Net )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( plus_plus_real @ ( F @ X ) @ C )
@ F2
@ Net ) ) ).
% has_derivative_add_const
thf(fact_796_log__mult,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( A2 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( log @ A2 @ ( times_times_real @ X2 @ Y2 ) )
= ( plus_plus_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y2 ) ) ) ) ) ) ) ).
% log_mult
thf(fact_797_log__def,axiom,
( log
= ( ^ [A3: real,X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ A3 ) ) ) ) ).
% log_def
thf(fact_798_add__less__zeroD,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
=> ( ( ord_less_real @ X2 @ zero_zero_real )
| ( ord_less_real @ Y2 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_799_add__less__zeroD,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ ( plus_plus_int @ X2 @ Y2 ) @ zero_zero_int )
=> ( ( ord_less_int @ X2 @ zero_zero_int )
| ( ord_less_int @ Y2 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_800_add__neg__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_801_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_802_add__neg__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_803_add__pos__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_804_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_805_add__pos__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_806_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C4: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_807_pos__add__strict,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_808_pos__add__strict,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_809_pos__add__strict,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_810_less__add__one,axiom,
! [A2: real] : ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ one_one_real ) ) ).
% less_add_one
thf(fact_811_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_812_less__add__one,axiom,
! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% less_add_one
thf(fact_813_add__mono1,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_814_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_815_add__mono1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_816_diff__less__eq,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A2 @ B ) @ C )
= ( ord_less_real @ A2 @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_817_diff__less__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_818_less__diff__eq,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_real @ A2 @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A2 @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_819_less__diff__eq,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_820_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: real,B: real] :
( ~ ( ord_less_real @ A2 @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_821_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ~ ( ord_less_nat @ A2 @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_822_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: int,B: int] :
( ~ ( ord_less_int @ A2 @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_823_eq__add__iff1,axiom,
! [A2: real,E2: real,C: real,B: real,D4: real] :
( ( ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D4 ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ E2 ) @ C )
= D4 ) ) ).
% eq_add_iff1
thf(fact_824_eq__add__iff1,axiom,
! [A2: int,E2: int,C: int,B: int,D4: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D4 ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E2 ) @ C )
= D4 ) ) ).
% eq_add_iff1
thf(fact_825_eq__add__iff2,axiom,
! [A2: real,E2: real,C: real,B: real,D4: real] :
( ( ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D4 ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A2 ) @ E2 ) @ D4 ) ) ) ).
% eq_add_iff2
thf(fact_826_eq__add__iff2,axiom,
! [A2: int,E2: int,C: int,B: int,D4: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D4 ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E2 ) @ D4 ) ) ) ).
% eq_add_iff2
thf(fact_827_square__diff__square__factored,axiom,
! [X2: real,Y2: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) )
= ( times_times_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( minus_minus_real @ X2 @ Y2 ) ) ) ).
% square_diff_square_factored
thf(fact_828_square__diff__square__factored,axiom,
! [X2: int,Y2: int] :
( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) )
= ( times_times_int @ ( plus_plus_int @ X2 @ Y2 ) @ ( minus_minus_int @ X2 @ Y2 ) ) ) ).
% square_diff_square_factored
thf(fact_829_powr__add,axiom,
! [X2: real,A2: real,B: real] :
( ( powr_real @ X2 @ ( plus_plus_real @ A2 @ B ) )
= ( times_times_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B ) ) ) ).
% powr_add
thf(fact_830_log__add__eq__powr,axiom,
! [B: real,X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( plus_plus_real @ ( log @ B @ X2 ) @ Y2 )
= ( log @ B @ ( times_times_real @ X2 @ ( powr_real @ B @ Y2 ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_831_add__log__eq__powr,axiom,
! [B: real,X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( plus_plus_real @ Y2 @ ( log @ B @ X2 ) )
= ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y2 ) @ X2 ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_832_DERIV__add,axiom,
! [F: real > real,D: real,X2: real,S: set_real,G: real > real,E: real] :
( ( has_fi5821293074295781190e_real @ F @ D @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( ( has_fi5821293074295781190e_real @ G @ E @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) )
@ ( plus_plus_real @ D @ E )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ) ).
% DERIV_add
thf(fact_833_log__powr,axiom,
! [X2: real,B: real,Y2: real] :
( ( X2 != zero_zero_real )
=> ( ( log @ B @ ( powr_real @ X2 @ Y2 ) )
= ( times_times_real @ Y2 @ ( log @ B @ X2 ) ) ) ) ).
% log_powr
thf(fact_834_log__base__powr,axiom,
! [A2: real,B: real,X2: real] :
( ( A2 != zero_zero_real )
=> ( ( log @ ( powr_real @ A2 @ B ) @ X2 )
= ( divide_divide_real @ ( log @ A2 @ X2 ) @ B ) ) ) ).
% log_base_powr
thf(fact_835_not__sum__squares__lt__zero,axiom,
! [X2: real,Y2: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_836_not__sum__squares__lt__zero,axiom,
! [X2: int,Y2: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_837_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_838_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_839_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_840_less__add__iff2,axiom,
! [A2: real,E2: real,C: real,B: real,D4: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D4 ) )
= ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A2 ) @ E2 ) @ D4 ) ) ) ).
% less_add_iff2
thf(fact_841_less__add__iff2,axiom,
! [A2: int,E2: int,C: int,B: int,D4: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D4 ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E2 ) @ D4 ) ) ) ).
% less_add_iff2
thf(fact_842_less__add__iff1,axiom,
! [A2: real,E2: real,C: real,B: real,D4: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D4 ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B ) @ E2 ) @ C ) @ D4 ) ) ).
% less_add_iff1
thf(fact_843_less__add__iff1,axiom,
! [A2: int,E2: int,C: int,B: int,D4: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D4 ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E2 ) @ C ) @ D4 ) ) ).
