TPTP Problem File: SLH0996^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Youngs_Inequality/0000_Youngs/prob_00124_005024__12913144_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1422 ( 420 unt; 154 typ; 0 def)
% Number of atoms : 4472 ( 960 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11784 ( 323 ~; 57 |; 214 &;9069 @)
% ( 0 <=>;2121 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 1627 (1627 >; 0 *; 0 +; 0 <<)
% Number of symbols : 142 ( 139 usr; 23 con; 0-4 aty)
% Number of variables : 4007 ( 189 ^;3677 !; 141 ?;4007 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:26:45.361
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
set_real_real: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
set_Product_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Numeral____Type__Onum0_J,type,
set_Numeral_num0: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__String__Oliteral_J,type,
set_literal: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
set_rat: $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__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__String__Oliteral,type,
literal: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (139)
thf(sy_c_Countable_Ofrom__nat_001t__Nat__Onat,type,
from_nat_nat: nat > nat ).
thf(sy_c_Countable_Ofrom__nat_001t__Rat__Orat,type,
from_nat_rat: nat > rat ).
thf(sy_c_Countable_Ofrom__nat_001t__String__Oliteral,type,
from_nat_literal: nat > literal ).
thf(sy_c_Countable_Onat__to__rat__surj,type,
nat_to_rat_surj: nat > rat ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Numeral____Type__Onum0,type,
finite6454714172617411596l_num0: set_Numeral_num0 > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(sy_c_Finite__Set_Ocard_001t__String__Oliteral,type,
finite_card_literal: set_literal > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Complex__Ocomplex_001t__Int__Oint,type,
monoto2404022921142102083ex_int: set_complex > ( complex > complex > $o ) > ( int > int > $o ) > ( complex > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Complex__Ocomplex_001t__Nat__Onat,type,
monoto2406513391651152359ex_nat: set_complex > ( complex > complex > $o ) > ( nat > nat > $o ) > ( complex > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
monoto7363281639122250051x_real: set_complex > ( complex > complex > $o ) > ( real > real > $o ) > ( complex > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
monoto6775660721739060364l_real: set_int > ( int > int > $o ) > ( ( real > real ) > ( real > real ) > $o ) > ( int > real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Int__Oint,type,
monotone_on_int_int: set_int > ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Nat__Onat,type,
monotone_on_int_nat: set_int > ( int > int > $o ) > ( nat > nat > $o ) > ( int > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Int__Oint_001t__Real__Oreal,type,
monotone_on_int_real: set_int > ( int > int > $o ) > ( real > real > $o ) > ( int > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
monoto2824216093323351088l_real: set_nat > ( nat > nat > $o ) > ( ( real > real ) > ( real > real ) > $o ) > ( nat > real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Int__Oint,type,
monotone_on_nat_int: set_nat > ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Real__Oreal,type,
monotone_on_nat_real: set_nat > ( nat > nat > $o ) > ( real > real > $o ) > ( nat > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Rat__Orat_001t__Int__Oint,type,
monotone_on_rat_int: set_rat > ( rat > rat > $o ) > ( int > int > $o ) > ( rat > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Rat__Orat_001t__Nat__Onat,type,
monotone_on_rat_nat: set_rat > ( rat > rat > $o ) > ( nat > nat > $o ) > ( rat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Rat__Orat_001t__Real__Oreal,type,
monotone_on_rat_real: set_rat > ( rat > rat > $o ) > ( real > real > $o ) > ( rat > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
monoto8965231823629880588l_real: set_real > ( real > real > $o ) > ( ( real > real ) > ( real > real ) > $o ) > ( real > real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Int__Oint,type,
monotone_on_real_int: set_real > ( real > real > $o ) > ( int > int > $o ) > ( real > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Nat__Onat,type,
monotone_on_real_nat: set_real > ( real > real > $o ) > ( nat > nat > $o ) > ( real > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Real__Oreal_001t__Real__Oreal,type,
monoto4017252874604999745l_real: set_real > ( real > real > $o ) > ( real > real > $o ) > ( real > real ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__String__Oliteral_001t__Int__Oint,type,
monoto6090175056727812057al_int: set_literal > ( literal > literal > $o ) > ( int > int > $o ) > ( literal > int ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__String__Oliteral_001t__Nat__Onat,type,
monoto6092665527236862333al_nat: set_literal > ( literal > literal > $o ) > ( nat > nat > $o ) > ( literal > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__String__Oliteral_001t__Real__Oreal,type,
monoto1443783125453340121l_real: set_literal > ( literal > literal > $o ) > ( real > real > $o ) > ( literal > real ) > $o ).
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__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
ord_less_real_real: ( real > real ) > ( real > real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Complex__Ocomplex,type,
ord_less_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
ord_less_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
ord_le6291385379474865793l_real: set_real_real > set_real_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__String__Oliteral_J,type,
ord_less_set_literal: set_literal > set_literal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__String__Oliteral,type,
ord_less_literal: literal > literal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
ord_le6948328307412524503l_real: ( real > real ) > ( real > real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Complex__Ocomplex,type,
ord_less_eq_complex: complex > complex > $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__Rat__Orat,type,
ord_less_eq_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
ord_le4198349162570665613l_real: set_real_real > set_real_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__String__Oliteral_J,type,
ord_le7307670543136651348iteral: set_literal > set_literal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__String__Oliteral,type,
ord_less_eq_literal: literal > literal > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_M_Eo_J,type,
top_top_real_real_o: ( real > real ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Complex__Ocomplex_M_Eo_J,type,
top_top_complex_o: complex > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Rat__Orat_M_Eo_J,type,
top_top_rat_o: rat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
top_top_real_o: real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__String__Oliteral_M_Eo_J,type,
top_top_literal_o: literal > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
top_to2071711978144146653l_real: set_real_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Complex__Ocomplex_J,type,
top_top_set_complex: set_complex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
top_top_set_int: set_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Numeral____Type__Onum0_J,type,
top_to3689904424835650196l_num0: set_Numeral_num0 ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Rat__Orat_J,type,
top_top_set_rat: set_rat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Oliteral_J,type,
top_top_set_literal: set_literal ).
thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Rat__Orat,type,
field_6020823756834552118ts_rat: set_rat ).
thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
field_5140801741446780682s_real: set_real ).
thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
collect_real_real: ( ( real > real ) > $o ) > set_real_real ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__String__Oliteral,type,
collect_literal: ( literal > $o ) > set_literal ).
thf(sy_c_Set_Oimage_001_062_It__Real__Oreal_Mt__Real__Oreal_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
image_745864523092522741l_real: ( ( real > real ) > real > real ) > set_real_real > set_real_real ).
thf(sy_c_Set_Oimage_001_062_It__Real__Oreal_Mt__Real__Oreal_J_001t__Nat__Onat,type,
image_real_real_nat: ( ( real > real ) > nat ) > set_real_real > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Real__Oreal_Mt__Real__Oreal_J_001t__Real__Oreal,type,
image_real_real_real: ( ( real > real ) > real ) > set_real_real > set_real ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
image_1468599708987790691omplex: ( complex > complex ) > set_complex > set_complex ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Nat__Onat,type,
image_complex_nat: ( complex > nat ) > set_complex > set_nat ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Rat__Orat,type,
image_complex_rat: ( complex > rat ) > set_complex > set_rat ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Real__Oreal,type,
image_complex_real: ( complex > real ) > set_complex > set_real ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__String__Oliteral,type,
image_8841419608667285983iteral: ( complex > literal ) > set_complex > set_literal ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
image_nat_real_real: ( nat > real > real ) > set_nat > set_real_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Complex__Ocomplex,type,
image_nat_complex: ( nat > complex ) > set_nat > set_complex ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Rat__Orat,type,
image_nat_rat: ( nat > rat ) > set_nat > set_rat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Oliteral,type,
image_nat_literal: ( nat > literal ) > set_nat > set_literal ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
image_rat_real_real: ( rat > real > real ) > set_rat > set_real_real ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Nat__Onat,type,
image_rat_nat: ( rat > nat ) > set_rat > set_nat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Real__Oreal,type,
image_rat_real: ( rat > real ) > set_rat > set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
image_real_real_real2: ( real > real > real ) > set_real > set_real_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Complex__Ocomplex,type,
image_real_complex: ( real > complex ) > set_real > set_complex ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
image_real_nat: ( real > nat ) > set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Rat__Orat,type,
image_real_rat: ( real > rat ) > set_real > set_rat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__String__Oliteral,type,
image_real_literal: ( real > literal ) > set_real > set_literal ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Complex__Ocomplex,type,
image_5274195009022015549omplex: ( literal > complex ) > set_literal > set_complex ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Nat__Onat,type,
image_literal_nat: ( literal > nat ) > set_literal > set_nat ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Rat__Orat,type,
image_literal_rat: ( literal > rat ) > set_literal > set_rat ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Real__Oreal,type,
image_literal_real: ( literal > real ) > set_literal > set_real ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__String__Oliteral,type,
image_8195128725298311301iteral: ( literal > literal ) > set_literal > set_literal ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
member_real_real: ( real > real ) > set_real_real > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__String__Oliteral,type,
member_literal: literal > set_literal > $o ).
thf(sy_v_f,type,
f: real > real ).
thf(sy_v_g____,type,
g: nat > real > real ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_x____,type,
x: real ).
thf(sy_v_y____,type,
y: real ).
% Relevant facts (1263)
thf(fact_0_image__eqI,axiom,
! [B: real,F: real > real,X: real,A: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A )
=> ( member_real @ B @ ( image_real_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1_image__eqI,axiom,
! [B: nat,F: real > nat,X: real,A: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A )
=> ( member_nat @ B @ ( image_real_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_2_image__eqI,axiom,
! [B: rat,F: nat > rat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_rat @ B @ ( image_nat_rat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_3_image__eqI,axiom,
! [B: complex,F: nat > complex,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_complex @ B @ ( image_nat_complex @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_4_image__eqI,axiom,
! [B: real,F: nat > real,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_real @ B @ ( image_nat_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_5_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_6_image__eqI,axiom,
! [B: real,F: ( real > real ) > real,X: real > real,A: set_real_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real_real @ X @ A )
=> ( member_real @ B @ ( image_real_real_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_7_image__eqI,axiom,
! [B: nat,F: ( real > real ) > nat,X: real > real,A: set_real_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real_real @ X @ A )
=> ( member_nat @ B @ ( image_real_real_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_8_image__eqI,axiom,
! [B: real > real,F: real > real > real,X: real,A: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A )
=> ( member_real_real @ B @ ( image_real_real_real2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_9_image__eqI,axiom,
! [B: real > real,F: nat > real > real,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_real_real @ B @ ( image_nat_real_real @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_10_UNIV__I,axiom,
! [X: real > real] : ( member_real_real @ X @ top_to2071711978144146653l_real ) ).
% UNIV_I
thf(fact_11_UNIV__I,axiom,
! [X: complex] : ( member_complex @ X @ top_top_set_complex ) ).
% UNIV_I
thf(fact_12_UNIV__I,axiom,
! [X: literal] : ( member_literal @ X @ top_top_set_literal ) ).
% UNIV_I
thf(fact_13_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_14_UNIV__I,axiom,
! [X: rat] : ( member_rat @ X @ top_top_set_rat ) ).
% UNIV_I
thf(fact_15_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_16_iso__tuple__UNIV__I,axiom,
! [X: real > real] : ( member_real_real @ X @ top_to2071711978144146653l_real ) ).
% iso_tuple_UNIV_I
thf(fact_17_iso__tuple__UNIV__I,axiom,
! [X: complex] : ( member_complex @ X @ top_top_set_complex ) ).
% iso_tuple_UNIV_I
thf(fact_18_iso__tuple__UNIV__I,axiom,
! [X: literal] : ( member_literal @ X @ top_top_set_literal ) ).
% iso_tuple_UNIV_I
thf(fact_19_iso__tuple__UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_20_iso__tuple__UNIV__I,axiom,
! [X: rat] : ( member_rat @ X @ top_top_set_rat ) ).
% iso_tuple_UNIV_I
thf(fact_21_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_22_surjD,axiom,
! [F: complex > complex,Y: complex] :
( ( ( image_1468599708987790691omplex @ F @ top_top_set_complex )
= top_top_set_complex )
=> ? [X2: complex] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_23_surjD,axiom,
! [F: complex > literal,Y: literal] :
( ( ( image_8841419608667285983iteral @ F @ top_top_set_complex )
= top_top_set_literal )
=> ? [X2: complex] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_24_surjD,axiom,
! [F: complex > real,Y: real] :
( ( ( image_complex_real @ F @ top_top_set_complex )
= top_top_set_real )
=> ? [X2: complex] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_25_surjD,axiom,
! [F: complex > rat,Y: rat] :
( ( ( image_complex_rat @ F @ top_top_set_complex )
= top_top_set_rat )
=> ? [X2: complex] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_26_surjD,axiom,
! [F: complex > nat,Y: nat] :
( ( ( image_complex_nat @ F @ top_top_set_complex )
= top_top_set_nat )
=> ? [X2: complex] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_27_surjD,axiom,
! [F: literal > complex,Y: complex] :
( ( ( image_5274195009022015549omplex @ F @ top_top_set_literal )
= top_top_set_complex )
=> ? [X2: literal] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_28_surjD,axiom,
! [F: literal > literal,Y: literal] :
( ( ( image_8195128725298311301iteral @ F @ top_top_set_literal )
= top_top_set_literal )
=> ? [X2: literal] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_29_surjD,axiom,
! [F: literal > real,Y: real] :
( ( ( image_literal_real @ F @ top_top_set_literal )
= top_top_set_real )
=> ? [X2: literal] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_30_surjD,axiom,
! [F: literal > rat,Y: rat] :
( ( ( image_literal_rat @ F @ top_top_set_literal )
= top_top_set_rat )
=> ? [X2: literal] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_31_surjD,axiom,
! [F: literal > nat,Y: nat] :
( ( ( image_literal_nat @ F @ top_top_set_literal )
= top_top_set_nat )
=> ? [X2: literal] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_32_surjE,axiom,
! [F: complex > complex,Y: complex] :
( ( ( image_1468599708987790691omplex @ F @ top_top_set_complex )
= top_top_set_complex )
=> ~ ! [X2: complex] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_33_surjE,axiom,
! [F: complex > literal,Y: literal] :
( ( ( image_8841419608667285983iteral @ F @ top_top_set_complex )
= top_top_set_literal )
=> ~ ! [X2: complex] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_34_surjE,axiom,
! [F: complex > real,Y: real] :
( ( ( image_complex_real @ F @ top_top_set_complex )
= top_top_set_real )
=> ~ ! [X2: complex] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_35_surjE,axiom,
! [F: complex > rat,Y: rat] :
( ( ( image_complex_rat @ F @ top_top_set_complex )
= top_top_set_rat )
=> ~ ! [X2: complex] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_36_surjE,axiom,
! [F: complex > nat,Y: nat] :
( ( ( image_complex_nat @ F @ top_top_set_complex )
= top_top_set_nat )
=> ~ ! [X2: complex] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_37_surjE,axiom,
! [F: literal > complex,Y: complex] :
( ( ( image_5274195009022015549omplex @ F @ top_top_set_literal )
= top_top_set_complex )
=> ~ ! [X2: literal] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_38_surjE,axiom,
! [F: literal > literal,Y: literal] :
( ( ( image_8195128725298311301iteral @ F @ top_top_set_literal )
= top_top_set_literal )
=> ~ ! [X2: literal] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_39_surjE,axiom,
! [F: literal > real,Y: real] :
( ( ( image_literal_real @ F @ top_top_set_literal )
= top_top_set_real )
=> ~ ! [X2: literal] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_40_surjE,axiom,
! [F: literal > rat,Y: rat] :
( ( ( image_literal_rat @ F @ top_top_set_literal )
= top_top_set_rat )
=> ~ ! [X2: literal] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_41_surjE,axiom,
! [F: literal > nat,Y: nat] :
( ( ( image_literal_nat @ F @ top_top_set_literal )
= top_top_set_nat )
=> ~ ! [X2: literal] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_42_surjI,axiom,
! [G: complex > complex,F: complex > complex] :
( ! [X2: complex] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_1468599708987790691omplex @ G @ top_top_set_complex )
= top_top_set_complex ) ) ).
% surjI
thf(fact_43_surjI,axiom,
! [G: complex > literal,F: literal > complex] :
( ! [X2: literal] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_8841419608667285983iteral @ G @ top_top_set_complex )
= top_top_set_literal ) ) ).
% surjI
thf(fact_44_surjI,axiom,
! [G: complex > real,F: real > complex] :
( ! [X2: real] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_complex_real @ G @ top_top_set_complex )
= top_top_set_real ) ) ).
% surjI
thf(fact_45_surjI,axiom,
! [G: complex > rat,F: rat > complex] :
( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_complex_rat @ G @ top_top_set_complex )
= top_top_set_rat ) ) ).
% surjI
thf(fact_46_surjI,axiom,
! [G: complex > nat,F: nat > complex] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_complex_nat @ G @ top_top_set_complex )
= top_top_set_nat ) ) ).
% surjI
thf(fact_47_surjI,axiom,
! [G: literal > complex,F: complex > literal] :
( ! [X2: complex] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_5274195009022015549omplex @ G @ top_top_set_literal )
= top_top_set_complex ) ) ).
% surjI
thf(fact_48_surjI,axiom,
! [G: literal > literal,F: literal > literal] :
( ! [X2: literal] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_8195128725298311301iteral @ G @ top_top_set_literal )
= top_top_set_literal ) ) ).
% surjI
thf(fact_49_surjI,axiom,
! [G: literal > real,F: real > literal] :
( ! [X2: real] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_literal_real @ G @ top_top_set_literal )
= top_top_set_real ) ) ).
% surjI
thf(fact_50_surjI,axiom,
! [G: literal > rat,F: rat > literal] :
( ! [X2: rat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_literal_rat @ G @ top_top_set_literal )
= top_top_set_rat ) ) ).
% surjI
thf(fact_51_surjI,axiom,
! [G: literal > nat,F: nat > literal] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_literal_nat @ G @ top_top_set_literal )
= top_top_set_nat ) ) ).
% surjI
thf(fact_52_rangeI,axiom,
! [F: complex > real,X: complex] : ( member_real @ ( F @ X ) @ ( image_complex_real @ F @ top_top_set_complex ) ) ).
% rangeI
thf(fact_53_rangeI,axiom,
! [F: complex > nat,X: complex] : ( member_nat @ ( F @ X ) @ ( image_complex_nat @ F @ top_top_set_complex ) ) ).
% rangeI
thf(fact_54_rangeI,axiom,
! [F: literal > real,X: literal] : ( member_real @ ( F @ X ) @ ( image_literal_real @ F @ top_top_set_literal ) ) ).
% rangeI
thf(fact_55_rangeI,axiom,
! [F: literal > nat,X: literal] : ( member_nat @ ( F @ X ) @ ( image_literal_nat @ F @ top_top_set_literal ) ) ).
% rangeI
thf(fact_56_rangeI,axiom,
! [F: real > real,X: real] : ( member_real @ ( F @ X ) @ ( image_real_real @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_57_rangeI,axiom,
! [F: real > nat,X: real] : ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ top_top_set_real ) ) ).
