TPTP Problem File: SLH0300^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Dedekind_Real/0000_Dedekind_Real/prob_01354_044256__5758392_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1343 ( 567 unt; 62 typ; 0 def)
% Number of atoms : 3579 (1138 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10419 ( 324 ~; 87 |; 139 &;8261 @)
% ( 0 <=>;1608 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 309 ( 309 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 55 usr; 13 con; 0-3 aty)
% Number of variables : 3549 ( 175 ^;3239 !; 135 ?;3549 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:30:26.414
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
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__Dedekind____Real__Opreal,type,
dedekind_preal: $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 (55)
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
archim2898591450579166408c_real: real > real ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Dedekind____Real__Opreal,type,
minus_7336623429200594941_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
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__Dedekind____Real__Opreal,type,
one_on9143529541772854033_preal: dedekind_preal ).
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__Dedekind____Real__Opreal,type,
plus_p3173629198307831117_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Dedekind____Real__Opreal,type,
times_3000655703912201937_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
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__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
ring_1_Ints_int: set_int ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
ring_1_Ints_real: set_real ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Dedekind____Real__Opreal,type,
ord_le5708704896291381698_preal: dedekind_preal > dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Dedekind____Real__Opreal,type,
ord_le5604041210740703414_preal: dedekind_preal > dedekind_preal > $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_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
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_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_u____,type,
u: dedekind_preal ).
thf(sy_v_v____,type,
v: dedekind_preal ).
% Relevant facts (1273)
thf(fact_0_less,axiom,
ord_le5708704896291381698_preal @ u @ v ).
% less
thf(fact_1_preal__add__less__cancel__left,axiom,
! [T: dedekind_preal,R: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) )
= ( ord_le5708704896291381698_preal @ R @ S ) ) ).
% preal_add_less_cancel_left
thf(fact_2_preal__add__less__cancel__right,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) )
= ( ord_le5708704896291381698_preal @ R @ S ) ) ).
% preal_add_less_cancel_right
thf(fact_3_add__eq__exists,axiom,
! [A: int,B: int] :
? [X: int] :
( ( plus_plus_int @ A @ X )
= B ) ).
% add_eq_exists
thf(fact_4_add__eq__exists,axiom,
! [A: real,B: real] :
? [X: real] :
( ( plus_plus_real @ A @ X )
= B ) ).
% add_eq_exists
thf(fact_5_less__add__left,axiom,
! [R: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R @ S )
=> ( ( plus_p3173629198307831117_preal @ R @ ( minus_7336623429200594941_preal @ S @ R ) )
= S ) ) ).
% less_add_left
thf(fact_6_preal__add__assoc,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ X2 @ Y ) @ Z )
= ( plus_p3173629198307831117_preal @ X2 @ ( plus_p3173629198307831117_preal @ Y @ Z ) ) ) ).
% preal_add_assoc
thf(fact_7_preal__add__commute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [X3: dedekind_preal,Y2: dedekind_preal] : ( plus_p3173629198307831117_preal @ Y2 @ X3 ) ) ) ).
% preal_add_commute
thf(fact_8_preal__trans__lemma,axiom,
! [X2: dedekind_preal,Y1: dedekind_preal,X1: dedekind_preal,Y: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X2 @ Y1 )
= ( plus_p3173629198307831117_preal @ X1 @ Y ) )
=> ( ( ( plus_p3173629198307831117_preal @ X2 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y ) )
=> ( ( plus_p3173629198307831117_preal @ X1 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ).
% preal_trans_lemma
thf(fact_9_preal__add__left__cancel,axiom,
! [C: dedekind_preal,A: dedekind_preal,B: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ C @ A )
= ( plus_p3173629198307831117_preal @ C @ B ) )
=> ( A = B ) ) ).
% preal_add_left_cancel
thf(fact_10_preal__add__less2__mono1,axiom,
! [R: dedekind_preal,S: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R @ S )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) ) ) ).
% preal_add_less2_mono1
thf(fact_11_preal__add__less2__mono2,axiom,
! [R: dedekind_preal,S: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R @ S )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) ) ) ).
% preal_add_less2_mono2
thf(fact_12_preal__add__right__cancel,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ R @ T )
= ( plus_p3173629198307831117_preal @ S @ T ) )
=> ( R = S ) ) ).
% preal_add_right_cancel
thf(fact_13_preal__self__less__add__left,axiom,
! [R: dedekind_preal,S: dedekind_preal] : ( ord_le5708704896291381698_preal @ R @ ( plus_p3173629198307831117_preal @ R @ S ) ) ).
% preal_self_less_add_left
thf(fact_14_preal__add__left__less__cancel,axiom,
! [T: dedekind_preal,R: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) )
=> ( ord_le5708704896291381698_preal @ R @ S ) ) ).
% preal_add_left_less_cancel
thf(fact_15_preal__add__right__less__cancel,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) )
=> ( ord_le5708704896291381698_preal @ R @ S ) ) ).
% preal_add_right_less_cancel
thf(fact_16_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_17_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_18_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_19_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_20_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_21_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_22_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_23_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_24_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_25_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_26_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_27_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_28_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_29_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_30_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_31_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_32_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_33_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_34_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_35_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_36_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_37_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_38_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_39_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_40_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_41_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_42_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_43_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_44_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_45_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_46_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_47_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_48_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_49_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_50_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_51_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_52_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_53_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_54_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_55_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
=> ( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_56_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_57_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_58_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_59_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_60_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_61_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_62_add_Oleft__commute,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ B @ ( plus_p3173629198307831117_preal @ A @ C ) )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% add.left_commute
thf(fact_63_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_64_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_65_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_66_add_Ocommute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] : ( plus_p3173629198307831117_preal @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_67_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_68_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_69_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_70_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_71_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_72_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_73_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_74_add_Oassoc,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% add.assoc
thf(fact_75_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_76_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_77_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_78_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_79_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_80_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_81_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_82_group__cancel_Oadd1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_83_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_84_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_p3173629198307831117_preal @ I @ K )
= ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_85_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_86_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_87_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_88_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_89_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_90_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_91_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_92_one__reorient,axiom,
! [X2: dedekind_preal] :
( ( one_on9143529541772854033_preal = X2 )
= ( X2 = one_on9143529541772854033_preal ) ) ).