% less_add_iff1
thf(fact_844_divide__add__eq__iff,axiom,
! [Z2: real,X2: real,Y2: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z2 ) @ Y2 )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y2 @ Z2 ) ) @ Z2 ) ) ) ).
% divide_add_eq_iff
thf(fact_845_add__divide__eq__iff,axiom,
! [Z2: real,X2: real,Y2: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y2 @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z2 ) @ Y2 ) @ Z2 ) ) ) ).
% add_divide_eq_iff
thf(fact_846_add__num__frac,axiom,
! [Y2: real,Z2: real,X2: real] :
( ( Y2 != zero_zero_real )
=> ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X2 @ Y2 ) )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z2 @ Y2 ) ) @ Y2 ) ) ) ).
% add_num_frac
thf(fact_847_add__frac__num,axiom,
! [Y2: real,X2: real,Z2: real] :
( ( Y2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y2 ) @ Z2 )
= ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z2 @ Y2 ) ) @ Y2 ) ) ) ).
% add_frac_num
thf(fact_848_add__frac__eq,axiom,
! [Y2: real,Z2: real,X2: real,W: real] :
( ( Y2 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ W @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z2 ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z2 ) ) ) ) ) ).
% add_frac_eq
thf(fact_849_add__divide__eq__if__simps_I1_J,axiom,
! [Z2: real,A2: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B @ Z2 ) )
= A2 ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A2 @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_850_add__divide__eq__if__simps_I2_J,axiom,
! [Z2: real,A2: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z2 ) @ B )
= B ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z2 ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A2 @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_851_square__diff__one__factored,axiom,
! [X2: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X2 @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_852_square__diff__one__factored,axiom,
! [X2: int] :
( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X2 @ one_one_int ) @ ( minus_minus_int @ X2 @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_853_div__add__self2,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_854_div__add__self2,axiom,
! [B: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_855_div__add__self1,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_856_div__add__self1,axiom,
! [B: int,A2: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A2 @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_857_less__half__sum,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ A2 @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_858_gt__half__sum,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_859_square__bound__lemma,axiom,
! [X2: real] : ( ord_less_real @ X2 @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( plus_plus_real @ one_one_real @ X2 ) ) ) ).
% square_bound_lemma
thf(fact_860_isCont__add,axiom,
! [A2: real,F: real > nat,G: real > nat] :
( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F )
=> ( ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ G )
=> ( topolo8373849641844647293al_nat @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X: real] : ( plus_plus_nat @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% isCont_add
thf(fact_861_isCont__add,axiom,
! [A2: real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ G )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X: real] : ( plus_plus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% isCont_add
thf(fact_862_DERIV__mult,axiom,
! [F: real > real,Da: real,X2: real,S: set_real,G: real > real,Db: real] :
( ( has_fi5821293074295781190e_real @ F @ Da @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( ( has_fi5821293074295781190e_real @ G @ Db @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( G @ X ) )
@ ( plus_plus_real @ ( times_times_real @ Da @ ( G @ X2 ) ) @ ( times_times_real @ Db @ ( F @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ) ).
% DERIV_mult
thf(fact_863_DERIV__mult_H,axiom,
! [F: real > real,D: real,X2: real,S: set_real,G: real > real,E: real] :
( ( has_fi5821293074295781190e_real @ F @ D @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( ( has_fi5821293074295781190e_real @ G @ E @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( G @ X ) )
@ ( plus_plus_real @ ( times_times_real @ ( F @ X2 ) @ E ) @ ( times_times_real @ D @ ( G @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ) ).
% DERIV_mult'
thf(fact_864_DERIV__shift,axiom,
! [F: real > real,Y2: real,X2: real,Z2: real] :
( ( has_fi5821293074295781190e_real @ F @ Y2 @ ( topolo2177554685111907308n_real @ ( plus_plus_real @ X2 @ Z2 ) @ top_top_set_real ) )
= ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( F @ ( plus_plus_real @ X @ Z2 ) )
@ Y2
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% DERIV_shift
thf(fact_865_has__derivative__mult,axiom,
! [F: real > real,F2: real > real,X2: real,S: set_real,G: real > real,G2: real > real] :
( ( has_de1759254742604945161l_real @ F @ F2 @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( ( has_de1759254742604945161l_real @ G @ G2 @ ( topolo2177554685111907308n_real @ X2 @ S ) )
=> ( has_de1759254742604945161l_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( G @ X ) )
@ ^ [H2: real] : ( plus_plus_real @ ( times_times_real @ ( F @ X2 ) @ ( G2 @ H2 ) ) @ ( times_times_real @ ( F2 @ H2 ) @ ( G @ X2 ) ) )
@ ( topolo2177554685111907308n_real @ X2 @ S ) ) ) ) ).
% has_derivative_mult
thf(fact_866_log__base__change,axiom,
! [A2: real,B: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( A2 != one_one_real )
=> ( ( log @ B @ X2 )
= ( divide_divide_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ B ) ) ) ) ) ).
% log_base_change
thf(fact_867_powr__less__iff,axiom,
! [B: real,X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ ( powr_real @ B @ Y2 ) @ X2 )
= ( ord_less_real @ Y2 @ ( log @ B @ X2 ) ) ) ) ) ).
% powr_less_iff
thf(fact_868_less__powr__iff,axiom,
! [B: real,X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ ( powr_real @ B @ Y2 ) )
= ( ord_less_real @ ( log @ B @ X2 ) @ Y2 ) ) ) ) ).
% less_powr_iff
thf(fact_869_log__less__iff,axiom,
! [B: real,X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ ( log @ B @ X2 ) @ Y2 )
= ( ord_less_real @ X2 @ ( powr_real @ B @ Y2 ) ) ) ) ) ).
% log_less_iff
thf(fact_870_less__log__iff,axiom,
! [B: real,X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ Y2 @ ( log @ B @ X2 ) )
= ( ord_less_real @ ( powr_real @ B @ Y2 ) @ X2 ) ) ) ) ).