% rangeI
thf(fact_58_rangeI,axiom,
! [F: rat > real,X: rat] : ( member_real @ ( F @ X ) @ ( image_rat_real @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_59_rangeI,axiom,
! [F: rat > nat,X: rat] : ( member_nat @ ( F @ X ) @ ( image_rat_nat @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_60_rangeI,axiom,
! [F: nat > rat,X: nat] : ( member_rat @ ( F @ X ) @ ( image_nat_rat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_61_rangeI,axiom,
! [F: nat > complex,X: nat] : ( member_complex @ ( F @ X ) @ ( image_nat_complex @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_62_surj__def,axiom,
! [F: complex > complex] :
( ( ( image_1468599708987790691omplex @ F @ top_top_set_complex )
= top_top_set_complex )
= ( ! [Y2: complex] :
? [X3: complex] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_63_surj__def,axiom,
! [F: complex > literal] :
( ( ( image_8841419608667285983iteral @ F @ top_top_set_complex )
= top_top_set_literal )
= ( ! [Y2: literal] :
? [X3: complex] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_64_surj__def,axiom,
! [F: complex > real] :
( ( ( image_complex_real @ F @ top_top_set_complex )
= top_top_set_real )
= ( ! [Y2: real] :
? [X3: complex] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_65_surj__def,axiom,
! [F: complex > rat] :
( ( ( image_complex_rat @ F @ top_top_set_complex )
= top_top_set_rat )
= ( ! [Y2: rat] :
? [X3: complex] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_66_surj__def,axiom,
! [F: complex > nat] :
( ( ( image_complex_nat @ F @ top_top_set_complex )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X3: complex] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_67_surj__def,axiom,
! [F: literal > complex] :
( ( ( image_5274195009022015549omplex @ F @ top_top_set_literal )
= top_top_set_complex )
= ( ! [Y2: complex] :
? [X3: literal] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_68_surj__def,axiom,
! [F: literal > literal] :
( ( ( image_8195128725298311301iteral @ F @ top_top_set_literal )
= top_top_set_literal )
= ( ! [Y2: literal] :
? [X3: literal] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_69_surj__def,axiom,
! [F: literal > real] :
( ( ( image_literal_real @ F @ top_top_set_literal )
= top_top_set_real )
= ( ! [Y2: real] :
? [X3: literal] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_70_surj__def,axiom,
! [F: literal > rat] :
( ( ( image_literal_rat @ F @ top_top_set_literal )
= top_top_set_rat )
= ( ! [Y2: rat] :
? [X3: literal] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_71_surj__def,axiom,
! [F: literal > nat] :
( ( ( image_literal_nat @ F @ top_top_set_literal )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X3: literal] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_72_range__eqI,axiom,
! [B: real,F: complex > real,X: complex] :
( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_complex_real @ F @ top_top_set_complex ) ) ) ).
% range_eqI
thf(fact_73_range__eqI,axiom,
! [B: nat,F: complex > nat,X: complex] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_complex_nat @ F @ top_top_set_complex ) ) ) ).
% range_eqI
thf(fact_74_range__eqI,axiom,
! [B: real,F: literal > real,X: literal] :
( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_literal_real @ F @ top_top_set_literal ) ) ) ).
% range_eqI
thf(fact_75_range__eqI,axiom,
! [B: nat,F: literal > nat,X: literal] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_literal_nat @ F @ top_top_set_literal ) ) ) ).
% range_eqI
thf(fact_76_range__eqI,axiom,
! [B: real,F: real > real,X: real] :
( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_real_real @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_77_range__eqI,axiom,
! [B: nat,F: real > nat,X: real] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_real_nat @ F @ top_top_set_real ) ) ) ).
% range_eqI
thf(fact_78_range__eqI,axiom,
! [B: real,F: rat > real,X: rat] :
( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_rat_real @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_79_range__eqI,axiom,
! [B: nat,F: rat > nat,X: rat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_rat_nat @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_80_range__eqI,axiom,
! [B: rat,F: nat > rat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_nat_rat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_81_range__eqI,axiom,
! [B: complex,F: nat > complex,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_complex @ B @ ( image_nat_complex @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_82_top_Oextremum__strict,axiom,
! [A2: set_complex] :
~ ( ord_less_set_complex @ top_top_set_complex @ A2 ) ).
% top.extremum_strict
thf(fact_83_top_Oextremum__strict,axiom,
! [A2: set_literal] :
~ ( ord_less_set_literal @ top_top_set_literal @ A2 ) ).
% top.extremum_strict
thf(fact_84_top_Oextremum__strict,axiom,
! [A2: set_real] :
~ ( ord_less_set_real @ top_top_set_real @ A2 ) ).
% top.extremum_strict
thf(fact_85_top_Oextremum__strict,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ top_top_set_rat @ A2 ) ).
% top.extremum_strict
thf(fact_86_top_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% top.extremum_strict
thf(fact_87_top_Oextremum__strict,axiom,
! [A2: set_real_real] :
~ ( ord_le6291385379474865793l_real @ top_to2071711978144146653l_real @ A2 ) ).
% top.extremum_strict
thf(fact_88_top_Onot__eq__extremum,axiom,
! [A2: set_complex] :
( ( A2 != top_top_set_complex )
= ( ord_less_set_complex @ A2 @ top_top_set_complex ) ) ).
% top.not_eq_extremum
thf(fact_89_top_Onot__eq__extremum,axiom,
! [A2: set_literal] :
( ( A2 != top_top_set_literal )
= ( ord_less_set_literal @ A2 @ top_top_set_literal ) ) ).
% top.not_eq_extremum
thf(fact_90_top_Onot__eq__extremum,axiom,
! [A2: set_real] :
( ( A2 != top_top_set_real )
= ( ord_less_set_real @ A2 @ top_top_set_real ) ) ).
% top.not_eq_extremum
thf(fact_91_top_Onot__eq__extremum,axiom,
! [A2: set_rat] :
( ( A2 != top_top_set_rat )
= ( ord_less_set_rat @ A2 @ top_top_set_rat ) ) ).
% top.not_eq_extremum
thf(fact_92_top_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != top_top_set_nat )
= ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_93_top_Onot__eq__extremum,axiom,
! [A2: set_real_real] :
( ( A2 != top_to2071711978144146653l_real )
= ( ord_le6291385379474865793l_real @ A2 @ top_to2071711978144146653l_real ) ) ).
% top.not_eq_extremum
thf(fact_94_top__set__def,axiom,
( top_top_set_complex
= ( collect_complex @ top_top_complex_o ) ) ).
% top_set_def
thf(fact_95_top__set__def,axiom,
( top_top_set_literal
= ( collect_literal @ top_top_literal_o ) ) ).
% top_set_def
thf(fact_96_top__set__def,axiom,
( top_top_set_real
= ( collect_real @ top_top_real_o ) ) ).
% top_set_def
thf(fact_97_top__set__def,axiom,
( top_top_set_rat
= ( collect_rat @ top_top_rat_o ) ) ).
% top_set_def
thf(fact_98_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_99_top__set__def,axiom,
( top_to2071711978144146653l_real
= ( collect_real_real @ top_top_real_real_o ) ) ).
% top_set_def
thf(fact_100_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_101_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_102_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_103_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_104_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_105_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_106_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_107_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_108_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_109_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_110_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_111_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_112_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_113_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_114_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_115_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_116_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_117_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_118_order__less__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_119_order__less__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_120_order__less__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_121_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_122_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_123_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_124_order__less__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_125_order__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_126_order__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_127_order__less__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_128_order__less__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_129_order__less__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_130_order__less__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_131_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_132_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_133_order__less__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_134_order__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_135_order__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_136_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_137_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_138_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_139_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_140_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_141_ord__less__eq__subst,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_142_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_143_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_144_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_145_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_146_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_147_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_148_ord__eq__less__subst,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_149_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_150_ord__eq__less__subst,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_151_ord__eq__less__subst,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_152_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_153_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_154_ord__eq__less__subst,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_155_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_156_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_157_order__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_158_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_159_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_160_order__less__asym_H,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_161_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_162_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_163_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_164_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_165_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_166_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_167_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_168_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_169_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_170_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_171_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_172_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_173_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_174_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_175_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_176_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_177_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_178_dual__order_Ostrict__trans,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_179_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_180_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_181_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_182_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_183_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_184_order_Ostrict__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_185_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_186_order_Ostrict__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_187_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_188_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_189_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_190_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_191_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_192_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_193_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_194_dual__order_Oasym,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ~ ( ord_less_real @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_195_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_196_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_197_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_198_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_199_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_200_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_201_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_202_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_203_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_204_ord__less__eq__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_205_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_206_ord__less__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_207_ord__eq__less__trans,axiom,
! [A2: real,B: real,C: real] :
( ( A2 = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_208_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_209_ord__eq__less__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_210_order_Oasym,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ~ ( ord_less_real @ B @ A2 ) ) ).
% order.asym
thf(fact_211_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_212_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_213_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_214_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_215_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_216_mem__Collect__eq,axiom,
! [A2: real > real,P: ( real > real ) > $o] :
( ( member_real_real @ A2 @ ( collect_real_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_217_mem__Collect__eq,axiom,
! [A2: real,P: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_218_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_219_mem__Collect__eq,axiom,
! [A2: complex,P: complex > $o] :
( ( member_complex @ A2 @ ( collect_complex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_220_mem__Collect__eq,axiom,
! [A2: rat,P: rat > $o] :
( ( member_rat @ A2 @ ( collect_rat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_221_mem__Collect__eq,axiom,
! [A2: literal,P: literal > $o] :
( ( member_literal @ A2 @ ( collect_literal @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_222_Collect__mem__eq,axiom,
! [A: set_real_real] :
( ( collect_real_real
@ ^ [X3: real > real] : ( member_real_real @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_223_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_224_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_225_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X3: complex] : ( member_complex @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_226_Collect__mem__eq,axiom,
! [A: set_rat] :
( ( collect_rat
@ ^ [X3: rat] : ( member_rat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_227_Collect__mem__eq,axiom,
! [A: set_literal] :
( ( collect_literal
@ ^ [X3: literal] : ( member_literal @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_228_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z2: real] :
( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Z2 @ Y ) ) ) ).
% dense
thf(fact_229_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_230_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_231_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_232_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_233_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_234_UNIV__witness,axiom,
? [X2: complex] : ( member_complex @ X2 @ top_top_set_complex ) ).
% UNIV_witness
thf(fact_235_UNIV__witness,axiom,
? [X2: literal] : ( member_literal @ X2 @ top_top_set_literal ) ).
% UNIV_witness
thf(fact_236_UNIV__witness,axiom,
? [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_237_UNIV__witness,axiom,
? [X2: rat] : ( member_rat @ X2 @ top_top_set_rat ) ).
% UNIV_witness
thf(fact_238_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_239_UNIV__witness,axiom,
? [X2: real > real] : ( member_real_real @ X2 @ top_to2071711978144146653l_real ) ).
% UNIV_witness
thf(fact_240_UNIV__eq__I,axiom,
! [A: set_complex] :
( ! [X2: complex] : ( member_complex @ X2 @ A )
=> ( top_top_set_complex = A ) ) ).
% UNIV_eq_I
thf(fact_241_UNIV__eq__I,axiom,
! [A: set_literal] :
( ! [X2: literal] : ( member_literal @ X2 @ A )
=> ( top_top_set_literal = A ) ) ).
% UNIV_eq_I
thf(fact_242_UNIV__eq__I,axiom,
! [A: set_real] :
( ! [X2: real] : ( member_real @ X2 @ A )
=> ( top_top_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_243_UNIV__eq__I,axiom,
! [A: set_rat] :
( ! [X2: rat] : ( member_rat @ X2 @ A )
=> ( top_top_set_rat = A ) ) ).
% UNIV_eq_I
thf(fact_244_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_245_UNIV__eq__I,axiom,
! [A: set_real_real] :
( ! [X2: real > real] : ( member_real_real @ X2 @ A )
=> ( top_to2071711978144146653l_real = A ) ) ).
% UNIV_eq_I
thf(fact_246_rev__image__eqI,axiom,
! [X: real,A: set_real,B: real,F: real > real] :
( ( member_real @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_real_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_247_rev__image__eqI,axiom,
! [X: real,A: set_real,B: nat,F: real > nat] :
( ( member_real @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_real_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_248_rev__image__eqI,axiom,
! [X: real,A: set_real,B: complex,F: real > complex] :
( ( member_real @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_complex @ B @ ( image_real_complex @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_249_rev__image__eqI,axiom,
! [X: real,A: set_real,B: rat,F: real > rat] :
( ( member_real @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_real_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_250_rev__image__eqI,axiom,
! [X: real,A: set_real,B: literal,F: real > literal] :
( ( member_real @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_literal @ B @ ( image_real_literal @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_251_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: real,F: nat > real] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_nat_real @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_252_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_253_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: complex,F: nat > complex] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_complex @ B @ ( image_nat_complex @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_254_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: rat,F: nat > rat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_nat_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_255_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: literal,F: nat > literal] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_literal @ B @ ( image_nat_literal @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_256_ball__imageD,axiom,
! [F: nat > real > real,A: set_nat,P: ( real > real ) > $o] :
( ! [X2: real > real] :
( ( member_real_real @ X2 @ ( image_nat_real_real @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_257_ball__imageD,axiom,
! [F: nat > rat,A: set_nat,P: rat > $o] :
( ! [X2: rat] :
( ( member_rat @ X2 @ ( image_nat_rat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_258_ball__imageD,axiom,
! [F: nat > real,A: set_nat,P: real > $o] :
( ! [X2: real] :
( ( member_real @ X2 @ ( image_nat_real @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_259_ball__imageD,axiom,
! [F: nat > complex,A: set_nat,P: complex > $o] :
( ! [X2: complex] :
( ( member_complex @ X2 @ ( image_nat_complex @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_260_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_261_ball__imageD,axiom,
! [F: rat > real > real,A: set_rat,P: ( real > real ) > $o] :
( ! [X2: real > real] :
( ( member_real_real @ X2 @ ( image_rat_real_real @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: rat] :
( ( member_rat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_262_ball__imageD,axiom,
! [F: nat > literal,A: set_nat,P: literal > $o] :
( ! [X2: literal] :
( ( member_literal @ X2 @ ( image_nat_literal @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_263_ball__imageD,axiom,
! [F: ( real > real ) > real,A: set_real_real,P: real > $o] :
( ! [X2: real] :
( ( member_real @ X2 @ ( image_real_real_real @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: real > real] :
( ( member_real_real @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_264_ball__imageD,axiom,
! [F: ( real > real ) > nat,A: set_real_real,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_real_real_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: real > real] :
( ( member_real_real @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_265_ball__imageD,axiom,
! [F: ( real > real ) > real > real,A: set_real_real,P: ( real > real ) > $o] :
( ! [X2: real > real] :
( ( member_real_real @ X2 @ ( image_745864523092522741l_real @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: real > real] :
( ( member_real_real @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_266_image__cong,axiom,
! [M2: set_real_real,N2: set_real_real,F: ( real > real ) > real,G: ( real > real ) > real] :
( ( M2 = N2 )
=> ( ! [X2: real > real] :
( ( member_real_real @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_real_real_real @ F @ M2 )
= ( image_real_real_real @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_267_image__cong,axiom,
! [M2: set_real_real,N2: set_real_real,F: ( real > real ) > nat,G: ( real > real ) > nat] :
( ( M2 = N2 )
=> ( ! [X2: real > real] :
( ( member_real_real @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_real_real_nat @ F @ M2 )
= ( image_real_real_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_268_image__cong,axiom,
! [M2: set_real_real,N2: set_real_real,F: ( real > real ) > real > real,G: ( real > real ) > real > real] :
( ( M2 = N2 )
=> ( ! [X2: real > real] :
( ( member_real_real @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_745864523092522741l_real @ F @ M2 )
= ( image_745864523092522741l_real @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_269_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > real > real,G: nat > real > real] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_real_real @ F @ M2 )
= ( image_nat_real_real @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_270_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > rat,G: nat > rat] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_rat @ F @ M2 )
= ( image_nat_rat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_271_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > real,G: nat > real] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_real @ F @ M2 )
= ( image_nat_real @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_272_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > complex,G: nat > complex] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_complex @ F @ M2 )
= ( image_nat_complex @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_273_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F @ M2 )
= ( image_nat_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_274_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > literal,G: nat > literal] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_literal @ F @ M2 )
= ( image_nat_literal @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_275_image__cong,axiom,
! [M2: set_rat,N2: set_rat,F: rat > real > real,G: rat > real > real] :
( ( M2 = N2 )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_rat_real_real @ F @ M2 )
= ( image_rat_real_real @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_276_bex__imageD,axiom,
! [F: nat > real > real,A: set_nat,P: ( real > real ) > $o] :
( ? [X5: real > real] :
( ( member_real_real @ X5 @ ( image_nat_real_real @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_277_bex__imageD,axiom,
! [F: nat > rat,A: set_nat,P: rat > $o] :
( ? [X5: rat] :
( ( member_rat @ X5 @ ( image_nat_rat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_278_bex__imageD,axiom,
! [F: nat > real,A: set_nat,P: real > $o] :
( ? [X5: real] :
( ( member_real @ X5 @ ( image_nat_real @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_279_bex__imageD,axiom,
! [F: nat > complex,A: set_nat,P: complex > $o] :
( ? [X5: complex] :
( ( member_complex @ X5 @ ( image_nat_complex @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_280_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_281_bex__imageD,axiom,
! [F: rat > real > real,A: set_rat,P: ( real > real ) > $o] :
( ? [X5: real > real] :
( ( member_real_real @ X5 @ ( image_rat_real_real @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: rat] :
( ( member_rat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_282_bex__imageD,axiom,
! [F: nat > literal,A: set_nat,P: literal > $o] :
( ? [X5: literal] :
( ( member_literal @ X5 @ ( image_nat_literal @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_283_bex__imageD,axiom,
! [F: ( real > real ) > real,A: set_real_real,P: real > $o] :
( ? [X5: real] :
( ( member_real @ X5 @ ( image_real_real_real @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: real > real] :
( ( member_real_real @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_284_bex__imageD,axiom,
! [F: ( real > real ) > nat,A: set_real_real,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_real_real_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: real > real] :
( ( member_real_real @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_285_bex__imageD,axiom,
! [F: ( real > real ) > real > real,A: set_real_real,P: ( real > real ) > $o] :
( ? [X5: real > real] :
( ( member_real_real @ X5 @ ( image_745864523092522741l_real @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: real > real] :
( ( member_real_real @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_286_image__iff,axiom,
! [Z: real > real,F: nat > real > real,A: set_nat] :
( ( member_real_real @ Z @ ( image_nat_real_real @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_287_image__iff,axiom,
! [Z: real > real,F: rat > real > real,A: set_rat] :
( ( member_real_real @ Z @ ( image_rat_real_real @ F @ A ) )
= ( ? [X3: rat] :
( ( member_rat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_288_image__iff,axiom,
! [Z: real > real,F: ( real > real ) > real > real,A: set_real_real] :
( ( member_real_real @ Z @ ( image_745864523092522741l_real @ F @ A ) )
= ( ? [X3: real > real] :
( ( member_real_real @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_289_image__iff,axiom,
! [Z: real,F: nat > real,A: set_nat] :
( ( member_real @ Z @ ( image_nat_real @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_290_image__iff,axiom,
! [Z: real,F: ( real > real ) > real,A: set_real_real] :
( ( member_real @ Z @ ( image_real_real_real @ F @ A ) )
= ( ? [X3: real > real] :
( ( member_real_real @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_291_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_292_image__iff,axiom,
! [Z: nat,F: ( real > real ) > nat,A: set_real_real] :
( ( member_nat @ Z @ ( image_real_real_nat @ F @ A ) )
= ( ? [X3: real > real] :
( ( member_real_real @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_293_image__iff,axiom,
! [Z: complex,F: nat > complex,A: set_nat] :
( ( member_complex @ Z @ ( image_nat_complex @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_294_image__iff,axiom,
! [Z: rat,F: nat > rat,A: set_nat] :
( ( member_rat @ Z @ ( image_nat_rat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_295_image__iff,axiom,
! [Z: literal,F: nat > literal,A: set_nat] :
( ( member_literal @ Z @ ( image_nat_literal @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_296_imageI,axiom,
! [X: real,A: set_real,F: real > real] :
( ( member_real @ X @ A )
=> ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A ) ) ) ).
% imageI
thf(fact_297_imageI,axiom,
! [X: real,A: set_real,F: real > nat] :
( ( member_real @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ A ) ) ) ).
% imageI
thf(fact_298_imageI,axiom,
! [X: real,A: set_real,F: real > complex] :
( ( member_real @ X @ A )
=> ( member_complex @ ( F @ X ) @ ( image_real_complex @ F @ A ) ) ) ).
% imageI
thf(fact_299_imageI,axiom,
! [X: real,A: set_real,F: real > rat] :
( ( member_real @ X @ A )
=> ( member_rat @ ( F @ X ) @ ( image_real_rat @ F @ A ) ) ) ).
% imageI
thf(fact_300_imageI,axiom,
! [X: real,A: set_real,F: real > literal] :
( ( member_real @ X @ A )
=> ( member_literal @ ( F @ X ) @ ( image_real_literal @ F @ A ) ) ) ).
% imageI
thf(fact_301_imageI,axiom,
! [X: nat,A: set_nat,F: nat > real] :
( ( member_nat @ X @ A )
=> ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ A ) ) ) ).