% one_reorient
thf(fact_93_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_94_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_95_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_96_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_97_diff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_98_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_99_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_100_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_101_Collect__mem__eq,axiom,
! [A3: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_102_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_103_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_104_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_105_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_106_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_107_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_108_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_109_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_110_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_111_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_112_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_113_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_114_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_115_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_116_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_117_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_118_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_119_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_120_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_121_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_122_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_123_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_124_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_125_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_126_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_127_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_128_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_129_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_130_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_131_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_132_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_133_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_134_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_135_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_136_diff__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_137_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_138_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_139_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_140_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_141_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_142_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_143_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_144_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_145_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_146_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_147_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_148_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_149_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_150_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_151_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_152_group__cancel_Osub1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_153_group__cancel_Osub1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_154_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_155_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_156_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_157_add__diff__add,axiom,
! [A: int,C: int,B: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% add_diff_add
thf(fact_158_add__diff__add,axiom,
! [A: real,C: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_159_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_160_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_161_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_162_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_163_dbl__dec__def,axiom,
( neg_nu6075765906172075777c_real
= ( ^ [X3: real] : ( minus_minus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% dbl_dec_def
thf(fact_164_frac__add,axiom,
! [X2: real,Y: real] :
( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X2 @ Y ) )
= ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
& ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X2 @ Y ) )
= ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).
% frac_add
thf(fact_165_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_166_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_167_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_168_diff__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_169_diff__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_170_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_171_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_172_order__less__imp__not__less,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ~ ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_173_order__less__imp__not__less,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_174_order__less__imp__not__less,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_175_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_176_order__less__imp__not__eq2,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_177_order__less__imp__not__eq2,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_178_order__less__imp__not__eq2,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_179_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_180_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_181_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_182_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_183_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_184_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_185_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_186_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_187_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_188_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_189_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_190_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_191_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_192_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_193_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_194_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_195_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_196_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_197_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_198_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_199_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_200_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_201_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_202_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_203_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_204_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_205_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_206_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_207_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_208_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_209_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_210_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_211_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_212_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_213_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_214_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_215_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_216_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_217_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_218_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_219_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_220_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_221_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_222_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_223_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_224_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_225_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_226_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_227_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_228_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_229_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_230_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_231_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_232_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_233_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_234_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_235_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_236_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_237_frac__1,axiom,
( ( archim2898591450579166408c_real @ one_one_real )
= zero_zero_real ) ).
% frac_1
thf(fact_238_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_239_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_240_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_241_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_242_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_243_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_244_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_245_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_246_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_247_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_248_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_249_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_250_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_251_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_252_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_253_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_254_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_255_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_256_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_257_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_258_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_259_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_260_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( minus_minus_int @ A2 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_261_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A2: real,B2: real] :
( ( minus_minus_real @ A2 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_262_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_263_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_264_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_265_frac__lt__1,axiom,
! [X2: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X2 ) @ one_one_real ) ).
% frac_lt_1
thf(fact_266_frac__1__eq,axiom,
! [X2: real] :
( ( archim2898591450579166408c_real @ ( plus_plus_real @ X2 @ one_one_real ) )
= ( archim2898591450579166408c_real @ X2 ) ) ).
% frac_1_eq
thf(fact_267_add__less__zeroD,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X2 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X2 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_268_add__less__zeroD,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X2 @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_269_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_270_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_271_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_272_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_273_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_274_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_275_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_276_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_277_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_278_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_279_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_280_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_281_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_282_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_283_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_284_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_285_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_286_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_287_lt__ex,axiom,
! [X2: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).
% lt_ex
thf(fact_288_lt__ex,axiom,
! [X2: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% lt_ex
thf(fact_289_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_290_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_291_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_292_dense,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ? [Z3: real] :
( ( ord_less_real @ X2 @ Z3 )
& ( ord_less_real @ Z3 @ Y ) ) ) ).
% dense
thf(fact_293_less__imp__neq,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_294_less__imp__neq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_295_less__imp__neq,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_296_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_297_order_Oasym,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A ) ) ).
% order.asym
thf(fact_298_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_299_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_300_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_301_ord__eq__less__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A = B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_302_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_303_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_304_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_305_ord__less__eq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( B = C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_306_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_307_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_308_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_309_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_310_antisym__conv3,axiom,
! [Y: dedekind_preal,X2: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ Y @ X2 )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_311_antisym__conv3,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_int @ Y @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_312_antisym__conv3,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_real @ Y @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_313_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_314_linorder__cases,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_315_linorder__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_316_linorder__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_317_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_318_dual__order_Oasym,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ~ ( ord_le5708704896291381698_preal @ A @ B ) ) ).
% dual_order.asym
thf(fact_319_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_320_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_321_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_322_dual__order_Oirrefl,axiom,
! [A: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A @ A ) ).
% dual_order.irrefl
thf(fact_323_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_324_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_325_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_326_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_327_linorder__less__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A: dedekind_preal,B: dedekind_preal] :
( ! [A4: dedekind_preal,B4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: dedekind_preal] : ( P @ A4 @ A4 )
=> ( ! [A4: dedekind_preal,B4: dedekind_preal] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_328_linorder__less__wlog,axiom,
! [P: int > int > $o,A: 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 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_329_linorder__less__wlog,axiom,
! [P: real > real > $o,A: 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 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_330_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: 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 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_331_order_Ostrict__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_332_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_333_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_334_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_335_not__less__iff__gr__or__eq,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X2 @ Y ) )
= ( ( ord_le5708704896291381698_preal @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_336_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ( ord_less_int @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_337_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( ( ord_less_real @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_338_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_339_dual__order_Ostrict__trans,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ( ord_le5708704896291381698_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_340_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_341_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_342_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_343_order_Ostrict__implies__not__eq,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_344_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_345_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_346_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_347_dual__order_Ostrict__implies__not__eq,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_348_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_349_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_350_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_351_linorder__neqE,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( X2 != Y )
=> ( ~ ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_352_linorder__neqE,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_353_linorder__neqE,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
=> ( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_354_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_355_order__less__asym,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ~ ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_356_order__less__asym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_357_order__less__asym,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_358_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_359_linorder__neq__iff,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( X2 != Y )
= ( ( ord_le5708704896291381698_preal @ X2 @ Y )
| ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_360_linorder__neq__iff,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
= ( ( ord_less_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_361_linorder__neq__iff,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
= ( ( ord_less_real @ X2 @ Y )
| ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_362_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_363_order__less__asym_H,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A ) ) ).