% less_log_iff
thf(fact_871_ln__mult,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ln_ln_real @ ( times_times_real @ X2 @ Y2 ) )
= ( plus_plus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) ) ) ) ) ).
% ln_mult
thf(fact_872_has__real__derivative__neg__dec__right,axiom,
! [F: real > real,L: real,X2: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H: real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( member_real @ ( plus_plus_real @ X2 @ H ) @ S2 )
=> ( ( ord_less_real @ H @ D5 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_873_has__real__derivative__pos__inc__right,axiom,
! [F: real > real,L: real,X2: real,S2: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S2 ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H: real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( member_real @ ( plus_plus_real @ X2 @ H ) @ S2 )
=> ( ( ord_less_real @ H @ D5 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_874_log__divide,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( A2 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( log @ A2 @ ( divide_divide_real @ X2 @ Y2 ) )
= ( minus_minus_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y2 ) ) ) ) ) ) ) ).
% log_divide
thf(fact_875_continuous__at__within__log,axiom,
! [A2: real,S: set_real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ F )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S ) @ G )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ A2 ) )
=> ( ( ( F @ A2 )
!= one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ A2 ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ S )
@ ^ [X: real] : ( log @ ( F @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ).
% continuous_at_within_log
thf(fact_876_DERIV__neg__dec__right,axiom,
! [F: real > real,L: real,X2: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H: real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( ord_less_real @ H @ D5 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_877_DERIV__pos__inc__right,axiom,
! [F: real > real,L: real,X2: real] :
( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [H: real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( ord_less_real @ H @ D5 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_878_log__eq__div__ln__mult__log,axiom,
! [A2: real,B: real,X2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( A2 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( log @ A2 @ X2 )
= ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A2 ) ) @ ( log @ B @ X2 ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_879_minus__log__eq__powr,axiom,
! [B: real,X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( minus_minus_real @ Y2 @ ( log @ B @ X2 ) )
= ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y2 ) @ X2 ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_880_isCont__log,axiom,
! [A2: real,F: real > real,G: real > real] :
( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ F )
=> ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) @ G )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ A2 ) )
=> ( ( ( F @ A2 )
!= one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ A2 ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real )
@ ^ [X: real] : ( log @ ( F @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ).
% isCont_log
thf(fact_881_sum__squares__eq__zero__iff,axiom,
! [X2: real,Y2: real] :
( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_882_sum__squares__eq__zero__iff,axiom,
! [X2: int,Y2: int] :
( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_883_double__eq__0__iff,axiom,
! [A2: real] :
( ( ( plus_plus_real @ A2 @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_884_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_885_sum__squares__gt__zero__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) )
= ( ( X2 != zero_zero_real )
| ( Y2 != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_886_sum__squares__gt__zero__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) )
= ( ( X2 != zero_zero_int )
| ( Y2 != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_887_mult__diff__mult,axiom,
! [X2: real,Y2: real,A2: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ Y2 ) @ ( times_times_real @ A2 @ B ) )
= ( plus_plus_real @ ( times_times_real @ X2 @ ( minus_minus_real @ Y2 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X2 @ A2 ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_888_mult__diff__mult,axiom,
! [X2: int,Y2: int,A2: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X2 @ Y2 ) @ ( times_times_int @ A2 @ B ) )
= ( plus_plus_int @ ( times_times_int @ X2 @ ( minus_minus_int @ Y2 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X2 @ A2 ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_889_add__scale__eq__noteq,axiom,
! [R: real,A2: real,B: real,C: real,D4: real] :
( ( R != zero_zero_real )
=> ( ( ( A2 = B )
& ( C != D4 ) )
=> ( ( plus_plus_real @ A2 @ ( times_times_real @ R @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R @ D4 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_890_add__scale__eq__noteq,axiom,
! [R: nat,A2: nat,B: nat,C: nat,D4: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A2 = B )
& ( C != D4 ) )
=> ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D4 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_891_add__scale__eq__noteq,axiom,
! [R: int,A2: int,B: int,C: int,D4: int] :
( ( R != zero_zero_int )
=> ( ( ( A2 = B )
& ( C != D4 ) )
=> ( ( plus_plus_int @ A2 @ ( times_times_int @ R @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R @ D4 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_892_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E3: real] :
( ( ord_less_real @ zero_zero_real @ E3 )
& ( ord_less_real @ E3 @ D1 )
& ( ord_less_real @ E3 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_893_add__0__iff,axiom,
! [B: real,A2: real] :
( ( B
= ( plus_plus_real @ B @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% add_0_iff
thf(fact_894_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_895_add__0__iff,axiom,
! [B: int,A2: int] :
( ( B
= ( plus_plus_int @ B @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_896_crossproduct__eq,axiom,
! [W: real,Y2: real,X2: real,Z2: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y2 ) @ ( times_times_real @ X2 @ Z2 ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z2 ) @ ( times_times_real @ X2 @ Y2 ) ) )
= ( ( W = X2 )
| ( Y2 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_897_crossproduct__eq,axiom,
! [W: nat,Y2: nat,X2: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y2 ) @ ( times_times_nat @ X2 @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X2 @ Y2 ) ) )
= ( ( W = X2 )
| ( Y2 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_898_crossproduct__eq,axiom,
! [W: int,Y2: int,X2: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y2 ) @ ( times_times_int @ X2 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X2 @ Y2 ) ) )
= ( ( W = X2 )
| ( Y2 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_899_crossproduct__noteq,axiom,
! [A2: real,B: real,C: real,D4: real] :
( ( ( A2 != B )
& ( C != D4 ) )
= ( ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B @ D4 ) )
!= ( plus_plus_real @ ( times_times_real @ A2 @ D4 ) @ ( times_times_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_900_crossproduct__noteq,axiom,
! [A2: nat,B: nat,C: nat,D4: nat] :
( ( ( A2 != B )
& ( C != D4 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D4 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A2 @ D4 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_901_crossproduct__noteq,axiom,
! [A2: int,B: int,C: int,D4: int] :
( ( ( A2 != B )
& ( C != D4 ) )
= ( ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D4 ) )
!= ( plus_plus_int @ ( times_times_int @ A2 @ D4 ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_902_add__diff__add,axiom,
! [A2: real,C: real,B: real,D4: real] :
( ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ D4 ) )
= ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ ( minus_minus_real @ C @ D4 ) ) ) ).