% imageI
thf(fact_302_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_303_imageI,axiom,
! [X: nat,A: set_nat,F: nat > complex] :
( ( member_nat @ X @ A )
=> ( member_complex @ ( F @ X ) @ ( image_nat_complex @ F @ A ) ) ) ).
% imageI
thf(fact_304_imageI,axiom,
! [X: nat,A: set_nat,F: nat > rat] :
( ( member_nat @ X @ A )
=> ( member_rat @ ( F @ X ) @ ( image_nat_rat @ F @ A ) ) ) ).
% imageI
thf(fact_305_imageI,axiom,
! [X: nat,A: set_nat,F: nat > literal] :
( ( member_nat @ X @ A )
=> ( member_literal @ ( F @ X ) @ ( image_nat_literal @ F @ A ) ) ) ).
% imageI
thf(fact_306_surj__from__nat,axiom,
( ( image_nat_literal @ from_nat_literal @ top_top_set_nat )
= top_top_set_literal ) ).
% surj_from_nat
thf(fact_307_surj__from__nat,axiom,
( ( image_nat_rat @ from_nat_rat @ top_top_set_nat )
= top_top_set_rat ) ).
% surj_from_nat
thf(fact_308_surj__from__nat,axiom,
( ( image_nat_nat @ from_nat_nat @ top_top_set_nat )
= top_top_set_nat ) ).
% surj_from_nat
thf(fact_309_nonneg,axiom,
! [I: nat,X5: real] : ( ord_less_eq_real @ zero_zero_real @ ( g @ I @ X5 ) ) ).
% nonneg
thf(fact_310_rat__denum,axiom,
? [F2: nat > rat] :
( ( image_nat_rat @ F2 @ top_top_set_nat )
= top_top_set_rat ) ).
% rat_denum
thf(fact_311_real__non__denum,axiom,
~ ? [F3: nat > real] :
( ( image_nat_real @ F3 @ top_top_set_nat )
= top_top_set_real ) ).
% real_non_denum
thf(fact_312_complex__non__denum,axiom,
~ ? [F3: nat > complex] :
( ( image_nat_complex @ F3 @ top_top_set_nat )
= top_top_set_complex ) ).
% complex_non_denum
thf(fact_313_Sup_OSUP__cong,axiom,
! [A: set_real_real,B3: set_real_real,C2: ( real > real ) > real,D: ( real > real ) > real,Sup: set_real > real] :
( ( A = B3 )
=> ( ! [X2: real > real] :
( ( member_real_real @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_real_real_real @ C2 @ A ) )
= ( Sup @ ( image_real_real_real @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_314_Sup_OSUP__cong,axiom,
! [A: set_real_real,B3: set_real_real,C2: ( real > real ) > nat,D: ( real > real ) > nat,Sup: set_nat > nat] :
( ( A = B3 )
=> ( ! [X2: real > real] :
( ( member_real_real @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_real_real_nat @ C2 @ A ) )
= ( Sup @ ( image_real_real_nat @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_315_Sup_OSUP__cong,axiom,
! [A: set_real_real,B3: set_real_real,C2: ( real > real ) > real > real,D: ( real > real ) > real > real,Sup: set_real_real > real > real] :
( ( A = B3 )
=> ( ! [X2: real > real] :
( ( member_real_real @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_745864523092522741l_real @ C2 @ A ) )
= ( Sup @ ( image_745864523092522741l_real @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_316_Sup_OSUP__cong,axiom,
! [A: set_nat,B3: set_nat,C2: nat > real > real,D: nat > real > real,Sup: set_real_real > real > real] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_real_real @ C2 @ A ) )
= ( Sup @ ( image_nat_real_real @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_317_Sup_OSUP__cong,axiom,
! [A: set_nat,B3: set_nat,C2: nat > rat,D: nat > rat,Sup: set_rat > rat] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_rat @ C2 @ A ) )
= ( Sup @ ( image_nat_rat @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_318_Sup_OSUP__cong,axiom,
! [A: set_nat,B3: set_nat,C2: nat > real,D: nat > real,Sup: set_real > real] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_real @ C2 @ A ) )
= ( Sup @ ( image_nat_real @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_319_Sup_OSUP__cong,axiom,
! [A: set_nat,B3: set_nat,C2: nat > complex,D: nat > complex,Sup: set_complex > complex] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_complex @ C2 @ A ) )
= ( Sup @ ( image_nat_complex @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_320_Sup_OSUP__cong,axiom,
! [A: set_nat,B3: set_nat,C2: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A ) )
= ( Sup @ ( image_nat_nat @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_321_Sup_OSUP__cong,axiom,
! [A: set_nat,B3: set_nat,C2: nat > literal,D: nat > literal,Sup: set_literal > literal] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_literal @ C2 @ A ) )
= ( Sup @ ( image_nat_literal @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_322_Sup_OSUP__cong,axiom,
! [A: set_rat,B3: set_rat,C2: rat > real > real,D: rat > real > real,Sup: set_real_real > real > real] :
( ( A = B3 )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_rat_real_real @ C2 @ A ) )
= ( Sup @ ( image_rat_real_real @ D @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_323_Inf_OINF__cong,axiom,
! [A: set_real_real,B3: set_real_real,C2: ( real > real ) > real,D: ( real > real ) > real,Inf: set_real > real] :
( ( A = B3 )
=> ( ! [X2: real > real] :
( ( member_real_real @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_real_real_real @ C2 @ A ) )
= ( Inf @ ( image_real_real_real @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_324_Inf_OINF__cong,axiom,
! [A: set_real_real,B3: set_real_real,C2: ( real > real ) > nat,D: ( real > real ) > nat,Inf: set_nat > nat] :
( ( A = B3 )
=> ( ! [X2: real > real] :
( ( member_real_real @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_real_real_nat @ C2 @ A ) )
= ( Inf @ ( image_real_real_nat @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_325_Inf_OINF__cong,axiom,
! [A: set_real_real,B3: set_real_real,C2: ( real > real ) > real > real,D: ( real > real ) > real > real,Inf: set_real_real > real > real] :
( ( A = B3 )
=> ( ! [X2: real > real] :
( ( member_real_real @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_745864523092522741l_real @ C2 @ A ) )
= ( Inf @ ( image_745864523092522741l_real @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_326_Inf_OINF__cong,axiom,
! [A: set_nat,B3: set_nat,C2: nat > real > real,D: nat > real > real,Inf: set_real_real > real > real] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_real_real @ C2 @ A ) )
= ( Inf @ ( image_nat_real_real @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_327_Inf_OINF__cong,axiom,
! [A: set_nat,B3: set_nat,C2: nat > rat,D: nat > rat,Inf: set_rat > rat] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_rat @ C2 @ A ) )
= ( Inf @ ( image_nat_rat @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_328_Inf_OINF__cong,axiom,
! [A: set_nat,B3: set_nat,C2: nat > real,D: nat > real,Inf: set_real > real] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_real @ C2 @ A ) )
= ( Inf @ ( image_nat_real @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_329_Inf_OINF__cong,axiom,
! [A: set_nat,B3: set_nat,C2: nat > complex,D: nat > complex,Inf: set_complex > complex] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_complex @ C2 @ A ) )
= ( Inf @ ( image_nat_complex @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_330_Inf_OINF__cong,axiom,
! [A: set_nat,B3: set_nat,C2: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A ) )
= ( Inf @ ( image_nat_nat @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_331_Inf_OINF__cong,axiom,
! [A: set_nat,B3: set_nat,C2: nat > literal,D: nat > literal,Inf: set_literal > literal] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_literal @ C2 @ A ) )
= ( Inf @ ( image_nat_literal @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_332_Inf_OINF__cong,axiom,
! [A: set_rat,B3: set_rat,C2: rat > real > real,D: rat > real > real,Inf: set_real_real > real > real] :
( ( A = B3 )
=> ( ! [X2: rat] :
( ( member_rat @ X2 @ B3 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_rat_real_real @ C2 @ A ) )
= ( Inf @ ( image_rat_real_real @ D @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_333_top__empty__eq,axiom,
( top_top_complex_o
= ( ^ [X3: complex] : ( member_complex @ X3 @ top_top_set_complex ) ) ) ).
% top_empty_eq
thf(fact_334_top__empty__eq,axiom,
( top_top_literal_o
= ( ^ [X3: literal] : ( member_literal @ X3 @ top_top_set_literal ) ) ) ).
% top_empty_eq
thf(fact_335_top__empty__eq,axiom,
( top_top_real_o
= ( ^ [X3: real] : ( member_real @ X3 @ top_top_set_real ) ) ) ).
% top_empty_eq
thf(fact_336_top__empty__eq,axiom,
( top_top_rat_o
= ( ^ [X3: rat] : ( member_rat @ X3 @ top_top_set_rat ) ) ) ).
% top_empty_eq
thf(fact_337_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_338_top__empty__eq,axiom,
( top_top_real_real_o
= ( ^ [X3: real > real] : ( member_real_real @ X3 @ top_to2071711978144146653l_real ) ) ) ).
% top_empty_eq
thf(fact_339_ex__gt__or__lt,axiom,
! [A2: real] :
? [B2: real] :
( ( ord_less_real @ A2 @ B2 )
| ( ord_less_real @ B2 @ A2 ) ) ).
% ex_gt_or_lt
thf(fact_340_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_341_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_342_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_343_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_344_order__refl,axiom,
! [X: real > real] : ( ord_le6948328307412524503l_real @ X @ X ) ).
% order_refl
thf(fact_345_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_346_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_347_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_348_dual__order_Orefl,axiom,
! [A2: real > real] : ( ord_le6948328307412524503l_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_349_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_350_nle__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B ) )
= ( ( ord_less_eq_real @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_351_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_352_nle__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_353_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_354_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_355_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_356_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
& ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_357_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_358_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real > real,Z3: real > real] : ( Y5 = Z3 ) )
= ( ^ [X3: real > real,Y2: real > real] :
( ( ord_le6948328307412524503l_real @ X3 @ Y2 )
& ( ord_le6948328307412524503l_real @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_359_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_360_ord__eq__le__trans,axiom,
! [A2: real,B: real,C: real] :
( ( A2 = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_361_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_362_ord__eq__le__trans,axiom,
! [A2: real > real,B: real > real,C: real > real] :
( ( A2 = B )
=> ( ( ord_le6948328307412524503l_real @ B @ C )
=> ( ord_le6948328307412524503l_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_363_ord__eq__le__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_364_ord__le__eq__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_365_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_366_ord__le__eq__trans,axiom,
! [A2: real > real,B: real > real,C: real > real] :
( ( ord_le6948328307412524503l_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_le6948328307412524503l_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_367_ord__le__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_368_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_369_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_370_order__antisym,axiom,
! [X: real > real,Y: real > real] :
( ( ord_le6948328307412524503l_real @ X @ Y )
=> ( ( ord_le6948328307412524503l_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_371_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_372_order_Otrans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_373_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_374_order_Otrans,axiom,
! [A2: real > real,B: real > real,C: real > real] :
( ( ord_le6948328307412524503l_real @ A2 @ B )
=> ( ( ord_le6948328307412524503l_real @ B @ C )
=> ( ord_le6948328307412524503l_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_375_order_Otrans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_376_order__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_377_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_378_order__trans,axiom,
! [X: real > real,Y: real > real,Z: real > real] :
( ( ord_le6948328307412524503l_real @ X @ Y )
=> ( ( ord_le6948328307412524503l_real @ Y @ Z )
=> ( ord_le6948328307412524503l_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_379_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_380_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_381_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_382_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_383_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_384_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_385_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real > real,Z3: real > real] : ( Y5 = Z3 ) )
= ( ^ [A4: real > real,B4: real > real] :
( ( ord_le6948328307412524503l_real @ B4 @ A4 )
& ( ord_le6948328307412524503l_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_386_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_387_dual__order_Oantisym,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_388_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_389_dual__order_Oantisym,axiom,
! [B: real > real,A2: real > real] :
( ( ord_le6948328307412524503l_real @ B @ A2 )
=> ( ( ord_le6948328307412524503l_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_390_dual__order_Oantisym,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_391_dual__order_Otrans,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_392_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_393_dual__order_Otrans,axiom,
! [B: real > real,A2: real > real,C: real > real] :
( ( ord_le6948328307412524503l_real @ B @ A2 )
=> ( ( ord_le6948328307412524503l_real @ C @ B )
=> ( ord_le6948328307412524503l_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_394_dual__order_Otrans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_395_antisym,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_396_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_397_antisym,axiom,
! [A2: real > real,B: real > real] :
( ( ord_le6948328307412524503l_real @ A2 @ B )
=> ( ( ord_le6948328307412524503l_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_398_antisym,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_399_le__funD,axiom,
! [F: real > real,G: real > real,X: real] :
( ( ord_le6948328307412524503l_real @ F @ G )
=> ( ord_less_eq_real @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_400_le__funE,axiom,
! [F: real > real,G: real > real,X: real] :
( ( ord_le6948328307412524503l_real @ F @ G )
=> ( ord_less_eq_real @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_401_le__funI,axiom,
! [F: real > real,G: real > real] :
( ! [X2: real] : ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_le6948328307412524503l_real @ F @ G ) ) ).
% le_funI
thf(fact_402_le__fun__def,axiom,
( ord_le6948328307412524503l_real
= ( ^ [F4: real > real,G2: real > real] :
! [X3: real] : ( ord_less_eq_real @ ( F4 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_403_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z3: real] : ( Y5 = Z3 ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_404_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_405_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real > real,Z3: real > real] : ( Y5 = Z3 ) )
= ( ^ [A4: real > real,B4: real > real] :
( ( ord_le6948328307412524503l_real @ A4 @ B4 )
& ( ord_le6948328307412524503l_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_406_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_407_order__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_408_order__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_409_order__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_410_order__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_411_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_412_order__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_413_order__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_414_order__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_415_order__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_416_order__subst1,axiom,
! [A2: real,F: ( real > real ) > real,B: real > real,C: real > real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_le6948328307412524503l_real @ B @ C )
=> ( ! [X2: real > real,Y3: real > real] :
( ( ord_le6948328307412524503l_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_417_order__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_418_order__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_419_order__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_420_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_421_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_422_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_423_order__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_424_order__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_425_order__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_426_order__subst2,axiom,
! [A2: real,B: real,F: real > real > real,C: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_le6948328307412524503l_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le6948328307412524503l_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le6948328307412524503l_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_427_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_428_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_429_order__eq__refl,axiom,
! [X: real > real,Y: real > real] :
( ( X = Y )
=> ( ord_le6948328307412524503l_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_430_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_431_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_432_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_433_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_434_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_435_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_436_ord__eq__le__subst,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_437_ord__eq__le__subst,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_438_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_439_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_440_ord__eq__le__subst,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_441_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_442_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_443_ord__eq__le__subst,axiom,
! [A2: real > real,F: real > real > real,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le6948328307412524503l_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le6948328307412524503l_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_444_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_445_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_446_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_447_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_448_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_449_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_450_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_451_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_452_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_453_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > real > real,C: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le6948328307412524503l_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le6948328307412524503l_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_454_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_455_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_456_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_457_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_458_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_459_order__antisym__conv,axiom,
! [Y: real > real,X: real > real] :
( ( ord_le6948328307412524503l_real @ Y @ X )
=> ( ( ord_le6948328307412524503l_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_460_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_461_complete__interval,axiom,
! [A2: real,B: real,P: real > $o] :
( ( ord_less_real @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: real] :
( ( ord_less_eq_real @ A2 @ C3 )
& ( ord_less_eq_real @ C3 @ B )
& ! [X5: real] :
( ( ( ord_less_eq_real @ A2 @ X5 )
& ( ord_less_real @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: real] :
( ! [X2: real] :
( ( ( ord_less_eq_real @ A2 @ X2 )
& ( ord_less_real @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_real @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_462_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A2 @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A2 @ X2 )
& ( ord_less_nat @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_463_complete__interval,axiom,
! [A2: int,B: int,P: int > $o] :
( ( ord_less_int @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A2 @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X5: int] :
( ( ( ord_less_eq_int @ A2 @ X5 )
& ( ord_less_int @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: int] :
( ! [X2: int] :
( ( ( ord_less_eq_int @ A2 @ X2 )
& ( ord_less_int @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_int @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_464_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_465_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_466_leD,axiom,
! [Y: real > real,X: real > real] :
( ( ord_le6948328307412524503l_real @ Y @ X )
=> ~ ( ord_less_real_real @ X @ Y ) ) ).
% leD
thf(fact_467_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_468_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_469_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_470_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_471_nless__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_real @ A2 @ B ) )
= ( ~ ( ord_less_eq_real @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_472_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_473_nless__le,axiom,
! [A2: real > real,B: real > real] :
( ( ~ ( ord_less_real_real @ A2 @ B ) )
= ( ~ ( ord_le6948328307412524503l_real @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_474_nless__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_int @ A2 @ B ) )
= ( ~ ( ord_less_eq_int @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_475_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_476_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_477_antisym__conv1,axiom,
! [X: real > real,Y: real > real] :
( ~ ( ord_less_real_real @ X @ Y )
=> ( ( ord_le6948328307412524503l_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_478_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_479_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_480_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_481_antisym__conv2,axiom,
! [X: real > real,Y: real > real] :
( ( ord_le6948328307412524503l_real @ X @ Y )
=> ( ( ~ ( ord_less_real_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_482_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_483_dense__ge,axiom,
! [Z: real,Y: real] :
( ! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_ge
thf(fact_484_dense__le,axiom,
! [Y: real,Z: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Z ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ).