% order_less_asym'
thf(fact_364_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_365_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_366_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_367_order__less__trans,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ( ord_le5708704896291381698_preal @ Y @ Z )
=> ( ord_le5708704896291381698_preal @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_368_order__less__trans,axiom,
! [X2: int,Y: int,Z: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_369_order__less__trans,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_370_order__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_371_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_372_ord__eq__less__subst,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_373_ord__eq__less__subst,axiom,
! [A: real,F: dedekind_preal > real,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_374_ord__eq__less__subst,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_375_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_376_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_377_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_378_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_379_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: real > dedekind_preal,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_380_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_381_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_382_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_383_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > real,C: real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_384_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_385_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > dedekind_preal,C: dedekind_preal] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_386_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_387_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_388_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_389_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > dedekind_preal,C: dedekind_preal] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_390_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_391_order__less__irrefl,axiom,
! [X2: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_392_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_393_order__less__irrefl,axiom,
! [X2: real] :
~ ( ord_less_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_394_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_395_order__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_396_order__less__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_397_order__less__subst1,axiom,
! [A: dedekind_preal,F: real > dedekind_preal,B: real,C: real] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_398_order__less__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_399_order__less__subst1,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_400_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_401_order__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_402_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_403_order__less__subst1,axiom,
! [A: real,F: dedekind_preal > real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_404_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_405_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_406_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_407_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > real,C: real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_408_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_409_order__less__subst2,axiom,
! [A: int,B: int,F: int > dedekind_preal,C: dedekind_preal] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_410_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_411_order__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_412_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_413_order__less__subst2,axiom,
! [A: real,B: real,F: real > dedekind_preal,C: dedekind_preal] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_414_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_415_order__less__not__sym,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ~ ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_416_order__less__not__sym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_417_order__less__not__sym,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ~ ( ord_less_real @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_418_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_419_order__less__imp__triv,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,P: $o] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ( ord_le5708704896291381698_preal @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_420_order__less__imp__triv,axiom,
! [X2: int,Y: int,P: $o] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_421_order__less__imp__triv,axiom,
! [X2: real,Y: real,P: $o] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_422_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_423_linorder__less__linear,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
| ( X2 = Y )
| ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_424_linorder__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_425_linorder__less__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_real @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_426_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_427_order__less__imp__not__eq,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_428_order__less__imp__not__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_429_order__less__imp__not__eq,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_430_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_431_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_432_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_433_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_434_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_435_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_436_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_437_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_438_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_439_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_440_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_441_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_442_frac__eq,axiom,
! [X2: real] :
( ( ( archim2898591450579166408c_real @ X2 )
= X2 )
= ( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% frac_eq
thf(fact_443_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_444_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_445_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_446_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_447_frac__gt__0__iff,axiom,
! [X2: real] :
( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X2 ) )
= ( ~ ( member_real @ X2 @ ring_1_Ints_real ) ) ) ).
% frac_gt_0_iff
thf(fact_448_floor__add,axiom,
! [X2: real,Y: real] :
( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ Y ) )
= ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
& ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
=> ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ Y ) )
= ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).
% floor_add
thf(fact_449_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_450_dual__order_Orefl,axiom,
! [A: dedekind_preal] : ( ord_le5604041210740703414_preal @ A @ A ) ).
% dual_order.refl
thf(fact_451_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_452_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_453_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_454_order__refl,axiom,
! [X2: dedekind_preal] : ( ord_le5604041210740703414_preal @ X2 @ X2 ) ).
% order_refl
thf(fact_455_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_456_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_457_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_458_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_459_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_460_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_461_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_462_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_463_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_464_preal__add__le__cancel__right,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) )
= ( ord_le5604041210740703414_preal @ R @ S ) ) ).
% preal_add_le_cancel_right
thf(fact_465_preal__add__le__cancel__left,axiom,
! [T: dedekind_preal,R: dedekind_preal,S: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) )
= ( ord_le5604041210740703414_preal @ R @ S ) ) ).
% preal_add_le_cancel_left
thf(fact_466_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_467_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_468_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_469_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_470_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_471_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_472_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_473_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_474_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_475_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_476_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_477_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_478_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_479_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_480_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_481_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_482_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_483_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_484_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_485_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_486_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_487_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_488_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_489_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_490_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_491_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_492_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_493_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_494_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_495_floor__zero,axiom,
( ( archim6058952711729229775r_real @ zero_zero_real )
= zero_zero_int ) ).
% floor_zero
thf(fact_496_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_497_floor__frac,axiom,
! [R: real] :
( ( archim6058952711729229775r_real @ ( archim2898591450579166408c_real @ R ) )
= zero_zero_int ) ).
% floor_frac
thf(fact_498_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_499_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_500_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_501_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_502_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_503_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_504_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_505_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_506_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_507_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_508_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_509_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_510_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_511_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_512_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_513_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_514_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_515_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_516_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_517_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_518_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_519_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_520_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_521_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_522_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_523_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_524_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_525_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_526_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_527_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_528_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_529_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_530_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_531_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_532_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_533_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_534_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_535_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_536_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_537_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_538_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_539_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_540_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_541_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_542_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_543_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_544_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_545_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_546_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_547_uminus__add__conv__diff,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
= ( minus_minus_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_548_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_549_diff__minus__eq__add,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_550_zero__le__floor,axiom,
! [X2: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X2 ) )
= ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% zero_le_floor
thf(fact_551_floor__less__zero,axiom,
! [X2: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ zero_zero_int )
= ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% floor_less_zero
thf(fact_552_zero__less__floor,axiom,
! [X2: real] :
( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X2 ) )
= ( ord_less_eq_real @ one_one_real @ X2 ) ) ).
% zero_less_floor
thf(fact_553_floor__le__zero,axiom,
! [X2: real] :
( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ zero_zero_int )
= ( ord_less_real @ X2 @ one_one_real ) ) ).
% floor_le_zero
thf(fact_554_floor__one,axiom,
( ( archim6058952711729229775r_real @ one_one_real )
= one_one_int ) ).
% floor_one
thf(fact_555_frac__eq__0__iff,axiom,
! [X2: real] :
( ( ( archim2898591450579166408c_real @ X2 )
= zero_zero_real )
= ( member_real @ X2 @ ring_1_Ints_real ) ) ).
% frac_eq_0_iff
thf(fact_556_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_557_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_558_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_559_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_560_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_561_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_562_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_563_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_564_one__le__floor,axiom,
! [X2: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X2 ) )
= ( ord_less_eq_real @ one_one_real @ X2 ) ) ).
% one_le_floor
thf(fact_565_floor__less__one,axiom,
! [X2: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int )
= ( ord_less_real @ X2 @ one_one_real ) ) ).
% floor_less_one
thf(fact_566_floor__diff__one,axiom,
! [X2: real] :
( ( archim6058952711729229775r_real @ ( minus_minus_real @ X2 @ one_one_real ) )
= ( minus_minus_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int ) ) ).
% floor_diff_one
thf(fact_567_floor__add2,axiom,
! [X2: real,Y: real] :
( ( ( member_real @ X2 @ ring_1_Ints_real )
| ( member_real @ Y @ ring_1_Ints_real ) )
=> ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ Y ) )
= ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y ) ) ) ) ).
% floor_add2
thf(fact_568_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_569_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_570_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_571_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_572_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_573_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_574_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_575_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_576_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_577_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_578_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_579_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_580_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_581_le__floor__add,axiom,
! [X2: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ Y ) ) ) ).
% le_floor_add
thf(fact_582_order__antisym__conv,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_583_order__antisym__conv,axiom,
! [Y: dedekind_preal,X2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y @ X2 )
=> ( ( ord_le5604041210740703414_preal @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_584_order__antisym__conv,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_585_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_586_floor__mono,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).