% add_diff_add
thf(fact_903_add__diff__add,axiom,
! [A2: int,C: int,B: int,D4: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D4 ) )
= ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ ( minus_minus_int @ C @ D4 ) ) ) ).
% add_diff_add
thf(fact_904_DERIV__fun__powr,axiom,
! [G: real > real,M2: real,X2: real,R: real] :
( ( has_fi5821293074295781190e_real @ G @ M2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( powr_real @ ( G @ X ) @ R )
@ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X2 ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M2 )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% DERIV_fun_powr
thf(fact_905_GMVT_H,axiom,
! [A2: real,B: real,F: real > real,G: real > real,G2: real > real,F2: real > real] :
( ( ord_less_real @ A2 @ B )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A2 @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A2 @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G ) ) )
=> ( ! [Z3: real] :
( ( ord_less_real @ A2 @ Z3 )
=> ( ( ord_less_real @ Z3 @ B )
=> ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
=> ( ! [Z3: real] :
( ( ord_less_real @ A2 @ Z3 )
=> ( ( ord_less_real @ Z3 @ B )
=> ( has_fi5821293074295781190e_real @ F @ ( F2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
=> ? [C4: real] :
( ( ord_less_real @ A2 @ C4 )
& ( ord_less_real @ C4 @ B )
& ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A2 ) ) @ ( G2 @ C4 ) )
= ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A2 ) ) @ ( F2 @ C4 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_906_log__minus__eq__powr,axiom,
! [B: real,X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( minus_minus_real @ ( log @ B @ X2 ) @ Y2 )
= ( log @ B @ ( times_times_real @ X2 @ ( powr_real @ B @ ( uminus_uminus_real @ Y2 ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_907_eq__diff__eq_H,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( X2
= ( minus_minus_real @ Y2 @ Z2 ) )
= ( Y2
= ( plus_plus_real @ X2 @ Z2 ) ) ) ).
% eq_diff_eq'
thf(fact_908_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_909_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_910_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_911_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_912_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_913_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_914_add_Oinverse__inverse,axiom,
! [A2: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_915_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_916_neg__equal__iff__equal,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_917_neg__equal__iff__equal,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_918_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_919_add__le__cancel__right,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_920_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_921_add__le__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_922_add__le__cancel__left,axiom,
! [C: real,A2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_923_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_924_add__le__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_925_neg__le__iff__le,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_926_neg__le__iff__le,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_927_neg__equal__zero,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= A2 )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_928_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_929_equal__neg__zero,axiom,
! [A2: real] :
( ( A2
= ( uminus_uminus_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_930_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_931_neg__equal__0__iff__equal,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_932_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_933_neg__0__equal__iff__equal,axiom,
! [A2: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A2 ) )
= ( zero_zero_real = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_934_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_935_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_936_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_937_neg__less__iff__less,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_938_neg__less__iff__less,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_939_vector__space__over__itself_Oscale__minus__right,axiom,
! [A2: real,X2: real] :
( ( times_times_real @ A2 @ ( uminus_uminus_real @ X2 ) )
= ( uminus_uminus_real @ ( times_times_real @ A2 @ X2 ) ) ) ).
% vector_space_over_itself.scale_minus_right
thf(fact_940_vector__space__over__itself_Oscale__minus__left,axiom,
! [A2: real,X2: real] :
( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ X2 )
= ( uminus_uminus_real @ ( times_times_real @ A2 @ X2 ) ) ) ).
% vector_space_over_itself.scale_minus_left
thf(fact_941_mult__minus__right,axiom,
! [A2: real,B: real] :
( ( times_times_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A2 @ B ) ) ) ).
% mult_minus_right
thf(fact_942_mult__minus__right,axiom,
! [A2: int,B: int] :
( ( times_times_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A2 @ B ) ) ) ).
% mult_minus_right
thf(fact_943_minus__mult__minus,axiom,
! [A2: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A2 @ B ) ) ).
% minus_mult_minus
thf(fact_944_minus__mult__minus,axiom,
! [A2: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A2 @ B ) ) ).
% minus_mult_minus
thf(fact_945_mult__minus__left,axiom,
! [A2: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A2 @ B ) ) ) ).
% mult_minus_left
thf(fact_946_mult__minus__left,axiom,
! [A2: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A2 @ B ) ) ) ).
% mult_minus_left
thf(fact_947_minus__add__distrib,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_948_minus__add__distrib,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_949_minus__add__cancel,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( plus_plus_real @ A2 @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_950_minus__add__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_951_add__minus__cancel,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ A2 @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_952_add__minus__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_953_minus__diff__eq,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B ) )
= ( minus_minus_real @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_954_minus__diff__eq,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B ) )
= ( minus_minus_int @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_955_div__minus__minus,axiom,
! [A2: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A2 @ B ) ) ).
% div_minus_minus
thf(fact_956_arsinh__minus__real,axiom,
! [X2: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X2 ) )
= ( uminus_uminus_real @ ( arsinh_real @ X2 ) ) ) ).