% dense_le
thf(fact_485_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_486_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_487_less__le__not__le,axiom,
( ord_less_real_real
= ( ^ [X3: real > real,Y2: real > real] :
( ( ord_le6948328307412524503l_real @ X3 @ Y2 )
& ~ ( ord_le6948328307412524503l_real @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_488_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_489_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_490_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_491_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_492_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_493_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_494_order_Oorder__iff__strict,axiom,
( ord_le6948328307412524503l_real
= ( ^ [A4: real > real,B4: real > real] :
( ( ord_less_real_real @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_495_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_496_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_497_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_498_order_Ostrict__iff__order,axiom,
( ord_less_real_real
= ( ^ [A4: real > real,B4: real > real] :
( ( ord_le6948328307412524503l_real @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_499_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_500_order_Ostrict__trans1,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_501_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_502_order_Ostrict__trans1,axiom,
! [A2: real > real,B: real > real,C: real > real] :
( ( ord_le6948328307412524503l_real @ A2 @ B )
=> ( ( ord_less_real_real @ B @ C )
=> ( ord_less_real_real @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_503_order_Ostrict__trans1,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_504_order_Ostrict__trans2,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_505_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_506_order_Ostrict__trans2,axiom,
! [A2: real > real,B: real > real,C: real > real] :
( ( ord_less_real_real @ A2 @ B )
=> ( ( ord_le6948328307412524503l_real @ B @ C )
=> ( ord_less_real_real @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_507_order_Ostrict__trans2,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_508_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_509_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_510_order_Ostrict__iff__not,axiom,
( ord_less_real_real
= ( ^ [A4: real > real,B4: real > real] :
( ( ord_le6948328307412524503l_real @ A4 @ B4 )
& ~ ( ord_le6948328307412524503l_real @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_511_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_512_dense__ge__bounded,axiom,
! [Z: real,X: real,Y: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_513_dense__le__bounded,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_514_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_515_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_516_dual__order_Oorder__iff__strict,axiom,
( ord_le6948328307412524503l_real
= ( ^ [B4: real > real,A4: real > real] :
( ( ord_less_real_real @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_517_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_518_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_519_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_520_dual__order_Ostrict__iff__order,axiom,
( ord_less_real_real
= ( ^ [B4: real > real,A4: real > real] :
( ( ord_le6948328307412524503l_real @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_521_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_522_dual__order_Ostrict__trans1,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_523_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_524_dual__order_Ostrict__trans1,axiom,
! [B: real > real,A2: real > real,C: real > real] :
( ( ord_le6948328307412524503l_real @ B @ A2 )
=> ( ( ord_less_real_real @ C @ B )
=> ( ord_less_real_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_525_dual__order_Ostrict__trans1,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_526_dual__order_Ostrict__trans2,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_527_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_528_dual__order_Ostrict__trans2,axiom,
! [B: real > real,A2: real > real,C: real > real] :
( ( ord_less_real_real @ B @ A2 )
=> ( ( ord_le6948328307412524503l_real @ C @ B )
=> ( ord_less_real_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_529_dual__order_Ostrict__trans2,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_530_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_531_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_532_dual__order_Ostrict__iff__not,axiom,
( ord_less_real_real
= ( ^ [B4: real > real,A4: real > real] :
( ( ord_le6948328307412524503l_real @ B4 @ A4 )
& ~ ( ord_le6948328307412524503l_real @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_533_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_534_order_Ostrict__implies__order,axiom,
! [A2: real,B: real] :
( ( ord_less_real @ A2 @ B )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_535_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_536_order_Ostrict__implies__order,axiom,
! [A2: real > real,B: real > real] :
( ( ord_less_real_real @ A2 @ B )
=> ( ord_le6948328307412524503l_real @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_537_order_Ostrict__implies__order,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_538_dual__order_Ostrict__implies__order,axiom,
! [B: real,A2: real] :
( ( ord_less_real @ B @ A2 )
=> ( ord_less_eq_real @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_539_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_540_dual__order_Ostrict__implies__order,axiom,
! [B: real > real,A2: real > real] :
( ( ord_less_real_real @ B @ A2 )
=> ( ord_le6948328307412524503l_real @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_541_dual__order_Ostrict__implies__order,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_542_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_543_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_544_order__le__less,axiom,
( ord_le6948328307412524503l_real
= ( ^ [X3: real > real,Y2: real > real] :
( ( ord_less_real_real @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_545_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_546_order__less__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_547_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_548_order__less__le,axiom,
( ord_less_real_real
= ( ^ [X3: real > real,Y2: real > real] :
( ( ord_le6948328307412524503l_real @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_549_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_550_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_551_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_552_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_553_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_554_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_555_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_556_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_557_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_558_order__less__imp__le,axiom,
! [X: real > real,Y: real > real] :
( ( ord_less_real_real @ X @ Y )
=> ( ord_le6948328307412524503l_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_559_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_560_order__le__neq__trans,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_561_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_562_order__le__neq__trans,axiom,
! [A2: real > real,B: real > real] :
( ( ord_le6948328307412524503l_real @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_real_real @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_563_order__le__neq__trans,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_564_order__neq__le__trans,axiom,
! [A2: real,B: real] :
( ( A2 != B )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_real @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_565_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_566_order__neq__le__trans,axiom,
! [A2: real > real,B: real > real] :
( ( A2 != B )
=> ( ( ord_le6948328307412524503l_real @ A2 @ B )
=> ( ord_less_real_real @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_567_order__neq__le__trans,axiom,
! [A2: int,B: int] :
( ( A2 != B )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_568_order__le__less__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_569_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_570_order__le__less__trans,axiom,
! [X: real > real,Y: real > real,Z: real > real] :
( ( ord_le6948328307412524503l_real @ X @ Y )
=> ( ( ord_less_real_real @ Y @ Z )
=> ( ord_less_real_real @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_571_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_572_order__less__le__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_573_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_574_order__less__le__trans,axiom,
! [X: real > real,Y: real > real,Z: real > real] :
( ( ord_less_real_real @ X @ Y )
=> ( ( ord_le6948328307412524503l_real @ Y @ Z )
=> ( ord_less_real_real @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_575_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_576_order__le__less__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_577_order__le__less__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_578_order__le__less__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_579_order__le__less__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_580_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_581_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_582_order__le__less__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_583_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_584_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_585_order__le__less__subst1,axiom,
! [A2: real > real,F: real > real > real,B: real,C: real] :
( ( ord_le6948328307412524503l_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_586_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_587_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_588_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_589_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_590_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_591_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_592_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_593_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_594_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_595_order__le__less__subst2,axiom,
! [A2: real,B: real,F: real > real > real,C: real > real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_real_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le6948328307412524503l_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_596_order__less__le__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_597_order__less__le__subst1,axiom,
! [A2: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_598_order__less__le__subst1,axiom,
! [A2: int,F: real > int,B: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_599_order__less__le__subst1,axiom,
! [A2: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_600_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_601_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_602_order__less__le__subst1,axiom,
! [A2: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_603_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_604_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_605_order__less__le__subst1,axiom,
! [A2: real > real,F: real > real > real,B: real,C: real] :
( ( ord_less_real_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le6948328307412524503l_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_606_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_607_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_608_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_609_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_610_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_611_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_612_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_613_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_614_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_615_order__less__le__subst2,axiom,
! [A2: real,B: real,F: real > real > real,C: real > real] :
( ( ord_less_real @ A2 @ B )
=> ( ( ord_le6948328307412524503l_real @ ( F @ B ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_616_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_617_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_618_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_619_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_620_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_621_order__le__imp__less__or__eq,axiom,
! [X: real > real,Y: real > real] :
( ( ord_le6948328307412524503l_real @ X @ Y )
=> ( ( ord_less_real_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_622_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_623_top__greatest,axiom,
! [A2: set_complex] : ( ord_le211207098394363844omplex @ A2 @ top_top_set_complex ) ).
% top_greatest
thf(fact_624_top__greatest,axiom,
! [A2: set_literal] : ( ord_le7307670543136651348iteral @ A2 @ top_top_set_literal ) ).
% top_greatest
thf(fact_625_top__greatest,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ top_top_set_real ) ).
% top_greatest
thf(fact_626_top__greatest,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ A2 @ top_top_set_rat ) ).
% top_greatest
thf(fact_627_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_628_top__greatest,axiom,
! [A2: set_real_real] : ( ord_le4198349162570665613l_real @ A2 @ top_to2071711978144146653l_real ) ).
% top_greatest
thf(fact_629_top_Oextremum__unique,axiom,
! [A2: set_complex] :
( ( ord_le211207098394363844omplex @ top_top_set_complex @ A2 )
= ( A2 = top_top_set_complex ) ) ).
% top.extremum_unique
thf(fact_630_top_Oextremum__unique,axiom,
! [A2: set_literal] :
( ( ord_le7307670543136651348iteral @ top_top_set_literal @ A2 )
= ( A2 = top_top_set_literal ) ) ).
% top.extremum_unique
thf(fact_631_top_Oextremum__unique,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
= ( A2 = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_632_top_Oextremum__unique,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ top_top_set_rat @ A2 )
= ( A2 = top_top_set_rat ) ) ).
% top.extremum_unique
thf(fact_633_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_634_top_Oextremum__unique,axiom,
! [A2: set_real_real] :
( ( ord_le4198349162570665613l_real @ top_to2071711978144146653l_real @ A2 )
= ( A2 = top_to2071711978144146653l_real ) ) ).
% top.extremum_unique
thf(fact_635_top_Oextremum__uniqueI,axiom,
! [A2: set_complex] :
( ( ord_le211207098394363844omplex @ top_top_set_complex @ A2 )
=> ( A2 = top_top_set_complex ) ) ).
% top.extremum_uniqueI
thf(fact_636_top_Oextremum__uniqueI,axiom,
! [A2: set_literal] :
( ( ord_le7307670543136651348iteral @ top_top_set_literal @ A2 )
=> ( A2 = top_top_set_literal ) ) ).
% top.extremum_uniqueI
thf(fact_637_top_Oextremum__uniqueI,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
=> ( A2 = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_638_top_Oextremum__uniqueI,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ top_top_set_rat @ A2 )
=> ( A2 = top_top_set_rat ) ) ).
% top.extremum_uniqueI
thf(fact_639_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_640_top_Oextremum__uniqueI,axiom,
! [A2: set_real_real] :
( ( ord_le4198349162570665613l_real @ top_to2071711978144146653l_real @ A2 )
=> ( A2 = top_to2071711978144146653l_real ) ) ).
% top.extremum_uniqueI
thf(fact_641_not__gr__zero,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_642_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_643_g__le__f,axiom,
! [I2: nat,X: real] : ( ord_less_eq_real @ ( g @ I2 @ X ) @ ( f @ X ) ) ).
% g_le_f
thf(fact_644_surj__nat__to__rat__surj,axiom,
( ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat )
= top_top_set_rat ) ).
% surj_nat_to_rat_surj
thf(fact_645_g__le,axiom,
! [I2: nat,J: nat,X: real] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_real @ ( g @ I2 @ X ) @ ( g @ J @ X ) ) ) ).
% g_le
thf(fact_646__C0_C,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( f @ X ) ) ).
% "0"
thf(fact_647_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_648_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_649_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_650_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_651_ex__least__nat__le,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N3 )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ~ ( P @ I ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_652_less__imp__le__nat,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% less_imp_le_nat
thf(fact_653_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_654_less__or__eq__imp__le,axiom,
! [M3: nat,N3: nat] :
( ( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% less_or_eq_imp_le
thf(fact_655_less__fun__def,axiom,
( ord_less_real_real
= ( ^ [F4: real > real,G2: real > real] :
( ( ord_le6948328307412524503l_real @ F4 @ G2 )
& ~ ( ord_le6948328307412524503l_real @ G2 @ F4 ) ) ) ) ).
% less_fun_def
thf(fact_656_le__neq__implies__less,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( M3 != N3 )
=> ( ord_less_nat @ M3 @ N3 ) ) ) ).
% le_neq_implies_less
thf(fact_657_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_658_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_659_nat__le__linear,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
| ( ord_less_eq_nat @ N3 @ M3 ) ) ).
% nat_le_linear
thf(fact_660_le__antisym,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M3 )
=> ( M3 = N3 ) ) ) ).
% le_antisym
thf(fact_661_eq__imp__le,axiom,
! [M3: nat,N3: nat] :
( ( M3 = N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% eq_imp_le
thf(fact_662_le__trans,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_663_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_664_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_665_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_666_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_667_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_668_subset__UNIV,axiom,
! [A: set_complex] : ( ord_le211207098394363844omplex @ A @ top_top_set_complex ) ).
% subset_UNIV
thf(fact_669_subset__UNIV,axiom,
! [A: set_literal] : ( ord_le7307670543136651348iteral @ A @ top_top_set_literal ) ).
% subset_UNIV
thf(fact_670_subset__UNIV,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% subset_UNIV
thf(fact_671_subset__UNIV,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ A @ top_top_set_rat ) ).
% subset_UNIV
thf(fact_672_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_673_subset__UNIV,axiom,
! [A: set_real_real] : ( ord_le4198349162570665613l_real @ A @ top_to2071711978144146653l_real ) ).
% subset_UNIV
thf(fact_674_subset__image__iff,axiom,
! [B3: set_real_real,F: nat > real > real,A: set_nat] :
( ( ord_le4198349162570665613l_real @ B3 @ ( image_nat_real_real @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B3
= ( image_nat_real_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_675_subset__image__iff,axiom,
! [B3: set_rat,F: nat > rat,A: set_nat] :
( ( ord_less_eq_set_rat @ B3 @ ( image_nat_rat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B3
= ( image_nat_rat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_676_subset__image__iff,axiom,
! [B3: set_real,F: nat > real,A: set_nat] :
( ( ord_less_eq_set_real @ B3 @ ( image_nat_real @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B3
= ( image_nat_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_677_subset__image__iff,axiom,
! [B3: set_complex,F: nat > complex,A: set_nat] :
( ( ord_le211207098394363844omplex @ B3 @ ( image_nat_complex @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B3
= ( image_nat_complex @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_678_subset__image__iff,axiom,
! [B3: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B3
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_679_subset__image__iff,axiom,
! [B3: set_real_real,F: rat > real > real,A: set_rat] :
( ( ord_le4198349162570665613l_real @ B3 @ ( image_rat_real_real @ F @ A ) )
= ( ? [AA: set_rat] :
( ( ord_less_eq_set_rat @ AA @ A )
& ( B3
= ( image_rat_real_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_680_subset__image__iff,axiom,
! [B3: set_literal,F: nat > literal,A: set_nat] :
( ( ord_le7307670543136651348iteral @ B3 @ ( image_nat_literal @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B3
= ( image_nat_literal @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_681_subset__image__iff,axiom,
! [B3: set_real,F: ( real > real ) > real,A: set_real_real] :
( ( ord_less_eq_set_real @ B3 @ ( image_real_real_real @ F @ A ) )
= ( ? [AA: set_real_real] :
( ( ord_le4198349162570665613l_real @ AA @ A )
& ( B3
= ( image_real_real_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_682_subset__image__iff,axiom,
! [B3: set_nat,F: ( real > real ) > nat,A: set_real_real] :
( ( ord_less_eq_set_nat @ B3 @ ( image_real_real_nat @ F @ A ) )
= ( ? [AA: set_real_real] :
( ( ord_le4198349162570665613l_real @ AA @ A )
& ( B3
= ( image_real_real_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_683_subset__image__iff,axiom,
! [B3: set_real_real,F: ( real > real ) > real > real,A: set_real_real] :
( ( ord_le4198349162570665613l_real @ B3 @ ( image_745864523092522741l_real @ F @ A ) )
= ( ? [AA: set_real_real] :
( ( ord_le4198349162570665613l_real @ AA @ A )
& ( B3
= ( image_745864523092522741l_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_684_image__subset__iff,axiom,
! [F: nat > real > real,A: set_nat,B3: set_real_real] :
( ( ord_le4198349162570665613l_real @ ( image_nat_real_real @ F @ A ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_real_real @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_685_image__subset__iff,axiom,
! [F: rat > real > real,A: set_rat,B3: set_real_real] :
( ( ord_le4198349162570665613l_real @ ( image_rat_real_real @ F @ A ) @ B3 )
= ( ! [X3: rat] :
( ( member_rat @ X3 @ A )
=> ( member_real_real @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_686_image__subset__iff,axiom,
! [F: ( real > real ) > real > real,A: set_real_real,B3: set_real_real] :
( ( ord_le4198349162570665613l_real @ ( image_745864523092522741l_real @ F @ A ) @ B3 )
= ( ! [X3: real > real] :
( ( member_real_real @ X3 @ A )
=> ( member_real_real @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_687_image__subset__iff,axiom,
! [F: nat > real,A: set_nat,B3: set_real] :
( ( ord_less_eq_set_real @ ( image_nat_real @ F @ A ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_real @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_688_image__subset__iff,axiom,
! [F: ( real > real ) > real,A: set_real_real,B3: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real_real @ F @ A ) @ B3 )
= ( ! [X3: real > real] :
( ( member_real_real @ X3 @ A )
=> ( member_real @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_689_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_690_image__subset__iff,axiom,
! [F: ( real > real ) > nat,A: set_real_real,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_real_real_nat @ F @ A ) @ B3 )
= ( ! [X3: real > real] :
( ( member_real_real @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_691_image__subset__iff,axiom,
! [F: nat > complex,A: set_nat,B3: set_complex] :
( ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_complex @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_692_image__subset__iff,axiom,
! [F: nat > rat,A: set_nat,B3: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_rat @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_693_image__subset__iff,axiom,
! [F: nat > literal,A: set_nat,B3: set_literal] :
( ( ord_le7307670543136651348iteral @ ( image_nat_literal @ F @ A ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_literal @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_694_subset__imageE,axiom,
! [B3: set_real_real,F: nat > real > real,A: set_nat] :
( ( ord_le4198349162570665613l_real @ B3 @ ( image_nat_real_real @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B3
!= ( image_nat_real_real @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_695_subset__imageE,axiom,
! [B3: set_rat,F: nat > rat,A: set_nat] :
( ( ord_less_eq_set_rat @ B3 @ ( image_nat_rat @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B3
!= ( image_nat_rat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_696_subset__imageE,axiom,
! [B3: set_real,F: nat > real,A: set_nat] :
( ( ord_less_eq_set_real @ B3 @ ( image_nat_real @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B3
!= ( image_nat_real @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_697_subset__imageE,axiom,
! [B3: set_complex,F: nat > complex,A: set_nat] :
( ( ord_le211207098394363844omplex @ B3 @ ( image_nat_complex @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B3
!= ( image_nat_complex @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_698_subset__imageE,axiom,
! [B3: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B3
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_699_subset__imageE,axiom,
! [B3: set_real_real,F: rat > real > real,A: set_rat] :
( ( ord_le4198349162570665613l_real @ B3 @ ( image_rat_real_real @ F @ A ) )
=> ~ ! [C4: set_rat] :
( ( ord_less_eq_set_rat @ C4 @ A )
=> ( B3
!= ( image_rat_real_real @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_700_subset__imageE,axiom,
! [B3: set_literal,F: nat > literal,A: set_nat] :
( ( ord_le7307670543136651348iteral @ B3 @ ( image_nat_literal @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B3
!= ( image_nat_literal @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_701_subset__imageE,axiom,
! [B3: set_real,F: ( real > real ) > real,A: set_real_real] :
( ( ord_less_eq_set_real @ B3 @ ( image_real_real_real @ F @ A ) )
=> ~ ! [C4: set_real_real] :
( ( ord_le4198349162570665613l_real @ C4 @ A )
=> ( B3
!= ( image_real_real_real @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_702_subset__imageE,axiom,
! [B3: set_nat,F: ( real > real ) > nat,A: set_real_real] :
( ( ord_less_eq_set_nat @ B3 @ ( image_real_real_nat @ F @ A ) )
=> ~ ! [C4: set_real_real] :
( ( ord_le4198349162570665613l_real @ C4 @ A )
=> ( B3
!= ( image_real_real_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_703_subset__imageE,axiom,
! [B3: set_real_real,F: ( real > real ) > real > real,A: set_real_real] :
( ( ord_le4198349162570665613l_real @ B3 @ ( image_745864523092522741l_real @ F @ A ) )
=> ~ ! [C4: set_real_real] :
( ( ord_le4198349162570665613l_real @ C4 @ A )
=> ( B3
!= ( image_745864523092522741l_real @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_704_image__subsetI,axiom,
! [A: set_real,F: real > real,B3: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_705_image__subsetI,axiom,
! [A: set_real,F: real > nat,B3: set_nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_706_image__subsetI,axiom,
! [A: set_real,F: real > complex,B3: set_complex] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_complex @ ( F @ X2 ) @ B3 ) )
=> ( ord_le211207098394363844omplex @ ( image_real_complex @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_707_image__subsetI,axiom,
! [A: set_real,F: real > rat,B3: set_rat] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_rat @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_rat @ ( image_real_rat @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_708_image__subsetI,axiom,
! [A: set_real,F: real > literal,B3: set_literal] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_literal @ ( F @ X2 ) @ B3 ) )
=> ( ord_le7307670543136651348iteral @ ( image_real_literal @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_709_image__subsetI,axiom,
! [A: set_nat,F: nat > real,B3: set_real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_real @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_710_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B3: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_711_image__subsetI,axiom,
! [A: set_nat,F: nat > complex,B3: set_complex] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_complex @ ( F @ X2 ) @ B3 ) )
=> ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_712_image__subsetI,axiom,
! [A: set_nat,F: nat > rat,B3: set_rat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_rat @ ( F @ X2 ) @ B3 ) )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_713_image__subsetI,axiom,
! [A: set_nat,F: nat > literal,B3: set_literal] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_literal @ ( F @ X2 ) @ B3 ) )
=> ( ord_le7307670543136651348iteral @ ( image_nat_literal @ F @ A ) @ B3 ) ) ).
% image_subsetI
thf(fact_714_image__mono,axiom,
! [A: set_nat,B3: set_nat,F: nat > real > real] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_le4198349162570665613l_real @ ( image_nat_real_real @ F @ A ) @ ( image_nat_real_real @ F @ B3 ) ) ) ).
% image_mono
thf(fact_715_image__mono,axiom,
! [A: set_nat,B3: set_nat,F: nat > rat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A ) @ ( image_nat_rat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_716_image__mono,axiom,
! [A: set_nat,B3: set_nat,F: nat > real] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A ) @ ( image_nat_real @ F @ B3 ) ) ) ).
% image_mono
thf(fact_717_image__mono,axiom,
! [A: set_nat,B3: set_nat,F: nat > complex] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A ) @ ( image_nat_complex @ F @ B3 ) ) ) ).
% image_mono
thf(fact_718_image__mono,axiom,
! [A: set_nat,B3: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_719_image__mono,axiom,
! [A: set_rat,B3: set_rat,F: rat > real > real] :
( ( ord_less_eq_set_rat @ A @ B3 )
=> ( ord_le4198349162570665613l_real @ ( image_rat_real_real @ F @ A ) @ ( image_rat_real_real @ F @ B3 ) ) ) ).
% image_mono
thf(fact_720_image__mono,axiom,
! [A: set_nat,B3: set_nat,F: nat > literal] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_le7307670543136651348iteral @ ( image_nat_literal @ F @ A ) @ ( image_nat_literal @ F @ B3 ) ) ) ).