% floor_mono
thf(fact_587_linorder__le__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_588_linorder__le__cases,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ( ord_le5604041210740703414_preal @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_589_linorder__le__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_590_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_591_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_592_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_593_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_594_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_595_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_596_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_597_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > real,C: real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_598_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_599_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_600_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_601_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_602_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_603_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_604_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_605_ord__eq__le__subst,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_606_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_607_ord__eq__le__subst,axiom,
! [A: real,F: dedekind_preal > real,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_608_ord__eq__le__subst,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_609_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_610_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: real > dedekind_preal,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_611_linorder__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_612_linorder__linear,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
| ( ord_le5604041210740703414_preal @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_613_linorder__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_614_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_615_order__eq__refl,axiom,
! [X2: int,Y: int] :
( ( X2 = Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_616_order__eq__refl,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( X2 = Y )
=> ( ord_le5604041210740703414_preal @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_617_order__eq__refl,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_618_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_619_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_620_order__subst2,axiom,
! [A: int,B: int,F: int > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_621_order__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_622_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_623_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_624_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_625_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > real,C: real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_626_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_627_order__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_628_order__subst2,axiom,
! [A: real,B: real,F: real > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_629_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_630_order__subst1,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_631_order__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_632_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_633_order__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_634_order__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_635_order__subst1,axiom,
! [A: dedekind_preal,F: real > dedekind_preal,B: real,C: real] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_636_order__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_637_order__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_638_order__subst1,axiom,
! [A: real,F: dedekind_preal > real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_639_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_640_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: dedekind_preal,Z2: dedekind_preal] : ( Y3 = Z2 ) )
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B2 )
& ( ord_le5604041210740703414_preal @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_641_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_642_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_643_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_644_antisym,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_645_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_646_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_647_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_648_dual__order_Otrans,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ C @ B )
=> ( ord_le5604041210740703414_preal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_649_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_650_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_651_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_652_dual__order_Oantisym,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_653_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_654_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_655_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_656_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: dedekind_preal,Z2: dedekind_preal] : ( Y3 = Z2 ) )
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A2 )
& ( ord_le5604041210740703414_preal @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_657_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_658_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_659_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_660_linorder__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A: dedekind_preal,B: dedekind_preal] :
( ! [A4: dedekind_preal,B4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: dedekind_preal,B4: dedekind_preal] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_661_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_662_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_663_order__trans,axiom,
! [X2: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_664_order__trans,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ( ( ord_le5604041210740703414_preal @ Y @ Z )
=> ( ord_le5604041210740703414_preal @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_665_order__trans,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_666_order__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_667_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_668_order_Otrans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% order.trans
thf(fact_669_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_670_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_671_order__antisym,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_672_order__antisym,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ( ( ord_le5604041210740703414_preal @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_673_order__antisym,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_674_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_675_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_676_ord__le__eq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( B = C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_677_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_678_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_679_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_680_ord__eq__le__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A = B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_681_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_682_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_683_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_684_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: dedekind_preal,Z2: dedekind_preal] : ( Y3 = Z2 ) )
= ( ^ [X3: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y2 )
& ( ord_le5604041210740703414_preal @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_685_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
& ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_686_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_687_le__cases3,axiom,
! [X2: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_688_le__cases3,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ~ ( ord_le5604041210740703414_preal @ Y @ Z ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y @ X2 )
=> ~ ( ord_le5604041210740703414_preal @ X2 @ Z ) )
=> ( ( ( ord_le5604041210740703414_preal @ X2 @ Z )
=> ~ ( ord_le5604041210740703414_preal @ Z @ Y ) )
=> ( ( ( ord_le5604041210740703414_preal @ Z @ Y )
=> ~ ( ord_le5604041210740703414_preal @ Y @ X2 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y @ Z )
=> ~ ( ord_le5604041210740703414_preal @ Z @ X2 ) )
=> ~ ( ( ord_le5604041210740703414_preal @ Z @ X2 )
=> ~ ( ord_le5604041210740703414_preal @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_689_le__cases3,axiom,
! [X2: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_690_le__cases3,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_691_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_692_nle__le,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ A @ B ) )
= ( ( ord_le5604041210740703414_preal @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_693_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_694_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_695_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_696_verit__comp__simplify1_I2_J,axiom,
! [A: dedekind_preal] : ( ord_le5604041210740703414_preal @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_697_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_698_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_699_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_700_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_701_Ints__minus,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( member_int @ ( uminus_uminus_int @ A ) @ ring_1_Ints_int ) ) ).
% Ints_minus
thf(fact_702_Ints__minus,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( member_real @ ( uminus_uminus_real @ A ) @ ring_1_Ints_real ) ) ).
% Ints_minus
thf(fact_703_minus__in__Ints__iff,axiom,
! [X2: int] :
( ( member_int @ ( uminus_uminus_int @ X2 ) @ ring_1_Ints_int )
= ( member_int @ X2 @ ring_1_Ints_int ) ) ).
% minus_in_Ints_iff
thf(fact_704_minus__in__Ints__iff,axiom,
! [X2: real] :
( ( member_real @ ( uminus_uminus_real @ X2 ) @ ring_1_Ints_real )
= ( member_real @ X2 @ ring_1_Ints_real ) ) ).
% minus_in_Ints_iff
thf(fact_705_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_706_verit__la__disequality,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( A = B )
| ~ ( ord_le5604041210740703414_preal @ A @ B )
| ~ ( ord_le5604041210740703414_preal @ B @ A ) ) ).
% verit_la_disequality
thf(fact_707_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_708_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_709_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_710_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_711_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_712_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_713_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_714_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_715_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_716_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_717_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_718_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_719_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_720_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_721_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_722_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_723_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_724_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_725_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_726_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_727_verit__comp__simplify1_I3_J,axiom,
! [B5: dedekind_preal,A5: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ B5 @ A5 ) )
= ( ord_le5708704896291381698_preal @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_728_verit__comp__simplify1_I3_J,axiom,
! [B5: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
= ( ord_less_real @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_729_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_730_Ints__0,axiom,
member_int @ zero_zero_int @ ring_1_Ints_int ).
% Ints_0
thf(fact_731_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
% Ints_0
thf(fact_732_Ints__add,axiom,
! [A: int,B: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( member_int @ B @ ring_1_Ints_int )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% Ints_add
thf(fact_733_Ints__add,axiom,
! [A: real,B: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_add
thf(fact_734_Ints__1,axiom,
member_int @ one_one_int @ ring_1_Ints_int ).
% Ints_1
thf(fact_735_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_736_Ints__diff,axiom,
! [A: int,B: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( member_int @ B @ ring_1_Ints_int )
=> ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% Ints_diff
thf(fact_737_Ints__diff,axiom,
! [A: real,B: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_diff
thf(fact_738_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_739_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_740_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_741_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_742_floor__less__cancel,axiom,
! [X2: real,Y: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y ) )
=> ( ord_less_real @ X2 @ Y ) ) ).