% arsinh_minus_real
thf(fact_957_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_958_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_959_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_960_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_961_le__add__same__cancel2,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ B @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_962_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_963_le__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_964_le__add__same__cancel1,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ A2 @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_965_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_966_le__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_967_add__le__same__cancel2,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B ) @ B )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_968_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_969_add__le__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_970_add__le__same__cancel1,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A2 ) @ B )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_971_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_972_add__le__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_973_diff__ge__0__iff__ge,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_eq_real @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_974_diff__ge__0__iff__ge,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_eq_int @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_975_neg__less__eq__nonneg,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_976_neg__less__eq__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_977_less__eq__neg__nonpos,axiom,
! [A2: real] :
( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_978_less__eq__neg__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_979_neg__le__0__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_980_neg__le__0__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_981_neg__0__le__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_982_neg__0__le__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_983_le__add__diff__inverse2,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_984_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_985_le__add__diff__inverse2,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_986_le__add__diff__inverse,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_987_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_988_le__add__diff__inverse,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_989_less__neg__neg,axiom,
! [A2: real] :
( ( ord_less_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_990_less__neg__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_991_neg__less__pos,axiom,
! [A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% neg_less_pos
thf(fact_992_neg__less__pos,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_pos
thf(fact_993_neg__0__less__iff__less,axiom,
! [A2: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_994_neg__0__less__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_995_neg__less__0__iff__less,axiom,
! [A2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_996_neg__less__0__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_997_ab__left__minus,axiom,
! [A2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_998_ab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_999_add_Oright__inverse,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_1000_add_Oright__inverse,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1001_diff__0,axiom,
! [A2: real] :
( ( minus_minus_real @ zero_zero_real @ A2 )
= ( uminus_uminus_real @ A2 ) ) ).
% diff_0
thf(fact_1002_diff__0,axiom,
! [A2: int] :
( ( minus_minus_int @ zero_zero_int @ A2 )
= ( uminus_uminus_int @ A2 ) ) ).
% diff_0
thf(fact_1003_mult__minus1,axiom,
! [Z2: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z2 )
= ( uminus_uminus_real @ Z2 ) ) ).
% mult_minus1
thf(fact_1004_mult__minus1,axiom,
! [Z2: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1
thf(fact_1005_mult__minus1__right,axiom,
! [Z2: real] :
( ( times_times_real @ Z2 @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z2 ) ) ).
% mult_minus1_right
thf(fact_1006_mult__minus1__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1_right
thf(fact_1007_uminus__add__conv__diff,axiom,
! [A2: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( minus_minus_real @ B @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_1008_uminus__add__conv__diff,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( minus_minus_int @ B @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_1009_diff__minus__eq__add,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A2 @ B ) ) ).
% diff_minus_eq_add
thf(fact_1010_diff__minus__eq__add,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A2 @ B ) ) ).
% diff_minus_eq_add
thf(fact_1011_div__minus1__right,axiom,
! [A2: int] :
( ( divide_divide_int @ A2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A2 ) ) ).
% div_minus1_right
thf(fact_1012_divide__minus1,axiom,
! [X2: real] :
( ( divide_divide_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X2 ) ) ).
% divide_minus1
thf(fact_1013_real__add__minus__iff,axiom,
! [X2: real,A2: real] :
( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A2 ) )
= zero_zero_real )
= ( X2 = A2 ) ) ).
% real_add_minus_iff
thf(fact_1014_powr__nonneg__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less_eq_real @ ( powr_real @ A2 @ X2 ) @ zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% powr_nonneg_iff
thf(fact_1015_zero__le__divide__1__iff,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% zero_le_divide_1_iff
thf(fact_1016_divide__le__0__1__iff,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_1017_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_1018_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_1019_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_1020_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_1021_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_1022_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_1023_ln__le__cancel__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) )
= ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1024_powr__one__gt__zero__iff,axiom,
! [X2: real] :
( ( ( powr_real @ X2 @ one_one_real )
= X2 )
= ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% powr_one_gt_zero_iff
thf(fact_1025_powr__one,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( powr_real @ X2 @ one_one_real )
= X2 ) ) ).
% powr_one
thf(fact_1026_powr__le__cancel__iff,axiom,
! [X2: real,A2: real,B: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ( ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B ) )
= ( ord_less_eq_real @ A2 @ B ) ) ) ).
% powr_le_cancel_iff
thf(fact_1027_le__divide__eq__1__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ord_less_eq_real @ A2 @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1028_le__divide__eq__1__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A2 ) )
= ( ord_less_eq_real @ B @ A2 ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1029_divide__le__eq__1__pos,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ord_less_eq_real @ B @ A2 ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1030_divide__le__eq__1__neg,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A2 ) @ one_one_real )
= ( ord_less_eq_real @ A2 @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1031_ln__le__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
= ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_1032_ln__ge__zero__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
= ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% ln_ge_zero_iff
thf(fact_1033_zero__le__log__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less_real @ one_one_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A2 @ X2 ) )
= ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_1034_log__le__zero__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less_real @ one_one_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ ( log @ A2 @ X2 ) @ zero_zero_real )
= ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_1035_one__le__log__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less_real @ one_one_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( log @ A2 @ X2 ) )
= ( ord_less_eq_real @ A2 @ X2 ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_1036_log__le__one__cancel__iff,axiom,
! [A2: real,X2: real] :
( ( ord_less_real @ one_one_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ ( log @ A2 @ X2 ) @ one_one_real )
= ( ord_less_eq_real @ X2 @ A2 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_1037_log__le__cancel__iff,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y2 ) )
= ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_1038_real__add__le__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
= ( ord_less_eq_real @ Y2 @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_le_0_iff
thf(fact_1039_real__0__le__add__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y2 ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y2 ) ) ).
% real_0_le_add_iff
thf(fact_1040_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N2: nat,M: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% nat_less_real_le
thf(fact_1041_real__minus__mult__self__le,axiom,
! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X2 @ X2 ) ) ).
% real_minus_mult_self_le
thf(fact_1042_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y6: real] :
( ( ord_less_real @ X @ Y6 )
| ( X = Y6 ) ) ) ) ).
% less_eq_real_def
thf(fact_1043_minus__diff__minus,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B ) ) ) ).