% image_mono
thf(fact_721_image__mono,axiom,
! [A: set_real_real,B3: set_real_real,F: ( real > real ) > real] :
( ( ord_le4198349162570665613l_real @ A @ B3 )
=> ( ord_less_eq_set_real @ ( image_real_real_real @ F @ A ) @ ( image_real_real_real @ F @ B3 ) ) ) ).
% image_mono
thf(fact_722_image__mono,axiom,
! [A: set_real_real,B3: set_real_real,F: ( real > real ) > nat] :
( ( ord_le4198349162570665613l_real @ A @ B3 )
=> ( ord_less_eq_set_nat @ ( image_real_real_nat @ F @ A ) @ ( image_real_real_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_723_image__mono,axiom,
! [A: set_real_real,B3: set_real_real,F: ( real > real ) > real > real] :
( ( ord_le4198349162570665613l_real @ A @ B3 )
=> ( ord_le4198349162570665613l_real @ ( image_745864523092522741l_real @ F @ A ) @ ( image_745864523092522741l_real @ F @ B3 ) ) ) ).
% image_mono
thf(fact_724_range__subsetD,axiom,
! [F: complex > real,B3: set_real,I2: complex] :
( ( ord_less_eq_set_real @ ( image_complex_real @ F @ top_top_set_complex ) @ B3 )
=> ( member_real @ ( F @ I2 ) @ B3 ) ) ).
% range_subsetD
thf(fact_725_range__subsetD,axiom,
! [F: complex > nat,B3: set_nat,I2: complex] :
( ( ord_less_eq_set_nat @ ( image_complex_nat @ F @ top_top_set_complex ) @ B3 )
=> ( member_nat @ ( F @ I2 ) @ B3 ) ) ).
% range_subsetD
thf(fact_726_range__subsetD,axiom,
! [F: complex > complex,B3: set_complex,I2: complex] :
( ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ F @ top_top_set_complex ) @ B3 )
=> ( member_complex @ ( F @ I2 ) @ B3 ) ) ).
% range_subsetD
thf(fact_727_range__subsetD,axiom,
! [F: complex > rat,B3: set_rat,I2: complex] :
( ( ord_less_eq_set_rat @ ( image_complex_rat @ F @ top_top_set_complex ) @ B3 )
=> ( member_rat @ ( F @ I2 ) @ B3 ) ) ).
% range_subsetD
thf(fact_728_range__subsetD,axiom,
! [F: complex > literal,B3: set_literal,I2: complex] :
( ( ord_le7307670543136651348iteral @ ( image_8841419608667285983iteral @ F @ top_top_set_complex ) @ B3 )
=> ( member_literal @ ( F @ I2 ) @ B3 ) ) ).
% range_subsetD
thf(fact_729_range__subsetD,axiom,
! [F: literal > real,B3: set_real,I2: literal] :
( ( ord_less_eq_set_real @ ( image_literal_real @ F @ top_top_set_literal ) @ B3 )
=> ( member_real @ ( F @ I2 ) @ B3 ) ) ).
% range_subsetD
thf(fact_730_range__subsetD,axiom,
! [F: literal > nat,B3: set_nat,I2: literal] :
( ( ord_less_eq_set_nat @ ( image_literal_nat @ F @ top_top_set_literal ) @ B3 )
=> ( member_nat @ ( F @ I2 ) @ B3 ) ) ).
% range_subsetD
thf(fact_731_range__subsetD,axiom,
! [F: literal > complex,B3: set_complex,I2: literal] :
( ( ord_le211207098394363844omplex @ ( image_5274195009022015549omplex @ F @ top_top_set_literal ) @ B3 )
=> ( member_complex @ ( F @ I2 ) @ B3 ) ) ).
% range_subsetD
thf(fact_732_range__subsetD,axiom,
! [F: literal > rat,B3: set_rat,I2: literal] :
( ( ord_less_eq_set_rat @ ( image_literal_rat @ F @ top_top_set_literal ) @ B3 )
=> ( member_rat @ ( F @ I2 ) @ B3 ) ) ).
% range_subsetD
thf(fact_733_range__subsetD,axiom,
! [F: literal > literal,B3: set_literal,I2: literal] :
( ( ord_le7307670543136651348iteral @ ( image_8195128725298311301iteral @ F @ top_top_set_literal ) @ B3 )
=> ( member_literal @ ( F @ I2 ) @ B3 ) ) ).
% range_subsetD
thf(fact_734_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_735_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_736_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_737_complete__real,axiom,
! [S: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S )
=> ( ? [Z4: real] :
! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ? [Y3: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S )
=> ( ord_less_eq_real @ X5 @ Y3 ) )
& ! [Z4: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z4 ) )
=> ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_738_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_739_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_740_zero__less__iff__neq__zero,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( N3 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_741_gr__implies__not__zero,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_742_not__less__zero,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_743_gr__zeroI,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr_zeroI
thf(fact_744_bgauge__existence__lemma,axiom,
! [S2: set_real_real,Q: real > ( real > real ) > $o] :
( ( ! [X3: real > real] :
( ( member_real_real @ X3 @ S2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( Q @ D3 @ X3 ) ) ) )
= ( ! [X3: real > real] :
? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( ( member_real_real @ X3 @ S2 )
=> ( Q @ D3 @ X3 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_745_bgauge__existence__lemma,axiom,
! [S2: set_real,Q: real > real > $o] :
( ( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( Q @ D3 @ X3 ) ) ) )
= ( ! [X3: real] :
? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( ( member_real @ X3 @ S2 )
=> ( Q @ D3 @ X3 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_746_bgauge__existence__lemma,axiom,
! [S2: set_nat,Q: real > nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( Q @ D3 @ X3 ) ) ) )
= ( ! [X3: nat] :
? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( ( member_nat @ X3 @ S2 )
=> ( Q @ D3 @ X3 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_747_bgauge__existence__lemma,axiom,
! [S2: set_complex,Q: real > complex > $o] :
( ( ! [X3: complex] :
( ( member_complex @ X3 @ S2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( Q @ D3 @ X3 ) ) ) )
= ( ! [X3: complex] :
? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( ( member_complex @ X3 @ S2 )
=> ( Q @ D3 @ X3 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_748_bgauge__existence__lemma,axiom,
! [S2: set_rat,Q: real > rat > $o] :
( ( ! [X3: rat] :
( ( member_rat @ X3 @ S2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( Q @ D3 @ X3 ) ) ) )
= ( ! [X3: rat] :
? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( ( member_rat @ X3 @ S2 )
=> ( Q @ D3 @ X3 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_749_bgauge__existence__lemma,axiom,
! [S2: set_literal,Q: real > literal > $o] :
( ( ! [X3: literal] :
( ( member_literal @ X3 @ S2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( Q @ D3 @ X3 ) ) ) )
= ( ! [X3: literal] :
? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( ( member_literal @ X3 @ S2 )
=> ( Q @ D3 @ X3 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_750_g,axiom,
monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat @ ord_le6948328307412524503l_real @ g ).
% g
thf(fact_751_seq__mono__lemma,axiom,
! [M3: nat,D4: nat > real,E2: nat > real] :
( ! [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ord_less_real @ ( D4 @ N4 ) @ ( E2 @ N4 ) ) )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ord_less_eq_real @ ( E2 @ N4 ) @ ( E2 @ M3 ) ) )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ord_less_real @ ( D4 @ N5 ) @ ( E2 @ M3 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_752_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_753_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_754_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_755_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
& ~ ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_756_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_757_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_758_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_759_pinf_I6_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ Z2 @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% pinf(6)
thf(fact_760_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_761_pinf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_762_pinf_I8_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ Z2 @ X5 )
=> ( ord_less_eq_real @ T @ X5 ) ) ).
% pinf(8)
thf(fact_763_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_764_pinf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_765_minf_I6_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z2 )
=> ( ord_less_eq_real @ X5 @ T ) ) ).
% minf(6)
thf(fact_766_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_767_minf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_768_minf_I8_J,axiom,
! [T: real] :
? [Z2: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z2 )
=> ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% minf(8)
thf(fact_769_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_770_minf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_771_subsetI,axiom,
! [A: set_real_real,B3: set_real_real] :
( ! [X2: real > real] :
( ( member_real_real @ X2 @ A )
=> ( member_real_real @ X2 @ B3 ) )
=> ( ord_le4198349162570665613l_real @ A @ B3 ) ) ).
% subsetI
thf(fact_772_subsetI,axiom,
! [A: set_real,B3: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( member_real @ X2 @ B3 ) )
=> ( ord_less_eq_set_real @ A @ B3 ) ) ).
% subsetI
thf(fact_773_subsetI,axiom,
! [A: set_nat,B3: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B3 ) )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% subsetI
thf(fact_774_subsetI,axiom,
! [A: set_complex,B3: set_complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A )
=> ( member_complex @ X2 @ B3 ) )
=> ( ord_le211207098394363844omplex @ A @ B3 ) ) ).
% subsetI
thf(fact_775_subsetI,axiom,
! [A: set_rat,B3: set_rat] :
( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( member_rat @ X2 @ B3 ) )
=> ( ord_less_eq_set_rat @ A @ B3 ) ) ).
% subsetI
thf(fact_776_subsetI,axiom,
! [A: set_literal,B3: set_literal] :
( ! [X2: literal] :
( ( member_literal @ X2 @ A )
=> ( member_literal @ X2 @ B3 ) )
=> ( ord_le7307670543136651348iteral @ A @ B3 ) ) ).
% subsetI
thf(fact_777_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_778_neq0__conv,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% neq0_conv
thf(fact_779_less__nat__zero__code,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_780_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_781_gr0I,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr0I
thf(fact_782_not__gr0,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_783_in__mono,axiom,
! [A: set_real_real,B3: set_real_real,X: real > real] :
( ( ord_le4198349162570665613l_real @ A @ B3 )
=> ( ( member_real_real @ X @ A )
=> ( member_real_real @ X @ B3 ) ) ) ).
% in_mono
thf(fact_784_in__mono,axiom,
! [A: set_real,B3: set_real,X: real] :
( ( ord_less_eq_set_real @ A @ B3 )
=> ( ( member_real @ X @ A )
=> ( member_real @ X @ B3 ) ) ) ).
% in_mono
thf(fact_785_in__mono,axiom,
! [A: set_nat,B3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_786_in__mono,axiom,
! [A: set_complex,B3: set_complex,X: complex] :
( ( ord_le211207098394363844omplex @ A @ B3 )
=> ( ( member_complex @ X @ A )
=> ( member_complex @ X @ B3 ) ) ) ).
% in_mono
thf(fact_787_in__mono,axiom,
! [A: set_rat,B3: set_rat,X: rat] :
( ( ord_less_eq_set_rat @ A @ B3 )
=> ( ( member_rat @ X @ A )
=> ( member_rat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_788_in__mono,axiom,
! [A: set_literal,B3: set_literal,X: literal] :
( ( ord_le7307670543136651348iteral @ A @ B3 )
=> ( ( member_literal @ X @ A )
=> ( member_literal @ X @ B3 ) ) ) ).
% in_mono
thf(fact_789_subsetD,axiom,
! [A: set_real_real,B3: set_real_real,C: real > real] :
( ( ord_le4198349162570665613l_real @ A @ B3 )
=> ( ( member_real_real @ C @ A )
=> ( member_real_real @ C @ B3 ) ) ) ).
% subsetD
thf(fact_790_subsetD,axiom,
! [A: set_real,B3: set_real,C: real] :
( ( ord_less_eq_set_real @ A @ B3 )
=> ( ( member_real @ C @ A )
=> ( member_real @ C @ B3 ) ) ) ).
% subsetD
thf(fact_791_subsetD,axiom,
! [A: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_792_subsetD,axiom,
! [A: set_complex,B3: set_complex,C: complex] :
( ( ord_le211207098394363844omplex @ A @ B3 )
=> ( ( member_complex @ C @ A )
=> ( member_complex @ C @ B3 ) ) ) ).
% subsetD
thf(fact_793_subsetD,axiom,
! [A: set_rat,B3: set_rat,C: rat] :
( ( ord_less_eq_set_rat @ A @ B3 )
=> ( ( member_rat @ C @ A )
=> ( member_rat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_794_subsetD,axiom,
! [A: set_literal,B3: set_literal,C: literal] :
( ( ord_le7307670543136651348iteral @ A @ B3 )
=> ( ( member_literal @ C @ A )
=> ( member_literal @ C @ B3 ) ) ) ).
% subsetD
thf(fact_795_not__less0,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less0
thf(fact_796_subset__eq,axiom,
( ord_le4198349162570665613l_real
= ( ^ [A6: set_real_real,B6: set_real_real] :
! [X3: real > real] :
( ( member_real_real @ X3 @ A6 )
=> ( member_real_real @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_797_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A6 )
=> ( member_real @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_798_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( member_nat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_799_subset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A6: set_complex,B6: set_complex] :
! [X3: complex] :
( ( member_complex @ X3 @ A6 )
=> ( member_complex @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_800_subset__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A6: set_rat,B6: set_rat] :
! [X3: rat] :
( ( member_rat @ X3 @ A6 )
=> ( member_rat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_801_subset__eq,axiom,
( ord_le7307670543136651348iteral
= ( ^ [A6: set_literal,B6: set_literal] :
! [X3: literal] :
( ( member_literal @ X3 @ A6 )
=> ( member_literal @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_802_less__zeroE,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_803_subset__iff,axiom,
( ord_le4198349162570665613l_real
= ( ^ [A6: set_real_real,B6: set_real_real] :
! [T2: real > real] :
( ( member_real_real @ T2 @ A6 )
=> ( member_real_real @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_804_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A6: set_real,B6: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A6 )
=> ( member_real @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_805_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_806_subset__iff,axiom,
( ord_le211207098394363844omplex
= ( ^ [A6: set_complex,B6: set_complex] :
! [T2: complex] :
( ( member_complex @ T2 @ A6 )
=> ( member_complex @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_807_subset__iff,axiom,
( ord_less_eq_set_rat
= ( ^ [A6: set_rat,B6: set_rat] :
! [T2: rat] :
( ( member_rat @ T2 @ A6 )
=> ( member_rat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_808_subset__iff,axiom,
( ord_le7307670543136651348iteral
= ( ^ [A6: set_literal,B6: set_literal] :
! [T2: literal] :
( ( member_literal @ T2 @ A6 )
=> ( member_literal @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_809_gr__implies__not0,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_810_infinite__descent0,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N3 ) ) ) ).
% infinite_descent0
thf(fact_811_monotone__on__subset,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: ( real > real ) > ( real > real ) > $o,F: nat > real > real,B3: set_nat] :
( ( monoto2824216093323351088l_real @ A @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( monoto2824216093323351088l_real @ B3 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_812_monotone__on__subset,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,B3: set_nat] :
( ( monotone_on_nat_nat @ A @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( monotone_on_nat_nat @ B3 @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_813_monotone__onD,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: ( real > real ) > ( real > real ) > $o,F: nat > real > real,X: nat,Y: nat] :
( ( monoto2824216093323351088l_real @ A @ Orda @ Ordb @ F )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_814_monotone__onD,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ Orda @ Ordb @ F )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% monotone_onD
thf(fact_815_monotone__onI,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: ( real > real ) > ( real > real ) > $o,F: nat > real > real] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) )
=> ( monoto2824216093323351088l_real @ A @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_816_monotone__onI,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Orda @ Ordb @ F ) ) ).
% monotone_onI
thf(fact_817_monotone__on__def,axiom,
( monoto2824216093323351088l_real
= ( ^ [A6: set_nat,Orda2: nat > nat > $o,Ordb2: ( real > real ) > ( real > real ) > $o,F4: nat > real > real] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A6 )
=> ( ( Orda2 @ X3 @ Y2 )
=> ( Ordb2 @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_818_monotone__on__def,axiom,
( monotone_on_nat_nat
= ( ^ [A6: set_nat,Orda2: nat > nat > $o,Ordb2: nat > nat > $o,F4: nat > nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A6 )
=> ( ( Orda2 @ X3 @ Y2 )
=> ( Ordb2 @ ( F4 @ X3 ) @ ( F4 @ Y2 ) ) ) ) ) ) ) ).
% monotone_on_def
thf(fact_819_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_820_infinite__descent,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) )
=> ( P @ N3 ) ) ).
% infinite_descent
thf(fact_821_nat__less__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N3 ) ) ).
% nat_less_induct
thf(fact_822_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_823_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_824_less__not__refl2,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ N3 @ M3 )
=> ( M3 != N3 ) ) ).
% less_not_refl2
thf(fact_825_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_826_nat__neq__iff,axiom,
! [M3: nat,N3: nat] :
( ( M3 != N3 )
= ( ( ord_less_nat @ M3 @ N3 )
| ( ord_less_nat @ N3 @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_827_ord_Omono__onD,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > real,R: real,S2: real] :
( ( monoto4017252874604999745l_real @ A @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_828_ord_Omono__onD,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > real,R: nat,S2: nat] :
( ( monotone_on_nat_real @ A @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_829_ord_Omono__onD,axiom,
! [A: set_complex,Less_eq: complex > complex > $o,F: complex > real,R: complex,S2: complex] :
( ( monoto7363281639122250051x_real @ A @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_complex @ R @ A )
=> ( ( member_complex @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_830_ord_Omono__onD,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > real,R: rat,S2: rat] :
( ( monotone_on_rat_real @ A @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_831_ord_Omono__onD,axiom,
! [A: set_literal,Less_eq: literal > literal > $o,F: literal > real,R: literal,S2: literal] :
( ( monoto1443783125453340121l_real @ A @ Less_eq @ ord_less_eq_real @ F )
=> ( ( member_literal @ R @ A )
=> ( ( member_literal @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_832_ord_Omono__onD,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > nat,R: real,S2: real] :
( ( monotone_on_real_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_833_ord_Omono__onD,axiom,
! [A: set_complex,Less_eq: complex > complex > $o,F: complex > nat,R: complex,S2: complex] :
( ( monoto2406513391651152359ex_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_complex @ R @ A )
=> ( ( member_complex @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_834_ord_Omono__onD,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > nat,R: rat,S2: rat] :
( ( monotone_on_rat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_835_ord_Omono__onD,axiom,
! [A: set_literal,Less_eq: literal > literal > $o,F: literal > nat,R: literal,S2: literal] :
( ( monoto6092665527236862333al_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_literal @ R @ A )
=> ( ( member_literal @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_836_ord_Omono__onD,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R: nat,S2: nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less_eq @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_837_ord_Omono__onI,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_838_ord_Omono__onI,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > real] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_839_ord_Omono__onI,axiom,
! [A: set_complex,Less_eq: complex > complex > $o,F: complex > real] :
( ! [R2: complex,S3: complex] :
( ( member_complex @ R2 @ A )
=> ( ( member_complex @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto7363281639122250051x_real @ A @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_840_ord_Omono__onI,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > real] :
( ! [R2: rat,S3: rat] :
( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_rat_real @ A @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_841_ord_Omono__onI,axiom,
! [A: set_literal,Less_eq: literal > literal > $o,F: literal > real] :
( ! [R2: literal,S3: literal] :
( ( member_literal @ R2 @ A )
=> ( ( member_literal @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto1443783125453340121l_real @ A @ Less_eq @ ord_less_eq_real @ F ) ) ).