% floor_less_cancel
thf(fact_743_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_744_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_745_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_746_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_747_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_748_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_749_group__cancel_Oneg1,axiom,
! [A3: int,K: int,A: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_750_group__cancel_Oneg1,axiom,
! [A3: real,K: real,A: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( uminus_uminus_real @ A3 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_751_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_752_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_753_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_754_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_755_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_756_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_757_minus__diff__minus,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_758_minus__diff__minus,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_759_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_760_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_761_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_762_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_763_order__le__imp__less__or__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_764_order__le__imp__less__or__eq,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ( ( ord_le5708704896291381698_preal @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_765_order__le__imp__less__or__eq,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_766_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_767_linorder__le__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_768_linorder__le__less__linear,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
| ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_769_linorder__le__less__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_real @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_770_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_771_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_772_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_773_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_774_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_775_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_776_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > dedekind_preal,C: dedekind_preal] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_777_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > dedekind_preal,C: dedekind_preal] :
( ( ord_less_real @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_778_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_779_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > real,C: real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_780_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_781_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_782_order__less__le__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_783_order__less__le__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_784_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_785_order__less__le__subst1,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_786_order__less__le__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_787_order__less__le__subst1,axiom,
! [A: real,F: dedekind_preal > real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_788_order__less__le__subst1,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_789_order__less__le__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_790_order__less__le__subst1,axiom,
! [A: dedekind_preal,F: real > dedekind_preal,B: real,C: real] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_791_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_792_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_793_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_794_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_795_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_796_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_797_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > real,C: real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_798_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_799_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_800_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_801_order__le__less__subst1,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_802_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_803_order__le__less__subst1,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_804_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_805_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_806_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_807_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: real > dedekind_preal,B: real,C: real] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_808_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_809_order__le__less__subst1,axiom,
! [A: real,F: dedekind_preal > real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_810_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_811_order__less__le__trans,axiom,
! [X2: int,Y: int,Z: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_812_order__less__le__trans,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ( ord_le5604041210740703414_preal @ Y @ Z )
=> ( ord_le5708704896291381698_preal @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_813_order__less__le__trans,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_814_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_815_order__le__less__trans,axiom,
! [X2: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_816_order__le__less__trans,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ( ( ord_le5708704896291381698_preal @ Y @ Z )
=> ( ord_le5708704896291381698_preal @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_817_order__le__less__trans,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_818_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_819_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_820_order__neq__le__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( A != B )
=> ( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5708704896291381698_preal @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_821_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_822_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_823_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_824_order__le__neq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( A != B )
=> ( ord_le5708704896291381698_preal @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_825_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_826_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_827_order__less__imp__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_828_order__less__imp__le,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ord_le5604041210740703414_preal @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_829_order__less__imp__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_830_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_831_linorder__not__less,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_832_linorder__not__less,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X2 @ Y ) )
= ( ord_le5604041210740703414_preal @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_833_linorder__not__less,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_834_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_835_linorder__not__le,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y ) )
= ( ord_less_int @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_836_linorder__not__le,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ X2 @ Y ) )
= ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_837_linorder__not__le,axiom,
! [X2: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X2 @ Y ) )
= ( ord_less_real @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_838_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_839_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_840_order__less__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X3: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_841_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_842_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_843_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_844_order__le__less,axiom,
( ord_le5604041210740703414_preal
= ( ^ [X3: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_845_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_846_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_847_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_848_dual__order_Ostrict__implies__order,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ord_le5604041210740703414_preal @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_849_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_850_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_851_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_852_order_Ostrict__implies__order,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_853_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_854_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_855_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_856_dual__order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B2: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A2 )
& ~ ( ord_le5604041210740703414_preal @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_857_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ~ ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_858_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_859_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_860_dual__order_Ostrict__trans2,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_861_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_862_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_863_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_864_dual__order_Ostrict__trans1,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5708704896291381698_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_865_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_866_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_867_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_868_dual__order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B2: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_869_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_870_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_871_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_872_dual__order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [B2: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_873_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_874_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_875_dense__le__bounded,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ! [W2: real] :
( ( ord_less_real @ X2 @ W2 )
=> ( ( ord_less_real @ W2 @ Y )
=> ( ord_less_eq_real @ W2 @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_876_dense__ge__bounded,axiom,
! [Z: real,X2: real,Y: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ! [W2: real] :
( ( ord_less_real @ Z @ W2 )
=> ( ( ord_less_real @ W2 @ X2 )
=> ( ord_less_eq_real @ Y @ W2 ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_877_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_878_order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B2 )
& ~ ( ord_le5604041210740703414_preal @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_879_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ~ ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_880_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_881_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_882_order_Ostrict__trans2,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_883_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_884_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_885_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_886_order_Ostrict__trans1,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_887_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_888_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_889_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_890_order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_891_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_892_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_893_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_894_order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_895_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_896_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_897_not__le__imp__less,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_eq_int @ Y @ X2 )
=> ( ord_less_int @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_898_not__le__imp__less,axiom,
! [Y: dedekind_preal,X2: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ Y @ X2 )
=> ( ord_le5708704896291381698_preal @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_899_not__le__imp__less,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_eq_real @ Y @ X2 )
=> ( ord_less_real @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_900_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_901_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ~ ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_902_less__le__not__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X3: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y2 )
& ~ ( ord_le5604041210740703414_preal @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_903_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
& ~ ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_904_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_905_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_906_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X: real] :
( ( ord_less_real @ Z @ X )
=> ( ord_less_eq_real @ Y @ X ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_907_antisym__conv2,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_908_antisym__conv2,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_909_antisym__conv2,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ~ ( ord_less_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_910_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_911_antisym__conv1,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_912_antisym__conv1,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ( ord_le5604041210740703414_preal @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_913_antisym__conv1,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_914_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_915_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_916_nless__le,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ A @ B ) )
= ( ~ ( ord_le5604041210740703414_preal @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_917_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_918_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_919_leI,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% leI
thf(fact_920_leI,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ord_le5604041210740703414_preal @ Y @ X2 ) ) ).
% leI
thf(fact_921_leI,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% leI
thf(fact_922_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_923_leD,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_int @ X2 @ Y ) ) ).
% leD
thf(fact_924_leD,axiom,
! [Y: dedekind_preal,X2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y @ X2 )
=> ~ ( ord_le5708704896291381698_preal @ X2 @ Y ) ) ).
% leD
thf(fact_925_leD,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_real @ X2 @ Y ) ) ).
% leD
thf(fact_926_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_927_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_928_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_929_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_930_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_931_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_932_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_933_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_934_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_935_add__right__mono,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ A @ C ) @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% add_right_mono
thf(fact_936_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_937_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_938_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_939_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_940_add__left__mono,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ C @ A ) @ ( plus_p3173629198307831117_preal @ C @ B ) ) ) ).