% minus_diff_minus
thf(fact_1044_minus__diff__minus,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B ) ) ) ).
% minus_diff_minus
thf(fact_1045_inverse__of__nat__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_1046_real__archimedian__rdiv__eq__0,axiom,
! [X2: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X2 ) @ C ) )
=> ( X2 = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1047_real__of__nat__div4,axiom,
! [N: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% real_of_nat_div4
thf(fact_1048_real__of__nat__div2,axiom,
! [N: nat,X2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) ) ).
% real_of_nat_div2
thf(fact_1049_minus__less__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_1050_minus__less__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_1051_less__minus__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A2 ) ) ) ).
% less_minus_iff
thf(fact_1052_less__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_1053_minus__mult__commute,axiom,
! [A2: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( times_times_real @ A2 @ ( uminus_uminus_real @ B ) ) ) ).
% minus_mult_commute
thf(fact_1054_minus__mult__commute,axiom,
! [A2: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( times_times_int @ A2 @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_1055_square__eq__iff,axiom,
! [A2: real,B: real] :
( ( ( times_times_real @ A2 @ A2 )
= ( times_times_real @ B @ B ) )
= ( ( A2 = B )
| ( A2
= ( uminus_uminus_real @ B ) ) ) ) ).
% square_eq_iff
thf(fact_1056_square__eq__iff,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ A2 )
= ( times_times_int @ B @ B ) )
= ( ( A2 = B )
| ( A2
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_1057_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_1058_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_1059_add_Oinverse__distrib__swap,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1060_add_Oinverse__distrib__swap,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1061_group__cancel_Oneg1,axiom,
! [A: real,K: real,A2: real] :
( ( A
= ( plus_plus_real @ K @ A2 ) )
=> ( ( uminus_uminus_real @ A )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_1062_group__cancel_Oneg1,axiom,
! [A: int,K: int,A2: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( uminus_uminus_int @ A )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_1063_minus__diff__commute,axiom,
! [B: real,A2: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A2 )
= ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_1064_minus__diff__commute,axiom,
! [B: int,A2: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A2 )
= ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_1065_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_1066_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_1067_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1068_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1069_minus__divide__right,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B ) )
= ( divide_divide_real @ A2 @ ( uminus_uminus_real @ B ) ) ) ).
% minus_divide_right
thf(fact_1070_minus__divide__divide,axiom,
! [A2: real,B: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A2 @ B ) ) ).
% minus_divide_divide
thf(fact_1071_minus__divide__left,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B ) )
= ( divide_divide_real @ ( uminus_uminus_real @ A2 ) @ B ) ) ).
% minus_divide_left
thf(fact_1072_div__minus__right,axiom,
! [A2: int,B: int] :
( ( divide_divide_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A2 ) @ B ) ) ).
% div_minus_right
thf(fact_1073_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% minf(8)
thf(fact_1074_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_1075_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_1076_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ord_less_eq_real @ X5 @ T ) ) ).
% minf(6)
thf(fact_1077_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_1078_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_1079_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ord_less_eq_real @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1080_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1081_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1082_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1083_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1084_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1085_order__le__imp__less__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1086_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1087_order__le__imp__less__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1088_linorder__le__less__linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1089_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1090_linorder__le__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1091_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1092_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1093_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1094_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1095_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1096_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1097_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1098_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1099_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1100_order__less__le__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1101_order__less__le__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1102_order__less__le__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1103_order__less__le__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1104_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1105_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1106_order__less__le__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1107_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1108_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1109_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1110_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1111_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1112_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1113_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1114_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1115_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1116_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1117_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1118_order__le__less__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1119_order__le__less__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1120_order__le__less__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1121_order__le__less__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1122_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1123_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1124_order__le__less__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1125_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1126_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1127_order__less__le__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1128_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1129_order__less__le__trans,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1130_order__le__less__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z2 )
=> ( ord_less_real @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1131_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1132_order__le__less__trans,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1133_order__neq__le__trans,axiom,
! [A2: real,B: real] :
( ( A2 != B )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1134_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1135_order__neq__le__trans,axiom,
! [A2: int,B: int] :
( ( A2 != B )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1136_order__le__neq__trans,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1137_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1138_order__le__neq__trans,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1139_order__less__imp__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1140_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1141_order__less__imp__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1142_linorder__not__less,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_1143_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_1144_linorder__not__less,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_1145_linorder__not__le,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y2 ) )
= ( ord_less_real @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_1146_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_1147_linorder__not__le,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y2 ) )
= ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_1148_order__less__le,axiom,
( ord_less_real
= ( ^ [X: real,Y6: real] :
( ( ord_less_eq_real @ X @ Y6 )
& ( X != Y6 ) ) ) ) ).
% order_less_le
thf(fact_1149_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y6: nat] :
( ( ord_less_eq_nat @ X @ Y6 )
& ( X != Y6 ) ) ) ) ).
% order_less_le
thf(fact_1150_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y6: int] :
( ( ord_less_eq_int @ X @ Y6 )
& ( X != Y6 ) ) ) ) ).
% order_less_le
thf(fact_1151_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y6: real] :
( ( ord_less_real @ X @ Y6 )
| ( X = Y6 ) ) ) ) ).
% order_le_less
thf(fact_1152_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y6: nat] :
( ( ord_less_nat @ X @ Y6 )
| ( X = Y6 ) ) ) ) ).
% order_le_less
thf(fact_1153_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y6: int] :
( ( ord_less_int @ X @ Y6 )
| ( X = Y6 ) ) ) ) ).
% order_le_less
thf(fact_1154_dual__order_Ostrict__implies__order,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_eq_real @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1155_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1156_dual__order_Ostrict__implies__order,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1157_order_Ostrict__implies__order,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1158_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1159_order_Ostrict__implies__order,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1160_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A3: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1161_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1162_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A3: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1163_dual__order_Ostrict__trans2,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1164_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1165_dual__order_Ostrict__trans2,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1166_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1167_dual__order_Ostrict__trans1,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1168_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1169_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1170_real__of__nat__div3,axiom,
! [N: nat,X2: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_1171_ln__add__one__self__le__self2,axiom,
! [X2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% ln_add_one_self_le_self2
thf(fact_1172_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X: real,Y6: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y6 ) ) ) ) ).