% ord.mono_onI
thf(fact_842_ord_Omono__onI,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > nat] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_843_ord_Omono__onI,axiom,
! [A: set_complex,Less_eq: complex > complex > $o,F: complex > nat] :
( ! [R2: complex,S3: complex] :
( ( member_complex @ R2 @ A )
=> ( ( member_complex @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto2406513391651152359ex_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_844_ord_Omono__onI,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > nat] :
( ! [R2: rat,S3: rat] :
( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_rat_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_845_ord_Omono__onI,axiom,
! [A: set_literal,Less_eq: literal > literal > $o,F: literal > nat] :
( ! [R2: literal,S3: literal] :
( ( member_literal @ R2 @ A )
=> ( ( member_literal @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto6092665527236862333al_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_846_ord_Omono__onI,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( Less_eq @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_847_ord_Omono__on__def,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > real] :
( ( monoto4017252874604999745l_real @ A @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_848_ord_Omono__on__def,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > real] :
( ( monotone_on_nat_real @ A @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_849_ord_Omono__on__def,axiom,
! [A: set_complex,Less_eq: complex > complex > $o,F: complex > real] :
( ( monoto7363281639122250051x_real @ A @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: complex,S4: complex] :
( ( ( member_complex @ R3 @ A )
& ( member_complex @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_850_ord_Omono__on__def,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > real] :
( ( monotone_on_rat_real @ A @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: rat,S4: rat] :
( ( ( member_rat @ R3 @ A )
& ( member_rat @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_851_ord_Omono__on__def,axiom,
! [A: set_literal,Less_eq: literal > literal > $o,F: literal > real] :
( ( monoto1443783125453340121l_real @ A @ Less_eq @ ord_less_eq_real @ F )
= ( ! [R3: literal,S4: literal] :
( ( ( member_literal @ R3 @ A )
& ( member_literal @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_852_ord_Omono__on__def,axiom,
! [A: set_real,Less_eq: real > real > $o,F: real > nat] :
( ( monotone_on_real_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_853_ord_Omono__on__def,axiom,
! [A: set_complex,Less_eq: complex > complex > $o,F: complex > nat] :
( ( monoto2406513391651152359ex_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: complex,S4: complex] :
( ( ( member_complex @ R3 @ A )
& ( member_complex @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_854_ord_Omono__on__def,axiom,
! [A: set_rat,Less_eq: rat > rat > $o,F: rat > nat] :
( ( monotone_on_rat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: rat,S4: rat] :
( ( ( member_rat @ R3 @ A )
& ( member_rat @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_855_ord_Omono__on__def,axiom,
! [A: set_literal,Less_eq: literal > literal > $o,F: literal > nat] :
( ( monoto6092665527236862333al_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: literal,S4: literal] :
( ( ( member_literal @ R3 @ A )
& ( member_literal @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_856_ord_Omono__on__def,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S4 @ A )
& ( Less_eq @ R3 @ S4 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_857_ord_Omono__on__subset,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat,B3: set_nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( monotone_on_nat_nat @ B3 @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_858_ord_Omono__on__subset,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > real > real,B3: set_nat] :
( ( monoto2824216093323351088l_real @ A @ Less_eq @ ord_le6948328307412524503l_real @ F )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( monoto2824216093323351088l_real @ B3 @ Less_eq @ ord_le6948328307412524503l_real @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_859_mono__onD,axiom,
! [A: set_complex,F: complex > real,R: complex,S2: complex] :
( ( monoto7363281639122250051x_real @ A @ ord_less_eq_complex @ ord_less_eq_real @ F )
=> ( ( member_complex @ R @ A )
=> ( ( member_complex @ S2 @ A )
=> ( ( ord_less_eq_complex @ R @ S2 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_860_mono__onD,axiom,
! [A: set_rat,F: rat > real,R: rat,S2: rat] :
( ( monotone_on_rat_real @ A @ ord_less_eq_rat @ ord_less_eq_real @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( ord_less_eq_rat @ R @ S2 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_861_mono__onD,axiom,
! [A: set_literal,F: literal > real,R: literal,S2: literal] :
( ( monoto1443783125453340121l_real @ A @ ord_less_eq_literal @ ord_less_eq_real @ F )
=> ( ( member_literal @ R @ A )
=> ( ( member_literal @ S2 @ A )
=> ( ( ord_less_eq_literal @ R @ S2 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_862_mono__onD,axiom,
! [A: set_complex,F: complex > nat,R: complex,S2: complex] :
( ( monoto2406513391651152359ex_nat @ A @ ord_less_eq_complex @ ord_less_eq_nat @ F )
=> ( ( member_complex @ R @ A )
=> ( ( member_complex @ S2 @ A )
=> ( ( ord_less_eq_complex @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_863_mono__onD,axiom,
! [A: set_rat,F: rat > nat,R: rat,S2: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( ord_less_eq_rat @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_864_mono__onD,axiom,
! [A: set_literal,F: literal > nat,R: literal,S2: literal] :
( ( monoto6092665527236862333al_nat @ A @ ord_less_eq_literal @ ord_less_eq_nat @ F )
=> ( ( member_literal @ R @ A )
=> ( ( member_literal @ S2 @ A )
=> ( ( ord_less_eq_literal @ R @ S2 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_865_mono__onD,axiom,
! [A: set_complex,F: complex > int,R: complex,S2: complex] :
( ( monoto2404022921142102083ex_int @ A @ ord_less_eq_complex @ ord_less_eq_int @ F )
=> ( ( member_complex @ R @ A )
=> ( ( member_complex @ S2 @ A )
=> ( ( ord_less_eq_complex @ R @ S2 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_866_mono__onD,axiom,
! [A: set_rat,F: rat > int,R: rat,S2: rat] :
( ( monotone_on_rat_int @ A @ ord_less_eq_rat @ ord_less_eq_int @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( ord_less_eq_rat @ R @ S2 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_867_mono__onD,axiom,
! [A: set_literal,F: literal > int,R: literal,S2: literal] :
( ( monoto6090175056727812057al_int @ A @ ord_less_eq_literal @ ord_less_eq_int @ F )
=> ( ( member_literal @ R @ A )
=> ( ( member_literal @ S2 @ A )
=> ( ( ord_less_eq_literal @ R @ S2 )
=> ( ord_less_eq_int @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_868_mono__onD,axiom,
! [A: set_real,F: real > real,R: real,S2: real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( ord_less_eq_real @ R @ S2 )
=> ( ord_less_eq_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% mono_onD
thf(fact_869_mono__onI,axiom,
! [A: set_complex,F: complex > real] :
( ! [R2: complex,S3: complex] :
( ( member_complex @ R2 @ A )
=> ( ( member_complex @ S3 @ A )
=> ( ( ord_less_eq_complex @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto7363281639122250051x_real @ A @ ord_less_eq_complex @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_870_mono__onI,axiom,
! [A: set_rat,F: rat > real] :
( ! [R2: rat,S3: rat] :
( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S3 @ A )
=> ( ( ord_less_eq_rat @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_rat_real @ A @ ord_less_eq_rat @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_871_mono__onI,axiom,
! [A: set_literal,F: literal > real] :
( ! [R2: literal,S3: literal] :
( ( member_literal @ R2 @ A )
=> ( ( member_literal @ S3 @ A )
=> ( ( ord_less_eq_literal @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto1443783125453340121l_real @ A @ ord_less_eq_literal @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_872_mono__onI,axiom,
! [A: set_complex,F: complex > nat] :
( ! [R2: complex,S3: complex] :
( ( member_complex @ R2 @ A )
=> ( ( member_complex @ S3 @ A )
=> ( ( ord_less_eq_complex @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto2406513391651152359ex_nat @ A @ ord_less_eq_complex @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_873_mono__onI,axiom,
! [A: set_rat,F: rat > nat] :
( ! [R2: rat,S3: rat] :
( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S3 @ A )
=> ( ( ord_less_eq_rat @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_rat_nat @ A @ ord_less_eq_rat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_874_mono__onI,axiom,
! [A: set_literal,F: literal > nat] :
( ! [R2: literal,S3: literal] :
( ( member_literal @ R2 @ A )
=> ( ( member_literal @ S3 @ A )
=> ( ( ord_less_eq_literal @ R2 @ S3 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto6092665527236862333al_nat @ A @ ord_less_eq_literal @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_875_mono__onI,axiom,
! [A: set_complex,F: complex > int] :
( ! [R2: complex,S3: complex] :
( ( member_complex @ R2 @ A )
=> ( ( member_complex @ S3 @ A )
=> ( ( ord_less_eq_complex @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto2404022921142102083ex_int @ A @ ord_less_eq_complex @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_876_mono__onI,axiom,
! [A: set_rat,F: rat > int] :
( ! [R2: rat,S3: rat] :
( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S3 @ A )
=> ( ( ord_less_eq_rat @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_rat_int @ A @ ord_less_eq_rat @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_877_mono__onI,axiom,
! [A: set_literal,F: literal > int] :
( ! [R2: literal,S3: literal] :
( ( member_literal @ R2 @ A )
=> ( ( member_literal @ S3 @ A )
=> ( ( ord_less_eq_literal @ R2 @ S3 )
=> ( ord_less_eq_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto6090175056727812057al_int @ A @ ord_less_eq_literal @ ord_less_eq_int @ F ) ) ).
% mono_onI
thf(fact_878_mono__onI,axiom,
! [A: set_real,F: real > real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_eq_real @ R2 @ S3 )
=> ( ord_less_eq_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% mono_onI
thf(fact_879_mono__on__subset,axiom,
! [A: set_real,F: real > real,B3: set_real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_real @ B3 @ A )
=> ( monoto4017252874604999745l_real @ B3 @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_880_mono__on__subset,axiom,
! [A: set_real,F: real > nat,B3: set_real] :
( ( monotone_on_real_nat @ A @ ord_less_eq_real @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_real @ B3 @ A )
=> ( monotone_on_real_nat @ B3 @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_881_mono__on__subset,axiom,
! [A: set_real,F: real > int,B3: set_real] :
( ( monotone_on_real_int @ A @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_real @ B3 @ A )
=> ( monotone_on_real_int @ B3 @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_882_mono__on__subset,axiom,
! [A: set_nat,F: nat > real,B3: set_nat] :
( ( monotone_on_nat_real @ A @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( monotone_on_nat_real @ B3 @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_883_mono__on__subset,axiom,
! [A: set_nat,F: nat > nat,B3: set_nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( monotone_on_nat_nat @ B3 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_884_mono__on__subset,axiom,
! [A: set_nat,F: nat > int,B3: set_nat] :
( ( monotone_on_nat_int @ A @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( monotone_on_nat_int @ B3 @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_885_mono__on__subset,axiom,
! [A: set_int,F: int > real,B3: set_int] :
( ( monotone_on_int_real @ A @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_set_int @ B3 @ A )
=> ( monotone_on_int_real @ B3 @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ) ).
% mono_on_subset
thf(fact_886_mono__on__subset,axiom,
! [A: set_int,F: int > nat,B3: set_int] :
( ( monotone_on_int_nat @ A @ ord_less_eq_int @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_int @ B3 @ A )
=> ( monotone_on_int_nat @ B3 @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_887_mono__on__subset,axiom,
! [A: set_int,F: int > int,B3: set_int] :
( ( monotone_on_int_int @ A @ ord_less_eq_int @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_set_int @ B3 @ A )
=> ( monotone_on_int_int @ B3 @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ) ).
% mono_on_subset
thf(fact_888_mono__on__subset,axiom,
! [A: set_real,F: real > real > real,B3: set_real] :
( ( monoto8965231823629880588l_real @ A @ ord_less_eq_real @ ord_le6948328307412524503l_real @ F )
=> ( ( ord_less_eq_set_real @ B3 @ A )
=> ( monoto8965231823629880588l_real @ B3 @ ord_less_eq_real @ ord_le6948328307412524503l_real @ F ) ) ) ).
% mono_on_subset
thf(fact_889_ord_Ostrict__mono__onD,axiom,
! [A: set_real,Less: real > real > $o,F: real > real,R: real,S2: real] :
( ( monoto4017252874604999745l_real @ A @ Less @ ord_less_real @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_890_ord_Ostrict__mono__onD,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > real,R: nat,S2: nat] :
( ( monotone_on_nat_real @ A @ Less @ ord_less_real @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_891_ord_Ostrict__mono__onD,axiom,
! [A: set_complex,Less: complex > complex > $o,F: complex > real,R: complex,S2: complex] :
( ( monoto7363281639122250051x_real @ A @ Less @ ord_less_real @ F )
=> ( ( member_complex @ R @ A )
=> ( ( member_complex @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_892_ord_Ostrict__mono__onD,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > real,R: rat,S2: rat] :
( ( monotone_on_rat_real @ A @ Less @ ord_less_real @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_893_ord_Ostrict__mono__onD,axiom,
! [A: set_literal,Less: literal > literal > $o,F: literal > real,R: literal,S2: literal] :
( ( monoto1443783125453340121l_real @ A @ Less @ ord_less_real @ F )
=> ( ( member_literal @ R @ A )
=> ( ( member_literal @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_894_ord_Ostrict__mono__onD,axiom,
! [A: set_real,Less: real > real > $o,F: real > nat,R: real,S2: real] :
( ( monotone_on_real_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_895_ord_Ostrict__mono__onD,axiom,
! [A: set_complex,Less: complex > complex > $o,F: complex > nat,R: complex,S2: complex] :
( ( monoto2406513391651152359ex_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_complex @ R @ A )
=> ( ( member_complex @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_896_ord_Ostrict__mono__onD,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > nat,R: rat,S2: rat] :
( ( monotone_on_rat_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_897_ord_Ostrict__mono__onD,axiom,
! [A: set_literal,Less: literal > literal > $o,F: literal > nat,R: literal,S2: literal] :
( ( monoto6092665527236862333al_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_literal @ R @ A )
=> ( ( member_literal @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_898_ord_Ostrict__mono__onD,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat,R: nat,S2: nat] :
( ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A )
=> ( ( member_nat @ S2 @ A )
=> ( ( Less @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_899_ord_Ostrict__mono__onI,axiom,
! [A: set_real,Less: real > real > $o,F: real > real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_900_ord_Ostrict__mono__onI,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > real] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_real @ A @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_901_ord_Ostrict__mono__onI,axiom,
! [A: set_complex,Less: complex > complex > $o,F: complex > real] :
( ! [R2: complex,S3: complex] :
( ( member_complex @ R2 @ A )
=> ( ( member_complex @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto7363281639122250051x_real @ A @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_902_ord_Ostrict__mono__onI,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > real] :
( ! [R2: rat,S3: rat] :
( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_rat_real @ A @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_903_ord_Ostrict__mono__onI,axiom,
! [A: set_literal,Less: literal > literal > $o,F: literal > real] :
( ! [R2: literal,S3: literal] :
( ( member_literal @ R2 @ A )
=> ( ( member_literal @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto1443783125453340121l_real @ A @ Less @ ord_less_real @ F ) ) ).
% ord.strict_mono_onI
thf(fact_904_ord_Ostrict__mono__onI,axiom,
! [A: set_real,Less: real > real > $o,F: real > nat] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_real_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_905_ord_Ostrict__mono__onI,axiom,
! [A: set_complex,Less: complex > complex > $o,F: complex > nat] :
( ! [R2: complex,S3: complex] :
( ( member_complex @ R2 @ A )
=> ( ( member_complex @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto2406513391651152359ex_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_906_ord_Ostrict__mono__onI,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > nat] :
( ! [R2: rat,S3: rat] :
( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_rat_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_907_ord_Ostrict__mono__onI,axiom,
! [A: set_literal,Less: literal > literal > $o,F: literal > nat] :
( ! [R2: literal,S3: literal] :
( ( member_literal @ R2 @ A )
=> ( ( member_literal @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto6092665527236862333al_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_908_ord_Ostrict__mono__onI,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R2: nat,S3: nat] :
( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S3 @ A )
=> ( ( Less @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_909_ord_Ostrict__mono__on__def,axiom,
! [A: set_real,Less: real > real > $o,F: real > real] :
( ( monoto4017252874604999745l_real @ A @ Less @ ord_less_real @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_910_ord_Ostrict__mono__on__def,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > real] :
( ( monotone_on_nat_real @ A @ Less @ ord_less_real @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_911_ord_Ostrict__mono__on__def,axiom,
! [A: set_complex,Less: complex > complex > $o,F: complex > real] :
( ( monoto7363281639122250051x_real @ A @ Less @ ord_less_real @ F )
= ( ! [R3: complex,S4: complex] :
( ( ( member_complex @ R3 @ A )
& ( member_complex @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_912_ord_Ostrict__mono__on__def,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > real] :
( ( monotone_on_rat_real @ A @ Less @ ord_less_real @ F )
= ( ! [R3: rat,S4: rat] :
( ( ( member_rat @ R3 @ A )
& ( member_rat @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_913_ord_Ostrict__mono__on__def,axiom,
! [A: set_literal,Less: literal > literal > $o,F: literal > real] :
( ( monoto1443783125453340121l_real @ A @ Less @ ord_less_real @ F )
= ( ! [R3: literal,S4: literal] :
( ( ( member_literal @ R3 @ A )
& ( member_literal @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_real @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_914_ord_Ostrict__mono__on__def,axiom,
! [A: set_real,Less: real > real > $o,F: real > nat] :
( ( monotone_on_real_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: real,S4: real] :
( ( ( member_real @ R3 @ A )
& ( member_real @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_915_ord_Ostrict__mono__on__def,axiom,
! [A: set_complex,Less: complex > complex > $o,F: complex > nat] :
( ( monoto2406513391651152359ex_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: complex,S4: complex] :
( ( ( member_complex @ R3 @ A )
& ( member_complex @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_916_ord_Ostrict__mono__on__def,axiom,
! [A: set_rat,Less: rat > rat > $o,F: rat > nat] :
( ( monotone_on_rat_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: rat,S4: rat] :
( ( ( member_rat @ R3 @ A )
& ( member_rat @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_917_ord_Ostrict__mono__on__def,axiom,
! [A: set_literal,Less: literal > literal > $o,F: literal > nat] :
( ( monoto6092665527236862333al_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: literal,S4: literal] :
( ( ( member_literal @ R3 @ A )
& ( member_literal @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_918_ord_Ostrict__mono__on__def,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: nat,S4: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S4 @ A )
& ( Less @ R3 @ S4 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S4 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_919_strict__mono__onD,axiom,
! [A: set_complex,F: complex > real,R: complex,S2: complex] :
( ( monoto7363281639122250051x_real @ A @ ord_less_complex @ ord_less_real @ F )
=> ( ( member_complex @ R @ A )
=> ( ( member_complex @ S2 @ A )
=> ( ( ord_less_complex @ R @ S2 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_920_strict__mono__onD,axiom,
! [A: set_rat,F: rat > real,R: rat,S2: rat] :
( ( monotone_on_rat_real @ A @ ord_less_rat @ ord_less_real @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( ord_less_rat @ R @ S2 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_921_strict__mono__onD,axiom,
! [A: set_literal,F: literal > real,R: literal,S2: literal] :
( ( monoto1443783125453340121l_real @ A @ ord_less_literal @ ord_less_real @ F )
=> ( ( member_literal @ R @ A )
=> ( ( member_literal @ S2 @ A )
=> ( ( ord_less_literal @ R @ S2 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_922_strict__mono__onD,axiom,
! [A: set_complex,F: complex > nat,R: complex,S2: complex] :
( ( monoto2406513391651152359ex_nat @ A @ ord_less_complex @ ord_less_nat @ F )
=> ( ( member_complex @ R @ A )
=> ( ( member_complex @ S2 @ A )
=> ( ( ord_less_complex @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_923_strict__mono__onD,axiom,
! [A: set_rat,F: rat > nat,R: rat,S2: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_rat @ ord_less_nat @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( ord_less_rat @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_924_strict__mono__onD,axiom,
! [A: set_literal,F: literal > nat,R: literal,S2: literal] :
( ( monoto6092665527236862333al_nat @ A @ ord_less_literal @ ord_less_nat @ F )
=> ( ( member_literal @ R @ A )
=> ( ( member_literal @ S2 @ A )
=> ( ( ord_less_literal @ R @ S2 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_925_strict__mono__onD,axiom,
! [A: set_complex,F: complex > int,R: complex,S2: complex] :
( ( monoto2404022921142102083ex_int @ A @ ord_less_complex @ ord_less_int @ F )
=> ( ( member_complex @ R @ A )
=> ( ( member_complex @ S2 @ A )
=> ( ( ord_less_complex @ R @ S2 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_926_strict__mono__onD,axiom,
! [A: set_rat,F: rat > int,R: rat,S2: rat] :
( ( monotone_on_rat_int @ A @ ord_less_rat @ ord_less_int @ F )
=> ( ( member_rat @ R @ A )
=> ( ( member_rat @ S2 @ A )
=> ( ( ord_less_rat @ R @ S2 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_927_strict__mono__onD,axiom,
! [A: set_literal,F: literal > int,R: literal,S2: literal] :
( ( monoto6090175056727812057al_int @ A @ ord_less_literal @ ord_less_int @ F )
=> ( ( member_literal @ R @ A )
=> ( ( member_literal @ S2 @ A )
=> ( ( ord_less_literal @ R @ S2 )
=> ( ord_less_int @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_928_strict__mono__onD,axiom,
! [A: set_real,F: real > real,R: real,S2: real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_real @ ord_less_real @ F )
=> ( ( member_real @ R @ A )
=> ( ( member_real @ S2 @ A )
=> ( ( ord_less_real @ R @ S2 )
=> ( ord_less_real @ ( F @ R ) @ ( F @ S2 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_929_strict__mono__onI,axiom,
! [A: set_complex,F: complex > real] :
( ! [R2: complex,S3: complex] :
( ( member_complex @ R2 @ A )
=> ( ( member_complex @ S3 @ A )
=> ( ( ord_less_complex @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto7363281639122250051x_real @ A @ ord_less_complex @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_930_strict__mono__onI,axiom,
! [A: set_rat,F: rat > real] :
( ! [R2: rat,S3: rat] :
( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S3 @ A )
=> ( ( ord_less_rat @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_rat_real @ A @ ord_less_rat @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_931_strict__mono__onI,axiom,
! [A: set_literal,F: literal > real] :
( ! [R2: literal,S3: literal] :
( ( member_literal @ R2 @ A )
=> ( ( member_literal @ S3 @ A )
=> ( ( ord_less_literal @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto1443783125453340121l_real @ A @ ord_less_literal @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_932_strict__mono__onI,axiom,
! [A: set_complex,F: complex > nat] :
( ! [R2: complex,S3: complex] :
( ( member_complex @ R2 @ A )
=> ( ( member_complex @ S3 @ A )
=> ( ( ord_less_complex @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto2406513391651152359ex_nat @ A @ ord_less_complex @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_933_strict__mono__onI,axiom,
! [A: set_rat,F: rat > nat] :
( ! [R2: rat,S3: rat] :
( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S3 @ A )
=> ( ( ord_less_rat @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_rat_nat @ A @ ord_less_rat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_934_strict__mono__onI,axiom,
! [A: set_literal,F: literal > nat] :
( ! [R2: literal,S3: literal] :
( ( member_literal @ R2 @ A )
=> ( ( member_literal @ S3 @ A )
=> ( ( ord_less_literal @ R2 @ S3 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto6092665527236862333al_nat @ A @ ord_less_literal @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_935_strict__mono__onI,axiom,
! [A: set_complex,F: complex > int] :
( ! [R2: complex,S3: complex] :
( ( member_complex @ R2 @ A )
=> ( ( member_complex @ S3 @ A )
=> ( ( ord_less_complex @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto2404022921142102083ex_int @ A @ ord_less_complex @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_936_strict__mono__onI,axiom,
! [A: set_rat,F: rat > int] :
( ! [R2: rat,S3: rat] :
( ( member_rat @ R2 @ A )
=> ( ( member_rat @ S3 @ A )
=> ( ( ord_less_rat @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monotone_on_rat_int @ A @ ord_less_rat @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_937_strict__mono__onI,axiom,
! [A: set_literal,F: literal > int] :
( ! [R2: literal,S3: literal] :
( ( member_literal @ R2 @ A )
=> ( ( member_literal @ S3 @ A )
=> ( ( ord_less_literal @ R2 @ S3 )
=> ( ord_less_int @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto6090175056727812057al_int @ A @ ord_less_literal @ ord_less_int @ F ) ) ).