% add_left_mono
thf(fact_941_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_942_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_943_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_944_add__mono,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal,D: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ C @ D )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ A @ C ) @ ( plus_p3173629198307831117_preal @ B @ D ) ) ) ) ).
% add_mono
thf(fact_945_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_946_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_947_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_948_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ I @ J )
& ( ord_le5604041210740703414_preal @ K @ L ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ I @ K ) @ ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_949_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_950_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_951_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_952_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( I = J )
& ( ord_le5604041210740703414_preal @ K @ L ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ I @ K ) @ ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_953_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_954_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_955_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_956_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ I @ J )
& ( K = L ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ I @ K ) @ ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_957_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_958_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_959_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_960_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_961_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_962_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_963_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_964_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_965_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_966_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_967_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_968_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_969_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_970_preal__eq__le__imp__le,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal,D: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ A @ B )
= ( plus_p3173629198307831117_preal @ C @ D ) )
=> ( ( ord_le5604041210740703414_preal @ C @ A )
=> ( ord_le5604041210740703414_preal @ B @ D ) ) ) ).
% preal_eq_le_imp_le
thf(fact_971_real__le__lemma,axiom,
! [U1: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,V1: dedekind_preal,X1: dedekind_preal,Y1: dedekind_preal,X22: dedekind_preal,Y22: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U1 @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V1 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X1 @ V1 )
= ( plus_p3173629198307831117_preal @ U1 @ Y1 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X22 @ V2 )
= ( plus_p3173629198307831117_preal @ U2 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X1 @ Y22 ) @ ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ) ).
% real_le_lemma
thf(fact_972_Ints__double__eq__0__iff,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_973_Ints__double__eq__0__iff,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_974_one__add__floor,axiom,
! [X2: real] :
( ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int )
= ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ one_one_real ) ) ) ).
% one_add_floor
thf(fact_975_frac__neg,axiom,
! [X2: real] :
( ( ( member_real @ X2 @ ring_1_Ints_real )
=> ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X2 ) )
= zero_zero_real ) )
& ( ~ ( member_real @ X2 @ ring_1_Ints_real )
=> ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X2 ) )
= ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X2 ) ) ) ) ) ).
% frac_neg
thf(fact_976_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_977_add__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_978_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_979_ab__group__add__class_Oab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_980_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_981_add_Oinverse__unique,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_982_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_983_eq__neg__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_984_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_985_neg__eq__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_986_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_987_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_988_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_989_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_990_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_991_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_992_Ints__odd__nonzero,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
!= zero_zero_int ) ) ).
% Ints_odd_nonzero
thf(fact_993_Ints__odd__nonzero,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
!= zero_zero_real ) ) ).
% Ints_odd_nonzero
thf(fact_994_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_995_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A2: real,B2: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_996_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_997_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A2: real,B2: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_998_group__cancel_Osub2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_999_group__cancel_Osub2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( minus_minus_real @ A @ B3 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_1000_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1001_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1002_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1003_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1004_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1005_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1006_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1007_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1008_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1009_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1010_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1011_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1012_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1013_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1014_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1015_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1016_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_1017_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1018_add__nonneg__eq__0__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1019_add__nonneg__eq__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X2 @ Y )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1020_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1021_add__nonpos__eq__0__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X2 @ Y )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1022_add__nonpos__eq__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X2 @ Y )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1023_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1024_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1025_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_1026_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1027_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1028_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1029_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1030_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1031_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1032_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1033_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1034_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1035_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1036_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1037_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1038_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1039_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1040_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1041_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1042_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_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(4)
thf(fact_1043_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_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(4)
thf(fact_1044_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_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(4)
thf(fact_1045_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1046_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_1047_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1048_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1049_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1050_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1051_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1052_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1053_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1054_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1055_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1056_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1057_add__le__add__imp__diff__le,axiom,
! [I: real,K: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1058_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1059_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1060_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1061_add__le__imp__le__diff,axiom,
! [I: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1062_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1063_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1064_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1065_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1066_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1067_real__trans__lemma,axiom,
! [X2: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V3: dedekind_preal,U3: dedekind_preal,X22: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,Y22: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_p3173629198307831117_preal @ X22 @ V2 )
= ( plus_p3173629198307831117_preal @ U2 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_1068_real__trans__lemma,axiom,
! [X2: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V3: dedekind_preal,U3: dedekind_preal,X22: int,V2: int,U2: int,Y22: int] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_plus_int @ X22 @ V2 )
= ( plus_plus_int @ U2 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_1069_real__trans__lemma,axiom,
! [X2: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V3: dedekind_preal,U3: dedekind_preal,X22: real,V2: real,U2: real,Y22: real] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_plus_real @ X22 @ V2 )
= ( plus_plus_real @ U2 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_1070_real__trans__lemma,axiom,
! [X2: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V3: dedekind_preal,U3: dedekind_preal,X22: nat,V2: nat,U2: nat,Y22: nat] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_plus_nat @ X22 @ V2 )
= ( plus_plus_nat @ U2 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_1071_frac__ge__0,axiom,
! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X2 ) ) ).
% frac_ge_0
thf(fact_1072_frac__unique__iff,axiom,
! [X2: real,A: real] :
( ( ( archim2898591450579166408c_real @ X2 )
= A )
= ( ( member_real @ ( minus_minus_real @ X2 @ A ) @ ring_1_Ints_real )
& ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ one_one_real ) ) ) ).
% frac_unique_iff
thf(fact_1073_Ints__odd__less__0,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% Ints_odd_less_0
thf(fact_1074_Ints__odd__less__0,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% Ints_odd_less_0
thf(fact_1075_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_1076_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_1077_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_1078_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_1079_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1080_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1081_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1082_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1083_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1084_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1085_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1086_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1087_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1088_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1089_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_1090_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1091_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1092_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1093_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1094_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1095_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_1096_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1097_verit__comp__simplify1_I1_J,axiom,
! [A: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1098_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1099_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1100_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1101_field__le__epsilon,axiom,
! [X2: real,Y: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y @ E ) ) )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% field_le_epsilon
thf(fact_1102_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_1103_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_1104_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_1105_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1106_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1107_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1108_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1109_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1110_verit__la__generic,axiom,
! [A: int,X2: int] :
( ( ord_less_eq_int @ A @ X2 )
| ( A = X2 )
| ( ord_less_eq_int @ X2 @ A ) ) ).