% minus_real_def
thf(fact_1173_powr__ge__pzero,axiom,
! [X2: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X2 @ Y2 ) ) ).
% powr_ge_pzero
thf(fact_1174_powr__mono2,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y2 @ A2 ) ) ) ) ) ).
% powr_mono2
thf(fact_1175_powr__mono,axiom,
! [A2: real,B: real,X2: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X2 )
=> ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B ) ) ) ) ).
% powr_mono
thf(fact_1176_ln__one__minus__pos__upper__bound,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_1177_reals__Archimedean3,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ! [Y4: real] :
? [N3: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).
% reals_Archimedean3
thf(fact_1178_real__0__less__add__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y2 ) )
= ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y2 ) ) ).
% real_0_less_add_iff
thf(fact_1179_real__add__less__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
= ( ord_less_real @ Y2 @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_less_0_iff
thf(fact_1180_Bolzano,axiom,
! [A2: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [A4: real,B4: real,C4: real] :
( ( P @ A4 @ B4 )
=> ( ( P @ B4 @ C4 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C4 )
=> ( P @ A4 @ C4 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [D6: real] :
( ( ord_less_real @ zero_zero_real @ D6 )
& ! [A4: real,B4: real] :
( ( ( ord_less_eq_real @ A4 @ X3 )
& ( ord_less_eq_real @ X3 @ B4 )
& ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D6 ) )
=> ( P @ A4 @ B4 ) ) ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Bolzano
thf(fact_1181_divide__powr__uminus,axiom,
! [A2: real,B: real,C: real] :
( ( divide_divide_real @ A2 @ ( powr_real @ B @ C ) )
= ( times_times_real @ A2 @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% divide_powr_uminus
thf(fact_1182_ln__bound,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% ln_bound
thf(fact_1183_powr__mono2_H,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( powr_real @ Y2 @ A2 ) @ ( powr_real @ X2 @ A2 ) ) ) ) ) ).
% powr_mono2'
thf(fact_1184_powr__less__mono2,axiom,
! [A2: real,X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y2 @ A2 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_1185_ln__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq_real @ one_one_real @ X2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% ln_ge_zero
thf(fact_1186_ge__one__powr__ge__zero,axiom,
! [X2: real,A2: real] :
( ( ord_less_eq_real @ one_one_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( powr_real @ X2 @ A2 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_1187_powr__mono__both,axiom,
! [A2: real,B: real,X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y2 @ B ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_1188_powr__le1,axiom,
! [A2: real,X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ X2 @ one_one_real )
=> ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ one_one_real ) ) ) ) ).
% powr_le1
thf(fact_1189_powr__mult,axiom,
! [X2: real,Y2: real,A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( powr_real @ ( times_times_real @ X2 @ Y2 ) @ A2 )
= ( times_times_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y2 @ A2 ) ) ) ) ) ).
% powr_mult
thf(fact_1190_powr__divide,axiom,
! [X2: real,Y2: real,A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( powr_real @ ( divide_divide_real @ X2 @ Y2 ) @ A2 )
= ( divide_divide_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y2 @ A2 ) ) ) ) ) ).
% powr_divide
thf(fact_1191_isCont__Lb__Ub,axiom,
! [A2: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X3: real] :
( ( ( ord_less_eq_real @ A2 @ X3 )
& ( ord_less_eq_real @ X3 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
=> ? [L2: real,M4: real] :
( ! [X5: real] :
( ( ( ord_less_eq_real @ A2 @ X5 )
& ( ord_less_eq_real @ X5 @ B ) )
=> ( ( ord_less_eq_real @ L2 @ ( F @ X5 ) )
& ( ord_less_eq_real @ ( F @ X5 ) @ M4 ) ) )
& ! [Y4: real] :
( ( ( ord_less_eq_real @ L2 @ Y4 )
& ( ord_less_eq_real @ Y4 @ M4 ) )
=> ? [X3: real] :
( ( ord_less_eq_real @ A2 @ X3 )
& ( ord_less_eq_real @ X3 @ B )
& ( ( F @ X3 )
= Y4 ) ) ) ) ) ) ).
% isCont_Lb_Ub
thf(fact_1192_DERIV__mirror,axiom,
! [F: real > real,Y2: real,X2: real] :
( ( has_fi5821293074295781190e_real @ F @ Y2 @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X2 ) @ top_top_set_real ) )
= ( has_fi5821293074295781190e_real
@ ^ [X: real] : ( F @ ( uminus_uminus_real @ X ) )
@ ( uminus_uminus_real @ Y2 )
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% DERIV_mirror
thf(fact_1193_ln__ge__zero__imp__ge__one,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1194_ln__add__one__self__le__self,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% ln_add_one_self_le_self
thf(fact_1195_DERIV__nonneg__imp__nondecreasing,axiom,
! [A2: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [Y4: real] :
( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
& ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_1196_DERIV__nonpos__imp__nonincreasing,axiom,
! [A2: real,B: real,F: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [Y4: real] :
( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
& ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) )
=> ( ord_less_eq_real @ ( F @ B ) @ ( F @ A2 ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_1197_isCont__inverse__function2,axiom,
! [A2: real,X2: real,B: real,G: real > real,F: real > real] :
( ( ord_less_real @ A2 @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A2 @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( ( G @ ( F @ Z3 ) )
= Z3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq_real @ A2 @ Z3 )
=> ( ( ord_less_eq_real @ Z3 @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_1198_Preliminaries_Oinverse__powr,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( powr_real @ ( divide_divide_real @ one_one_real @ A2 ) @ B )
= ( powr_real @ A2 @ ( uminus_uminus_real @ B ) ) ) ) ).