% strict_mono_onI
thf(fact_938_strict__mono__onI,axiom,
! [A: set_real,F: real > real] :
( ! [R2: real,S3: real] :
( ( member_real @ R2 @ A )
=> ( ( member_real @ S3 @ A )
=> ( ( ord_less_real @ R2 @ S3 )
=> ( ord_less_real @ ( F @ R2 ) @ ( F @ S3 ) ) ) ) )
=> ( monoto4017252874604999745l_real @ A @ ord_less_real @ ord_less_real @ F ) ) ).
% strict_mono_onI
thf(fact_939_strict__mono__on__eqD,axiom,
! [A: set_rat,F: rat > real,X: rat,Y: rat] :
( ( monotone_on_rat_real @ A @ ord_less_rat @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_940_strict__mono__on__eqD,axiom,
! [A: set_literal,F: literal > real,X: literal,Y: literal] :
( ( monoto1443783125453340121l_real @ A @ ord_less_literal @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_literal @ X @ A )
=> ( ( member_literal @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_941_strict__mono__on__eqD,axiom,
! [A: set_rat,F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_rat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_942_strict__mono__on__eqD,axiom,
! [A: set_literal,F: literal > nat,X: literal,Y: literal] :
( ( monoto6092665527236862333al_nat @ A @ ord_less_literal @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_literal @ X @ A )
=> ( ( member_literal @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_943_strict__mono__on__eqD,axiom,
! [A: set_rat,F: rat > int,X: rat,Y: rat] :
( ( monotone_on_rat_int @ A @ ord_less_rat @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_944_strict__mono__on__eqD,axiom,
! [A: set_literal,F: literal > int,X: literal,Y: literal] :
( ( monoto6090175056727812057al_int @ A @ ord_less_literal @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_literal @ X @ A )
=> ( ( member_literal @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_945_strict__mono__on__eqD,axiom,
! [A: set_real,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_real @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_946_strict__mono__on__eqD,axiom,
! [A: set_real,F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ A @ ord_less_real @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_947_strict__mono__on__eqD,axiom,
! [A: set_real,F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ A @ ord_less_real @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_948_strict__mono__on__eqD,axiom,
! [A: set_nat,F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ A @ ord_less_nat @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_949_monotoneD,axiom,
! [Orda: nat > nat > $o,Ordb: ( real > real ) > ( real > real ) > $o,F: nat > real > real,X: nat,Y: nat] :
( ( monoto2824216093323351088l_real @ top_top_set_nat @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_950_monotoneD,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_951_monotoneI,axiom,
! [Orda: nat > nat > $o,Ordb: ( real > real ) > ( real > real ) > $o,F: nat > real > real] :
( ! [X2: nat,Y3: nat] :
( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto2824216093323351088l_real @ top_top_set_nat @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_952_monotoneI,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( Orda @ X2 @ Y3 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_953_strict__mono__on__imp__mono__on,axiom,
! [A: set_real,F: real > real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_real @ ord_less_real @ F )
=> ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_954_strict__mono__on__imp__mono__on,axiom,
! [A: set_real,F: real > nat] :
( ( monotone_on_real_nat @ A @ ord_less_real @ ord_less_nat @ F )
=> ( monotone_on_real_nat @ A @ ord_less_eq_real @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_955_strict__mono__on__imp__mono__on,axiom,
! [A: set_real,F: real > int] :
( ( monotone_on_real_int @ A @ ord_less_real @ ord_less_int @ F )
=> ( monotone_on_real_int @ A @ ord_less_eq_real @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_956_strict__mono__on__imp__mono__on,axiom,
! [A: set_nat,F: nat > real] :
( ( monotone_on_nat_real @ A @ ord_less_nat @ ord_less_real @ F )
=> ( monotone_on_nat_real @ A @ ord_less_eq_nat @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_957_strict__mono__on__imp__mono__on,axiom,
! [A: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_958_strict__mono__on__imp__mono__on,axiom,
! [A: set_nat,F: nat > int] :
( ( monotone_on_nat_int @ A @ ord_less_nat @ ord_less_int @ F )
=> ( monotone_on_nat_int @ A @ ord_less_eq_nat @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_959_strict__mono__on__imp__mono__on,axiom,
! [A: set_int,F: int > real] :
( ( monotone_on_int_real @ A @ ord_less_int @ ord_less_real @ F )
=> ( monotone_on_int_real @ A @ ord_less_eq_int @ ord_less_eq_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_960_strict__mono__on__imp__mono__on,axiom,
! [A: set_int,F: int > nat] :
( ( monotone_on_int_nat @ A @ ord_less_int @ ord_less_nat @ F )
=> ( monotone_on_int_nat @ A @ ord_less_eq_int @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_961_strict__mono__on__imp__mono__on,axiom,
! [A: set_int,F: int > int] :
( ( monotone_on_int_int @ A @ ord_less_int @ ord_less_int @ F )
=> ( monotone_on_int_int @ A @ ord_less_eq_int @ ord_less_eq_int @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_962_strict__mono__on__imp__mono__on,axiom,
! [A: set_real,F: real > real > real] :
( ( monoto8965231823629880588l_real @ A @ ord_less_real @ ord_less_real_real @ F )
=> ( monoto8965231823629880588l_real @ A @ ord_less_eq_real @ ord_le6948328307412524503l_real @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_963_strict__mono__on__leD,axiom,
! [A: set_rat,F: rat > real,X: rat,Y: rat] :
( ( monotone_on_rat_real @ A @ ord_less_rat @ ord_less_real @ F )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y @ A )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_964_strict__mono__on__leD,axiom,
! [A: set_literal,F: literal > real,X: literal,Y: literal] :
( ( monoto1443783125453340121l_real @ A @ ord_less_literal @ ord_less_real @ F )
=> ( ( member_literal @ X @ A )
=> ( ( member_literal @ Y @ A )
=> ( ( ord_less_eq_literal @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_965_strict__mono__on__leD,axiom,
! [A: set_rat,F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_rat @ ord_less_nat @ F )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y @ A )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_966_strict__mono__on__leD,axiom,
! [A: set_literal,F: literal > nat,X: literal,Y: literal] :
( ( monoto6092665527236862333al_nat @ A @ ord_less_literal @ ord_less_nat @ F )
=> ( ( member_literal @ X @ A )
=> ( ( member_literal @ Y @ A )
=> ( ( ord_less_eq_literal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_967_strict__mono__on__leD,axiom,
! [A: set_rat,F: rat > int,X: rat,Y: rat] :
( ( monotone_on_rat_int @ A @ ord_less_rat @ ord_less_int @ F )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y @ A )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_968_strict__mono__on__leD,axiom,
! [A: set_literal,F: literal > int,X: literal,Y: literal] :
( ( monoto6090175056727812057al_int @ A @ ord_less_literal @ ord_less_int @ F )
=> ( ( member_literal @ X @ A )
=> ( ( member_literal @ Y @ A )
=> ( ( ord_less_eq_literal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_969_strict__mono__on__leD,axiom,
! [A: set_real,F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_real @ ord_less_real @ F )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_970_strict__mono__on__leD,axiom,
! [A: set_real,F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ A @ ord_less_real @ ord_less_nat @ F )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_971_strict__mono__on__leD,axiom,
! [A: set_real,F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ A @ ord_less_real @ ord_less_int @ F )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_972_strict__mono__on__leD,axiom,
! [A: set_nat,F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ A @ ord_less_nat @ ord_less_real @ F )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_973_mono__on__greaterD,axiom,
! [A: set_rat,G: rat > real,X: rat,Y: rat] :
( ( monotone_on_rat_real @ A @ ord_less_eq_rat @ ord_less_eq_real @ G )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y @ A )
=> ( ( ord_less_real @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_rat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_974_mono__on__greaterD,axiom,
! [A: set_literal,G: literal > real,X: literal,Y: literal] :
( ( monoto1443783125453340121l_real @ A @ ord_less_eq_literal @ ord_less_eq_real @ G )
=> ( ( member_literal @ X @ A )
=> ( ( member_literal @ Y @ A )
=> ( ( ord_less_real @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_literal @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_975_mono__on__greaterD,axiom,
! [A: set_rat,G: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ A @ ord_less_eq_rat @ ord_less_eq_nat @ G )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y @ A )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_rat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_976_mono__on__greaterD,axiom,
! [A: set_literal,G: literal > nat,X: literal,Y: literal] :
( ( monoto6092665527236862333al_nat @ A @ ord_less_eq_literal @ ord_less_eq_nat @ G )
=> ( ( member_literal @ X @ A )
=> ( ( member_literal @ Y @ A )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_literal @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_977_mono__on__greaterD,axiom,
! [A: set_rat,G: rat > int,X: rat,Y: rat] :
( ( monotone_on_rat_int @ A @ ord_less_eq_rat @ ord_less_eq_int @ G )
=> ( ( member_rat @ X @ A )
=> ( ( member_rat @ Y @ A )
=> ( ( ord_less_int @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_rat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_978_mono__on__greaterD,axiom,
! [A: set_literal,G: literal > int,X: literal,Y: literal] :
( ( monoto6090175056727812057al_int @ A @ ord_less_eq_literal @ ord_less_eq_int @ G )
=> ( ( member_literal @ X @ A )
=> ( ( member_literal @ Y @ A )
=> ( ( ord_less_int @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_literal @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_979_mono__on__greaterD,axiom,
! [A: set_real,G: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ G )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_real @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_real @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_980_mono__on__greaterD,axiom,
! [A: set_real,G: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ A @ ord_less_eq_real @ ord_less_eq_nat @ G )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_real @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_981_mono__on__greaterD,axiom,
! [A: set_real,G: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ A @ ord_less_eq_real @ ord_less_eq_int @ G )
=> ( ( member_real @ X @ A )
=> ( ( member_real @ Y @ A )
=> ( ( ord_less_int @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_real @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_982_mono__on__greaterD,axiom,
! [A: set_nat,G: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ A @ ord_less_eq_nat @ ord_less_eq_real @ G )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_real @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_983_mono__imp__mono__on,axiom,
! [F: complex > real,A: set_complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_real @ F )
=> ( monoto7363281639122250051x_real @ A @ ord_less_eq_complex @ ord_less_eq_real @ F ) ) ).
% mono_imp_mono_on
thf(fact_984_mono__imp__mono__on,axiom,
! [F: literal > real,A: set_literal] :
( ( monoto1443783125453340121l_real @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_real @ F )
=> ( monoto1443783125453340121l_real @ A @ ord_less_eq_literal @ ord_less_eq_real @ F ) ) ).
% mono_imp_mono_on
thf(fact_985_mono__imp__mono__on,axiom,
! [F: rat > real,A: set_rat] :
( ( monotone_on_rat_real @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_real @ F )
=> ( monotone_on_rat_real @ A @ ord_less_eq_rat @ ord_less_eq_real @ F ) ) ).
% mono_imp_mono_on
thf(fact_986_mono__imp__mono__on,axiom,
! [F: complex > nat,A: set_complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F )
=> ( monoto2406513391651152359ex_nat @ A @ ord_less_eq_complex @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_987_mono__imp__mono__on,axiom,
! [F: literal > nat,A: set_literal] :
( ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_nat @ F )
=> ( monoto6092665527236862333al_nat @ A @ ord_less_eq_literal @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_988_mono__imp__mono__on,axiom,
! [F: rat > nat,A: set_rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( monotone_on_rat_nat @ A @ ord_less_eq_rat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_989_mono__imp__mono__on,axiom,
! [F: complex > int,A: set_complex] :
( ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_int @ F )
=> ( monoto2404022921142102083ex_int @ A @ ord_less_eq_complex @ ord_less_eq_int @ F ) ) ).
% mono_imp_mono_on
thf(fact_990_mono__imp__mono__on,axiom,
! [F: literal > int,A: set_literal] :
( ( monoto6090175056727812057al_int @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_int @ F )
=> ( monoto6090175056727812057al_int @ A @ ord_less_eq_literal @ ord_less_eq_int @ F ) ) ).
% mono_imp_mono_on
thf(fact_991_mono__imp__mono__on,axiom,
! [F: rat > int,A: set_rat] :
( ( monotone_on_rat_int @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_int @ F )
=> ( monotone_on_rat_int @ A @ ord_less_eq_rat @ ord_less_eq_int @ F ) ) ).
% mono_imp_mono_on
thf(fact_992_mono__imp__mono__on,axiom,
! [F: real > real,A: set_real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( monoto4017252874604999745l_real @ A @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% mono_imp_mono_on
thf(fact_993_monoI,axiom,
! [F: complex > real] :
( ! [X2: complex,Y3: complex] :
( ( ord_less_eq_complex @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_real @ F ) ) ).
% monoI
thf(fact_994_monoI,axiom,
! [F: literal > real] :
( ! [X2: literal,Y3: literal] :
( ( ord_less_eq_literal @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto1443783125453340121l_real @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_real @ F ) ) ).
% monoI
thf(fact_995_monoI,axiom,
! [F: rat > real] :
( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_rat_real @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_real @ F ) ) ).
% monoI
thf(fact_996_monoI,axiom,
! [F: complex > nat] :
( ! [X2: complex,Y3: complex] :
( ( ord_less_eq_complex @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_997_monoI,axiom,
! [F: literal > nat] :
( ! [X2: literal,Y3: literal] :
( ( ord_less_eq_literal @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_998_monoI,axiom,
! [F: rat > nat] :
( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_999_monoI,axiom,
! [F: complex > int] :
( ! [X2: complex,Y3: complex] :
( ( ord_less_eq_complex @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_int @ F ) ) ).
% monoI
thf(fact_1000_monoI,axiom,
! [F: literal > int] :
( ! [X2: literal,Y3: literal] :
( ( ord_less_eq_literal @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto6090175056727812057al_int @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_int @ F ) ) ).
% monoI
thf(fact_1001_monoI,axiom,
! [F: rat > int] :
( ! [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_rat_int @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_int @ F ) ) ).
% monoI
thf(fact_1002_monoI,axiom,
! [F: real > real] :
( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F ) ) ).
% monoI
thf(fact_1003_monoE,axiom,
! [F: complex > real,X: complex,Y: complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1004_monoE,axiom,
! [F: literal > real,X: literal,Y: literal] :
( ( monoto1443783125453340121l_real @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_literal @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1005_monoE,axiom,
! [F: rat > real,X: rat,Y: rat] :
( ( monotone_on_rat_real @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1006_monoE,axiom,
! [F: complex > nat,X: complex,Y: complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1007_monoE,axiom,
! [F: literal > nat,X: literal,Y: literal] :
( ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_literal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1008_monoE,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1009_monoE,axiom,
! [F: complex > int,X: complex,Y: complex] :
( ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1010_monoE,axiom,
! [F: literal > int,X: literal,Y: literal] :
( ( monoto6090175056727812057al_int @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_literal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1011_monoE,axiom,
! [F: rat > int,X: rat,Y: rat] :
( ( monotone_on_rat_int @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1012_monoE,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1013_monoD,axiom,
! [F: complex > real,X: complex,Y: complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1014_monoD,axiom,
! [F: literal > real,X: literal,Y: literal] :
( ( monoto1443783125453340121l_real @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_literal @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1015_monoD,axiom,
! [F: rat > real,X: rat,Y: rat] :
( ( monotone_on_rat_real @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1016_monoD,axiom,
! [F: complex > nat,X: complex,Y: complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1017_monoD,axiom,
! [F: literal > nat,X: literal,Y: literal] :
( ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_literal @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1018_monoD,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1019_monoD,axiom,
! [F: complex > int,X: complex,Y: complex] :
( ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_eq_complex @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1020_monoD,axiom,
! [F: literal > int,X: literal,Y: literal] :
( ( monoto6090175056727812057al_int @ top_top_set_literal @ ord_less_eq_literal @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_literal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1021_monoD,axiom,
! [F: rat > int,X: rat,Y: rat] :
( ( monotone_on_rat_int @ top_top_set_rat @ ord_less_eq_rat @ ord_less_eq_int @ F )
=> ( ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1022_monoD,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_real @ F )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1023_strict__monoD,axiom,
! [F: complex > real,X: complex,Y: complex] :
( ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_complex @ ord_less_real @ F )
=> ( ( ord_less_complex @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1024_strict__monoD,axiom,
! [F: literal > real,X: literal,Y: literal] :
( ( monoto1443783125453340121l_real @ top_top_set_literal @ ord_less_literal @ ord_less_real @ F )
=> ( ( ord_less_literal @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1025_strict__monoD,axiom,
! [F: rat > real,X: rat,Y: rat] :
( ( monotone_on_rat_real @ top_top_set_rat @ ord_less_rat @ ord_less_real @ F )
=> ( ( ord_less_rat @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1026_strict__monoD,axiom,
! [F: complex > nat,X: complex,Y: complex] :
( ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_complex @ ord_less_nat @ F )
=> ( ( ord_less_complex @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1027_strict__monoD,axiom,
! [F: literal > nat,X: literal,Y: literal] :
( ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_literal @ ord_less_nat @ F )
=> ( ( ord_less_literal @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1028_strict__monoD,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F )
=> ( ( ord_less_rat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1029_strict__monoD,axiom,
! [F: complex > int,X: complex,Y: complex] :
( ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_complex @ ord_less_int @ F )
=> ( ( ord_less_complex @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1030_strict__monoD,axiom,
! [F: literal > int,X: literal,Y: literal] :
( ( monoto6090175056727812057al_int @ top_top_set_literal @ ord_less_literal @ ord_less_int @ F )
=> ( ( ord_less_literal @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1031_strict__monoD,axiom,
! [F: rat > int,X: rat,Y: rat] :
( ( monotone_on_rat_int @ top_top_set_rat @ ord_less_rat @ ord_less_int @ F )
=> ( ( ord_less_rat @ X @ Y )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1032_strict__monoD,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ F )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_1033_strict__monoI,axiom,
! [F: complex > real] :
( ! [X2: complex,Y3: complex] :
( ( ord_less_complex @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto7363281639122250051x_real @ top_top_set_complex @ ord_less_complex @ ord_less_real @ F ) ) ).