% verit_la_generic
thf(fact_1111_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% linordered_field_no_lb
thf(fact_1112_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_1113_imp__le__cong,axiom,
! [X2: int,X6: int,P: $o,P4: $o] :
( ( X2 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1114_conj__le__cong,axiom,
! [X2: int,X6: int,P: $o,P4: $o] :
( ( X2 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1115_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X5: int] :
( ( ( ord_less_eq_int @ A @ X5 )
& ( ord_less_int @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D3: int] :
( ! [X: int] :
( ( ( ord_less_eq_int @ A @ X )
& ( ord_less_int @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1116_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X5: real] :
( ( ( ord_less_eq_real @ A @ X5 )
& ( ord_less_real @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D3: real] :
( ! [X: real] :
( ( ( ord_less_eq_real @ A @ X )
& ( ord_less_real @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_real @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1117_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D3: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1118_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_1119_pinf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X5 )
=> ~ ( ord_le5604041210740703414_preal @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1120_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_1121_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_1122_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_1123_pinf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X5 )
=> ( ord_le5604041210740703414_preal @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1124_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_1125_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_1126_minf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Z3 )
=> ~ ( ord_le5708704896291381698_preal @ T @ X5 ) ) ).
% minf(7)
thf(fact_1127_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_1128_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ~ ( ord_less_real @ T @ X5 ) ) ).
% minf(7)
thf(fact_1129_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_1130_minf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Z3 )
=> ( ord_le5708704896291381698_preal @ X5 @ T ) ) ).
% minf(5)
thf(fact_1131_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_1132_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ord_less_real @ X5 @ T ) ) ).
% minf(5)
thf(fact_1133_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_1134_minf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1135_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1136_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1137_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1138_minf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1139_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1140_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1141_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1142_minf_I2_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1143_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1144_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1145_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1146_minf_I1_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1147_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1148_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1149_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1150_pinf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X5 )
=> ( ord_le5708704896291381698_preal @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1151_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1152_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ord_less_real @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1153_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1154_pinf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X5 )
=> ~ ( ord_le5708704896291381698_preal @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1155_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1156_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ~ ( ord_less_real @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1157_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1158_pinf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1159_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1160_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1161_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1162_pinf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1163_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1164_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1165_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1166_pinf_I2_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1167_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1168_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1169_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1170_pinf_I1_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1171_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1172_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: real] :
! [X: real] :
( ( ord_less_real @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: real] :
! [X5: real] :
( ( ord_less_real @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1173_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1174_ex__gt__or__lt,axiom,
! [A: real] :
? [B4: real] :
( ( ord_less_real @ A @ B4 )
| ( ord_less_real @ B4 @ A ) ) ).
% ex_gt_or_lt
thf(fact_1175_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_1176_minf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Z3 )
=> ~ ( ord_le5604041210740703414_preal @ T @ X5 ) ) ).
% minf(8)
thf(fact_1177_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_1178_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_1179_minf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z3: dedekind_preal] :
! [X5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Z3 )
=> ( ord_le5604041210740703414_preal @ X5 @ T ) ) ).
% minf(6)
thf(fact_1180_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_1181_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_1182_bset_I6_J,axiom,
! [D4: int,B3: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% bset(6)
thf(fact_1183_bset_I8_J,axiom,
! [D4: int,T: int,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ T @ X5 )
=> ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% bset(8)
thf(fact_1184_aset_I6_J,axiom,
! [D4: int,T: int,A3: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A3 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% aset(6)
thf(fact_1185_aset_I8_J,axiom,
! [D4: int,A3: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A3 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ T @ X5 )
=> ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% aset(8)
thf(fact_1186_aset_I2_J,axiom,
! [D4: int,A3: set_int,P: int > $o,Q: int > $o] :
( ! [X: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A3 )
=> ( X
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X )
=> ( P @ ( plus_plus_int @ X @ D4 ) ) ) )
=> ( ! [X: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A3 )
=> ( X
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X )
=> ( Q @ ( plus_plus_int @ X @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A3 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
| ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_1187_aset_I1_J,axiom,
! [D4: int,A3: set_int,P: int > $o,Q: int > $o] :
( ! [X: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A3 )
=> ( X
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X )
=> ( P @ ( plus_plus_int @ X @ D4 ) ) ) )
=> ( ! [X: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A3 )
=> ( X
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X )
=> ( Q @ ( plus_plus_int @ X @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A3 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
& ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_1188_bset_I2_J,axiom,
! [D4: int,B3: set_int,P: int > $o,Q: int > $o] :
( ! [X: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X )
=> ( P @ ( minus_minus_int @ X @ D4 ) ) ) )
=> ( ! [X: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X )
=> ( Q @ ( minus_minus_int @ X @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
| ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_1189_bset_I1_J,axiom,
! [D4: int,B3: set_int,P: int > $o,Q: int > $o] :
( ! [X: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X )
=> ( P @ ( minus_minus_int @ X @ D4 ) ) ) )
=> ( ! [X: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X )
=> ( Q @ ( minus_minus_int @ X @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
& ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_1190_aset_I7_J,axiom,
! [D4: int,A3: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A3 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% aset(7)
thf(fact_1191_aset_I5_J,axiom,
! [D4: int,T: int,A3: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ A3 )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A3 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_1192_aset_I4_J,axiom,
! [D4: int,T: int,A3: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ A3 )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A3 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( X5 != T )
=> ( ( plus_plus_int @ X5 @ D4 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_1193_aset_I3_J,axiom,
! [D4: int,T: int,A3: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A3 )
=> ( X5
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( X5 = T )
=> ( ( plus_plus_int @ X5 @ D4 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_1194_bset_I7_J,axiom,
! [D4: int,T: int,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ B3 )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% bset(7)
thf(fact_1195_bset_I5_J,axiom,
! [D4: int,B3: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_1196_bset_I4_J,axiom,
! [D4: int,T: int,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ B3 )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( X5 != T )
=> ( ( minus_minus_int @ X5 @ D4 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_1197_bset_I3_J,axiom,
! [D4: int,T: int,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
=> ! [X5: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B3 )
=> ( X5
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( X5 = T )
=> ( ( minus_minus_int @ X5 @ D4 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_1198_cpmi,axiom,
! [D4: int,P: int > $o,P4: int > $o,B3: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ! [X: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B3 )
=> ( X
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X )
=> ( P @ ( minus_minus_int @ X @ D4 ) ) ) )
=> ( ! [X: int,K2: int] :
( ( P4 @ X )
= ( P4 @ ( minus_minus_int @ X @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
& ( P4 @ X3 ) )
| ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
& ? [Y2: int] :
( ( member_int @ Y2 @ B3 )
& ( P @ ( plus_plus_int @ Y2 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_1199_cppi,axiom,
! [D4: int,P: int > $o,P4: int > $o,A3: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ! [X: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A3 )
=> ( X
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X )
=> ( P @ ( plus_plus_int @ X @ D4 ) ) ) )
=> ( ! [X: int,K2: int] :
( ( P4 @ X )
= ( P4 @ ( minus_minus_int @ X @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
& ( P4 @ X3 ) )
| ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
& ? [Y2: int] :
( ( member_int @ Y2 @ A3 )
& ( P @ ( minus_minus_int @ Y2 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_1200_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1201_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1202_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1203_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1204_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1205_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1206_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1207_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1208_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1209_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1210_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X: int,K2: int] :
( ( P1 @ X )
= ( P1 @ ( minus_minus_int @ X @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( P @ X )
= ( P1 @ X ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1211_plusinfinity,axiom,
! [D: int,P4: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X: int,K2: int] :
( ( P4 @ X )
= ( P4 @ ( minus_minus_int @ X @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [X_12: int] : ( P4 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1212_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X: int] :
( ( P @ X )
=> ( P @ ( plus_plus_int @ X @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1213_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X: int] :
( ( P @ X )
=> ( P @ ( minus_minus_int @ X @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1214_periodic__finite__ex,axiom,
! [D: int,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X: int,K2: int] :
( ( P @ X )
= ( P @ ( minus_minus_int @ X @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ( P @ X3 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_1215_real__mult__congruent2__lemma,axiom,
! [X1: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal,Y1: dedekind_preal,X2: dedekind_preal,Y: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X1 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) )
=> ( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X2 @ X1 ) @ ( times_3000655703912201937_preal @ Y @ Y1 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X2 @ Y22 ) @ ( times_3000655703912201937_preal @ Y @ X22 ) ) )
= ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X2 @ X22 ) @ ( times_3000655703912201937_preal @ Y @ Y22 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X2 @ Y1 ) @ ( times_3000655703912201937_preal @ Y @ X1 ) ) ) ) ) ).