% Preliminaries.inverse_powr
thf(fact_1199_powr__neg__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( powr_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
= ( divide_divide_real @ one_one_real @ X2 ) ) ) ).
% powr_neg_one
thf(fact_1200_isCont__artanh,axiom,
! [X2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
=> ( ( ord_less_real @ X2 @ one_one_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ artanh_real ) ) ) ).
% isCont_artanh
thf(fact_1201_ln__le__minus__one,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_1202_ln__diff__le,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X2 @ Y2 ) @ Y2 ) ) ) ) ).
% ln_diff_le
thf(fact_1203_powr__mult__base,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( times_times_real @ X2 @ ( powr_real @ X2 @ Y2 ) )
= ( powr_real @ X2 @ ( plus_plus_real @ one_one_real @ Y2 ) ) ) ) ).
% powr_mult_base
thf(fact_1204_DERIV__neg__imp__decreasing,axiom,
! [A2: real,B: real,F: real > real] :
( ( ord_less_real @ A2 @ B )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [Y4: real] :
( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
& ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
=> ( ord_less_real @ ( F @ B ) @ ( F @ A2 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_1205_DERIV__pos__imp__increasing,axiom,
! [A2: real,B: real,F: real > real] :
( ( ord_less_real @ A2 @ B )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ? [Y4: real] :
( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
& ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ ( F @ B ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_1206_powr__le__iff,axiom,
! [B: real,X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ ( powr_real @ B @ Y2 ) @ X2 )
= ( ord_less_eq_real @ Y2 @ ( log @ B @ X2 ) ) ) ) ) ).
% powr_le_iff
thf(fact_1207_le__powr__iff,axiom,
! [B: real,X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ X2 @ ( powr_real @ B @ Y2 ) )
= ( ord_less_eq_real @ ( log @ B @ X2 ) @ Y2 ) ) ) ) ).
% le_powr_iff
thf(fact_1208_Transcendental_Olog__le__iff,axiom,
! [B: real,X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ ( log @ B @ X2 ) @ Y2 )
= ( ord_less_eq_real @ X2 @ ( powr_real @ B @ Y2 ) ) ) ) ) ).
% Transcendental.log_le_iff
thf(fact_1209_le__log__iff,axiom,
! [B: real,X2: real,Y2: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ Y2 @ ( log @ B @ X2 ) )
= ( ord_less_eq_real @ ( powr_real @ B @ Y2 ) @ X2 ) ) ) ) ).
% le_log_iff
thf(fact_1210_ln__powr__bound2,axiom,
! [X2: real,A2: real] :
( ( ord_less_real @ one_one_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X2 ) @ A2 ) @ ( times_times_real @ ( powr_real @ A2 @ A2 ) @ X2 ) ) ) ) ).
% ln_powr_bound2
thf(fact_1211_ln__powr__bound,axiom,
! [X2: real,A2: real] :
( ( ord_less_eq_real @ one_one_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( divide_divide_real @ ( powr_real @ X2 @ A2 ) @ A2 ) ) ) ) ).
% ln_powr_bound
thf(fact_1212_MVT2,axiom,
! [A2: real,B: real,F: real > real,F2: real > real] :
( ( ord_less_real @ A2 @ B )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A2 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ( has_fi5821293074295781190e_real @ F @ ( F2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
=> ? [Z3: real] :
( ( ord_less_real @ A2 @ Z3 )
& ( ord_less_real @ Z3 @ B )
& ( ( minus_minus_real @ ( F @ B ) @ ( F @ A2 ) )
= ( times_times_real @ ( minus_minus_real @ B @ A2 ) @ ( F2 @ Z3 ) ) ) ) ) ) ).
% MVT2
thf(fact_1213_Youngs__inequality,axiom,
! [P5: real,Q3: real,A2: real,B: real] :
( ( ord_less_real @ one_one_real @ P5 )
=> ( ( ord_less_real @ one_one_real @ Q3 )
=> ( ( ( plus_plus_real @ ( divide_divide_real @ one_one_real @ P5 ) @ ( divide_divide_real @ one_one_real @ Q3 ) )
= one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A2 @ B ) @ ( plus_plus_real @ ( divide_divide_real @ ( powr_real @ A2 @ P5 ) @ P5 ) @ ( divide_divide_real @ ( powr_real @ B @ Q3 ) @ Q3 ) ) ) ) ) ) ) ) ).
% Youngs_inequality
thf(fact_1214_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1215_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1216_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1217_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1218_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1219_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1220_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1221_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1222_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1223_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1224_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1225_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1226_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1227_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1228_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1229_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1230_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1231_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1232_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1233_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1234_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1235_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1236_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1237_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1238_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1239_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1240_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1241_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1242_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1243_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1244_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1245_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1246_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_1247_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1248_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1249_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1250_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1251_zdiff__int__split,axiom,
! [P: int > $o,X2: nat,Y2: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y2 ) ) )
= ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
& ( ( ord_less_nat @ X2 @ Y2 )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1252_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1253_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1254_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1255_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1256_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1257_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1258_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1259_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1260_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1261_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1262_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1263_div__times__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1264_times__div__less__eq__dividend,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1265_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1266_zdiv__int,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_1267_div__le__mono,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1268_div__le__mono2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1269_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_1270_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1271_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1272_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1273_div__less__iff__less__mult,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y2: int] :
( ( if_int @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y2: int] :
( ( if_int @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y2: real] :
( ( if_real @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y2: real] :
( ( if_real @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
has_fi5821293074295781190e_real @ ( powr_real @ a ) @ ( times_times_real @ ( powr_real @ a @ x ) @ ( ln_ln_real @ a ) ) @ ( topolo2177554685111907308n_real @ x @ top_top_set_real ) ).
%------------------------------------------------------------------------------