% real_mult_congruent2_lemma
thf(fact_1216_preal__add__mult__distrib2,axiom,
! [W: dedekind_preal,X2: dedekind_preal,Y: dedekind_preal] :
( ( times_3000655703912201937_preal @ W @ ( plus_p3173629198307831117_preal @ X2 @ Y ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W @ X2 ) @ ( times_3000655703912201937_preal @ W @ Y ) ) ) ).
% preal_add_mult_distrib2
thf(fact_1217_preal__add__mult__distrib,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,W: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ X2 @ Y ) @ W )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X2 @ W ) @ ( times_3000655703912201937_preal @ Y @ W ) ) ) ).
% preal_add_mult_distrib
thf(fact_1218_preal__mult__1,axiom,
! [Z: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ Z )
= Z ) ).
% preal_mult_1
thf(fact_1219_decr__lemma,axiom,
! [D: int,X2: int,Z: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% decr_lemma
thf(fact_1220_incr__lemma,axiom,
! [D: int,Z: int,X2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z @ ( plus_plus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% incr_lemma
thf(fact_1221_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1222_preal__mult__commute,axiom,
( times_3000655703912201937_preal
= ( ^ [X3: dedekind_preal,Y2: dedekind_preal] : ( times_3000655703912201937_preal @ Y2 @ X3 ) ) ) ).
% preal_mult_commute
thf(fact_1223_preal__mult__assoc,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ X2 @ Y ) @ Z )
= ( times_3000655703912201937_preal @ X2 @ ( times_3000655703912201937_preal @ Y @ Z ) ) ) ).
% preal_mult_assoc
thf(fact_1224_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N2: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% int_less_real_le
thf(fact_1225_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N2: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_1226_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1227_zabs__def,axiom,
( abs_abs_int
= ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_1228_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1229_ln__inj__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X2 )
= ( ln_ln_real @ Y ) )
= ( X2 = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_1230_ln__less__cancel__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X2 @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1231_real__add__minus__iff,axiom,
! [X2: real,A: real] :
( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X2 = A ) ) ).
% real_add_minus_iff
thf(fact_1232_ln__le__cancel__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X2 @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1233_artanh__minus__real,axiom,
! [X2: real] :
( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
=> ( ( artanh_real @ ( uminus_uminus_real @ X2 ) )
= ( uminus_uminus_real @ ( artanh_real @ X2 ) ) ) ) ).
% artanh_minus_real
thf(fact_1234_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_1235_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_1236_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_1237_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_1238_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_1239_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_1240_ln__mult,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( times_times_real @ X2 @ Y ) )
= ( plus_plus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_1241_abs__real__def,axiom,
( abs_abs_real
= ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).
% abs_real_def
thf(fact_1242_real__of__int__floor__add__one__ge,axiom,
! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_1243_floor__divide__real__eq__div,axiom,
! [B: int,A: real] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
= ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% floor_divide_real_eq_div
thf(fact_1244_real__of__int__div4,axiom,
! [N: int,X2: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) ) ).
% real_of_int_div4
thf(fact_1245_real__of__int__floor__add__one__gt,axiom,
! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_1246_floor__eq2,axiom,
! [N: int,X2: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X2 )
=> ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X2 )
= N ) ) ) ).
% floor_eq2
thf(fact_1247_floor__eq,axiom,
! [N: int,X2: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X2 )
=> ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X2 )
= N ) ) ) ).
% floor_eq
thf(fact_1248_real__of__int__floor__gt__diff__one,axiom,
! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_1249_real__of__int__floor__ge__diff__one,axiom,
! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_1250_real__of__int__div3,axiom,
! [N: int,X2: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) ) @ one_one_real ) ).
% real_of_int_div3
thf(fact_1251_real__of__int__div2,axiom,
! [N: int,X2: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) ) ) ).
% real_of_int_div2
thf(fact_1252_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_1253_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X3: real,Y2: real] : ( plus_plus_real @ X3 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).
% minus_real_def
thf(fact_1254_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_1255_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_1256_ln__diff__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X2 @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_1257_ln__div,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( divide_divide_real @ X2 @ Y ) )
= ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_1258_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_1259_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_1260_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_1261_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_1262_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_1263_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_1264_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_1265_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_1266_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_1267_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_1268_real__add__less__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_less_0_iff
thf(fact_1269_real__0__less__add__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1270_real__add__le__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_le_0_iff
thf(fact_1271_real__0__le__add__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1272_verit__less__mono__div__int2,axiom,
! [A3: int,B3: int,N: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).
% verit_less_mono_div_int2
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_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,Y: real] :
( ( if_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
$true ).
thf(conj_1,hypothesis,
! [R2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S2 )
=> ( ( plus_p3173629198307831117_preal @ R2 @ ( minus_7336623429200594941_preal @ S2 @ R2 ) )
= S2 ) ) ).
thf(conj_2,conjecture,
? [M3: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ u @ one_on9143529541772854033_preal )
= ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ M3 @ one_on9143529541772854033_preal ) @ v ) )
| ( ( plus_p3173629198307831117_preal @ u @ one_on9143529541772854033_preal )
= ( plus_p3173629198307831117_preal @ one_on9143529541772854033_preal @ v ) )
| ( ( plus_p3173629198307831117_preal @ u @ ( plus_p3173629198307831117_preal @ M3 @ one_on9143529541772854033_preal ) )
= ( plus_p3173629198307831117_preal @ one_on9143529541772854033_preal @ v ) ) ) ).
%------------------------------------------------------------------------------