% strict_monoI
thf(fact_1034_strict__monoI,axiom,
! [F: literal > real] :
( ! [X2: literal,Y3: literal] :
( ( ord_less_literal @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto1443783125453340121l_real @ top_top_set_literal @ ord_less_literal @ ord_less_real @ F ) ) ).
% strict_monoI
thf(fact_1035_strict__monoI,axiom,
! [F: rat > real] :
( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_rat_real @ top_top_set_rat @ ord_less_rat @ ord_less_real @ F ) ) ).
% strict_monoI
thf(fact_1036_strict__monoI,axiom,
! [F: complex > nat] :
( ! [X2: complex,Y3: complex] :
( ( ord_less_complex @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto2406513391651152359ex_nat @ top_top_set_complex @ ord_less_complex @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_1037_strict__monoI,axiom,
! [F: literal > nat] :
( ! [X2: literal,Y3: literal] :
( ( ord_less_literal @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_literal @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_1038_strict__monoI,axiom,
! [F: rat > nat] :
( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_1039_strict__monoI,axiom,
! [F: complex > int] :
( ! [X2: complex,Y3: complex] :
( ( ord_less_complex @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto2404022921142102083ex_int @ top_top_set_complex @ ord_less_complex @ ord_less_int @ F ) ) ).
% strict_monoI
thf(fact_1040_strict__monoI,axiom,
! [F: literal > int] :
( ! [X2: literal,Y3: literal] :
( ( ord_less_literal @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto6090175056727812057al_int @ top_top_set_literal @ ord_less_literal @ ord_less_int @ F ) ) ).
% strict_monoI
thf(fact_1041_strict__monoI,axiom,
! [F: rat > int] :
( ! [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monotone_on_rat_int @ top_top_set_rat @ ord_less_rat @ ord_less_int @ F ) ) ).
% strict_monoI
thf(fact_1042_strict__monoI,axiom,
! [F: real > real] :
( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ F ) ) ).
% strict_monoI
thf(fact_1043_strict__mono__eq,axiom,
! [F: literal > real,X: literal,Y: literal] :
( ( monoto1443783125453340121l_real @ top_top_set_literal @ ord_less_literal @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_1044_strict__mono__eq,axiom,
! [F: rat > real,X: rat,Y: rat] :
( ( monotone_on_rat_real @ top_top_set_rat @ ord_less_rat @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_1045_strict__mono__eq,axiom,
! [F: literal > nat,X: literal,Y: literal] :
( ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_literal @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_1046_strict__mono__eq,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_1047_strict__mono__eq,axiom,
! [F: literal > int,X: literal,Y: literal] :
( ( monoto6090175056727812057al_int @ top_top_set_literal @ ord_less_literal @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_1048_strict__mono__eq,axiom,
! [F: rat > int,X: rat,Y: rat] :
( ( monotone_on_rat_int @ top_top_set_rat @ ord_less_rat @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_1049_strict__mono__eq,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_1050_strict__mono__eq,axiom,
! [F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_real @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_1051_strict__mono__eq,axiom,
! [F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_real @ ord_less_int @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_1052_strict__mono__eq,axiom,
! [F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_1053_strict__mono__less,axiom,
! [F: literal > real,X: literal,Y: literal] :
( ( monoto1443783125453340121l_real @ top_top_set_literal @ ord_less_literal @ ord_less_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_literal @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_1054_strict__mono__less,axiom,
! [F: rat > real,X: rat,Y: rat] :
( ( monotone_on_rat_real @ top_top_set_rat @ ord_less_rat @ ord_less_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_rat @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_1055_strict__mono__less,axiom,
! [F: literal > nat,X: literal,Y: literal] :
( ( monoto6092665527236862333al_nat @ top_top_set_literal @ ord_less_literal @ ord_less_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_literal @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_1056_strict__mono__less,axiom,
! [F: rat > nat,X: rat,Y: rat] :
( ( monotone_on_rat_nat @ top_top_set_rat @ ord_less_rat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_rat @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_1057_strict__mono__less,axiom,
! [F: literal > int,X: literal,Y: literal] :
( ( monoto6090175056727812057al_int @ top_top_set_literal @ ord_less_literal @ ord_less_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_literal @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_1058_strict__mono__less,axiom,
! [F: rat > int,X: rat,Y: rat] :
( ( monotone_on_rat_int @ top_top_set_rat @ ord_less_rat @ ord_less_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_rat @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_1059_strict__mono__less,axiom,
! [F: real > real,X: real,Y: real] :
( ( monoto4017252874604999745l_real @ top_top_set_real @ ord_less_real @ ord_less_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_1060_strict__mono__less,axiom,
! [F: real > nat,X: real,Y: real] :
( ( monotone_on_real_nat @ top_top_set_real @ ord_less_real @ ord_less_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_1061_strict__mono__less,axiom,
! [F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_real @ ord_less_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_1062_strict__mono__less,axiom,
! [F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_nat @ ord_less_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_1063_mono__invE,axiom,
! [F: real > int,X: real,Y: real] :
( ( monotone_on_real_int @ top_top_set_real @ ord_less_eq_real @ ord_less_eq_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_real @ X @ Y ) ) ) ).
% mono_invE
thf(fact_1064_mono__invE,axiom,
! [F: nat > real,X: nat,Y: nat] :
( ( monotone_on_nat_real @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_1065_mono__invE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_1066_mono__invE,axiom,
! [F: nat > real > real,X: nat,Y: nat] :
( ( monoto2824216093323351088l_real @ top_top_set_nat @ ord_less_eq_nat @ ord_le6948328307412524503l_real @ F )
=> ( ( ord_less_real_real @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_1067_mono__invE,axiom,
! [F: nat > int,X: nat,Y: nat] :
( ( monotone_on_nat_int @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_1068_mono__invE,axiom,
! [F: int > real,X: int,Y: int] :
( ( monotone_on_int_real @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_real @ F )
=> ( ( ord_less_real @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% mono_invE
thf(fact_1069_mono__invE,axiom,
! [F: int > nat,X: int,Y: int] :
( ( monotone_on_int_nat @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% mono_invE
thf(fact_1070_mono__invE,axiom,
! [F: int > real > real,X: int,Y: int] :
( ( monoto6775660721739060364l_real @ top_top_set_int @ ord_less_eq_int @ ord_le6948328307412524503l_real @ F )
=> ( ( ord_less_real_real @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% mono_invE
thf(fact_1071_mono__invE,axiom,
! [F: int > int,X: int,Y: int] :
( ( monotone_on_int_int @ top_top_set_int @ ord_less_eq_int @ ord_less_eq_int @ F )
=> ( ( ord_less_int @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_int @ X @ Y ) ) ) ).
% mono_invE
thf(fact_1072_nat__descend__induct,axiom,
! [N3: nat,P: nat > $o,M3: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N3 @ K )
=> ( P @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N3 )
=> ( ! [I: nat] :
( ( ord_less_nat @ K @ I )
=> ( P @ I ) )
=> ( P @ K ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_1073_Rats__eq__range__nat__to__rat__surj,axiom,
( field_6020823756834552118ts_rat
= ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat ) ) ).
% Rats_eq_range_nat_to_rat_surj
thf(fact_1074_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N3 @ ( F @ N3 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_1075_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1076_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1077_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1078_Rats__no__top__le,axiom,
! [X: real] :
? [X2: real] :
( ( member_real @ X2 @ field_5140801741446780682s_real )
& ( ord_less_eq_real @ X @ X2 ) ) ).
% Rats_no_top_le
thf(fact_1079_Rats__no__bot__less,axiom,
! [X: real] :
? [X2: real] :
( ( member_real @ X2 @ field_5140801741446780682s_real )
& ( ord_less_real @ X2 @ X ) ) ).
% Rats_no_bot_less
thf(fact_1080_Rats__dense__in__real,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [X2: real] :
( ( member_real @ X2 @ field_5140801741446780682s_real )
& ( ord_less_real @ X @ X2 )
& ( ord_less_real @ X2 @ Y ) ) ) ).
% Rats_dense_in_real
thf(fact_1081_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N4: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% pos_int_cases
thf(fact_1082_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
& ( K2
= ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1083_mono__times__nat,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ N3 ) ) ) ).
% mono_times_nat
thf(fact_1084_mult__is__0,axiom,
! [M3: nat,N3: nat] :
( ( ( times_times_nat @ M3 @ N3 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
| ( N3 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1085_mult__0__right,axiom,
! [M3: nat] :
( ( times_times_nat @ M3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1086_mult__cancel1,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ( times_times_nat @ K2 @ M3 )
= ( times_times_nat @ K2 @ N3 ) )
= ( ( M3 = N3 )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1087_mult__cancel2,axiom,
! [M3: nat,K2: nat,N3: nat] :
( ( ( times_times_nat @ M3 @ K2 )
= ( times_times_nat @ N3 @ K2 ) )
= ( ( M3 = N3 )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1088_nat__0__less__mult__iff,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M3 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1089_mult__less__cancel2,axiom,
! [M3: nat,K2: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ M3 @ K2 ) @ ( times_times_nat @ N3 @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M3 @ N3 ) ) ) ).
% mult_less_cancel2
thf(fact_1090_mult__le__cancel2,axiom,
! [M3: nat,K2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M3 @ K2 ) @ ( times_times_nat @ N3 @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ) ).
% mult_le_cancel2
thf(fact_1091_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1092_int__if,axiom,
! [P: $o,A2: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_1093_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1094_mult__0,axiom,
! [N3: nat] :
( ( times_times_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1095_mult__le__mono2,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I2 ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_1096_mult__le__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1097_mult__le__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1098_le__square,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ).
% le_square
thf(fact_1099_le__cube,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ) ).
% le_cube
thf(fact_1100_mult__less__mono2,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I2 ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1101_mult__less__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1102_zle__int,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% zle_int
thf(fact_1103_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1104_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M3 @ N3 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1105_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y4: real] :
? [N4: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1106_zmult__zless__mono2,axiom,
! [I2: int,J: int,K2: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I2 ) @ ( times_times_int @ K2 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1107_int__ops_I7_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1108_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1109_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K2: nat] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1110_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ( times_times_nat @ K2 @ M3 )
= ( times_times_nat @ K2 @ N3 ) )
= ( ( K2 = zero_zero_nat )
| ( M3 = N3 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1111_nat__mult__eq__cancel1,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M3 )
= ( times_times_nat @ K2 @ N3 ) )
= ( M3 = N3 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1112_nat__mult__less__cancel1,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1113_nat__mult__le__cancel1,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1114_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1115_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ~ ! [N4: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% neg_int_cases
thf(fact_1116_negative__eq__positive,axiom,
! [N3: nat,M3: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) )
= ( semiri1314217659103216013at_int @ M3 ) )
= ( ( N3 = zero_zero_nat )
& ( M3 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1117_not__int__zless__negative,axiom,
! [N3: nat,M3: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M3 ) ) ) ).
% not_int_zless_negative
thf(fact_1118_int__cases4,axiom,
! [M3: int] :
( ! [N4: nat] :
( M3
!= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( M3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% int_cases4
thf(fact_1119_int__zle__neg,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M3 ) ) )
= ( ( N3 = zero_zero_nat )
& ( M3 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1120_int__cases3,axiom,
! [K2: int] :
( ( K2 != zero_zero_int )
=> ( ! [N4: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N4 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
=> ~ ! [N4: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% int_cases3
thf(fact_1121_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1122_real__eq__0__iff__le__ge__0,axiom,
! [X: real] :
( ( X = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_1123_nat__less__iff,axiom,
! [W2: int,M3: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M3 )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M3 ) ) ) ) ).
% nat_less_iff
thf(fact_1124_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1125_nat__0__iff,axiom,
! [I2: int] :
( ( ( nat2 @ I2 )
= zero_zero_nat )
= ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1126_zless__nat__conj,axiom,
! [W2: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W2 @ Z ) ) ) ).
% zless_nat_conj
thf(fact_1127_nat__zminus__int,axiom,
! [N3: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1128_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_1129_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1130_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1131_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1132_nat__mono__iff,axiom,
! [Z: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_mono_iff
thf(fact_1133_nat__le__iff,axiom,
! [X: int,N3: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N3 )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nat_le_iff
thf(fact_1134_zless__nat__eq__int__zless,axiom,
! [M3: nat,Z: int] :
( ( ord_less_nat @ M3 @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_1135_nat__eq__iff2,axiom,
! [M3: nat,W2: int] :
( ( M3
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M3 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M3 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1136_nat__eq__iff,axiom,
! [W2: int,M3: nat] :
( ( ( nat2 @ W2 )
= M3 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M3 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M3 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1137_split__nat,axiom,
! [P: nat > $o,I2: int] :
( ( P @ ( nat2 @ I2 ) )
= ( ! [N: nat] :
( ( I2
= ( semiri1314217659103216013at_int @ N ) )
=> ( P @ N ) )
& ( ( ord_less_int @ I2 @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1138_nat__le__eq__zle,axiom,
! [W2: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_1139_le__nat__iff,axiom,
! [K2: int,N3: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_nat @ N3 @ ( nat2 @ K2 ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N3 ) @ K2 ) ) ) ).
% le_nat_iff
thf(fact_1140_nat__less__eq__zless,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W2 @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_1141_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D5: real,E: real] :
( ( ord_less_real @ D5 @ E )
=> ( ( P @ D5 )
=> ( P @ E ) ) )
=> ( ! [N4: nat] :
( ( N4 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_1142_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N: nat] :
( ( N != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_1143_real__arch__invD,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [N4: nat] :
( ( N4 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E2 ) ) ) ).
% real_arch_invD
thf(fact_1144_nat__mult__eq__1__iff,axiom,
! [M3: nat,N3: nat] :
( ( ( times_times_nat @ M3 @ N3 )
= one_one_nat )
= ( ( M3 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1145_nat__1__eq__mult__iff,axiom,
! [M3: nat,N3: nat] :
( ( one_one_nat
= ( times_times_nat @ M3 @ N3 ) )
= ( ( M3 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1146_less__one,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ one_one_nat )
= ( N3 = zero_zero_nat ) ) ).
% less_one
thf(fact_1147_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1148_nat__mult__1__right,axiom,
! [N3: nat] :
( ( times_times_nat @ N3 @ one_one_nat )
= N3 ) ).
% nat_mult_1_right
thf(fact_1149_nat__mult__1,axiom,
! [N3: nat] :
( ( times_times_nat @ one_one_nat @ N3 )
= N3 ) ).
% nat_mult_1
thf(fact_1150_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1151_mult__eq__self__implies__10,axiom,
! [M3: nat,N3: nat] :
( ( M3
= ( times_times_nat @ M3 @ N3 ) )
=> ( ( N3 = one_one_nat )
| ( M3 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1152_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_1153_pos__zmult__eq__1__iff,axiom,
! [M3: int,N3: int] :
( ( ord_less_int @ zero_zero_int @ M3 )
=> ( ( ( times_times_int @ M3 @ N3 )
= one_one_int )
= ( ( M3 = one_one_int )
& ( N3 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1154_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q2: nat > $o] :
( ! [X2: nat > real] :
( ( P @ X2 )
=> ( P @ ( F @ X2 ) ) )
=> ( ! [X2: nat > real] :
( ( P @ X2 )
=> ! [I3: nat] :
( ( Q2 @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I3 ) )
& ( ord_less_eq_real @ ( X2 @ I3 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I: nat] : ( ord_less_eq_nat @ ( L2 @ X5 @ I ) @ one_one_nat )
& ! [X5: nat > real,I: nat] :
( ( ( P @ X5 )
& ( Q2 @ I )
& ( ( X5 @ I )
= zero_zero_real ) )
=> ( ( L2 @ X5 @ I )
= zero_zero_nat ) )
& ! [X5: nat > real,I: nat] :
( ( ( P @ X5 )
& ( Q2 @ I )
& ( ( X5 @ I )
= one_one_real ) )
=> ( ( L2 @ X5 @ I )
= one_one_nat ) )
& ! [X5: nat > real,I: nat] :
( ( ( P @ X5 )
& ( Q2 @ I )
& ( ( L2 @ X5 @ I )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I ) @ ( F @ X5 @ I ) ) )
& ! [X5: nat > real,I: nat] :
( ( ( P @ X5 )
& ( Q2 @ I )
& ( ( L2 @ X5 @ I )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X5 @ I ) @ ( X5 @ I ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1155_real__of__nat__ge__one__iff,axiom,
! [N3: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
= ( ord_less_eq_nat @ one_one_nat @ N3 ) ) ).
% real_of_nat_ge_one_iff
thf(fact_1156_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_1157_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1158_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1159_Suc__le__mono,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N3 @ M3 ) ) ).
% Suc_le_mono
thf(fact_1160_lessI,axiom,
! [N3: nat] : ( ord_less_nat @ N3 @ ( suc @ N3 ) ) ).
% lessI
thf(fact_1161_Suc__mono,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N3 ) ) ) ).
% Suc_mono
thf(fact_1162_Suc__less__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_less_eq
thf(fact_1163_less__Suc0,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( suc @ zero_zero_nat ) )
= ( N3 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1164_zero__less__Suc,axiom,
! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N3 ) ) ).
% zero_less_Suc
thf(fact_1165_mult__eq__1__iff,axiom,
! [M3: nat,N3: nat] :
( ( ( times_times_nat @ M3 @ N3 )
= ( suc @ zero_zero_nat ) )
= ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1166_one__eq__mult__iff,axiom,
! [M3: nat,N3: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M3 @ N3 ) )
= ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1167_one__le__mult__iff,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M3 @ N3 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M3 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N3 ) ) ) ).
% one_le_mult_iff
thf(fact_1168_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1169_negative__zless,axiom,
! [N3: nat,M3: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) ).
% negative_zless
thf(fact_1170_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1171_not0__implies__Suc,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ? [M5: nat] :
( N3
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1172_Zero__not__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_1173_Zero__neq__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_1174_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1175_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1176_diff__induct,axiom,
! [P: nat > nat > $o,M3: nat,N3: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M3 @ N3 ) ) ) ) ).
% diff_induct
thf(fact_1177_nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N3 ) ) ) ).
% nat_induct
thf(fact_1178_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1179_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1180_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1181_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1182_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1183_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N4: nat] :
( ~ ( P @ N4 )
& ( P @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1184_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1185_n__not__Suc__n,axiom,
! [N3: nat] :
( N3
!= ( suc @ N3 ) ) ).
% n_not_Suc_n
thf(fact_1186_Nat_OlessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( K2
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1187_Suc__lessD,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ N3 )
=> ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_lessD
thf(fact_1188_Suc__lessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1189_Suc__lessI,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ( ( suc @ M3 )
!= N3 )
=> ( ord_less_nat @ ( suc @ M3 ) @ N3 ) ) ) ).
% Suc_lessI
thf(fact_1190_less__SucE,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_nat @ M3 @ N3 )
=> ( M3 = N3 ) ) ) ).
% less_SucE
thf(fact_1191_less__SucI,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ M3 @ ( suc @ N3 ) ) ) ).
% less_SucI
thf(fact_1192_Ex__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N3 ) )
& ( P @ I4 ) ) )
= ( ( P @ N3 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1193_less__Suc__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
= ( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ).
% less_Suc_eq
thf(fact_1194_not__less__eq,axiom,
! [M3: nat,N3: nat] :
( ( ~ ( ord_less_nat @ M3 @ N3 ) )
= ( ord_less_nat @ N3 @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_1195_All__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N3 ) )
=> ( P @ I4 ) ) )
= ( ( P @ N3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1196_Suc__less__eq2,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ N3 ) @ M3 )
= ( ? [M6: nat] :
( ( M3
= ( suc @ M6 ) )
& ( ord_less_nat @ N3 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1197_less__antisym,axiom,
! [N3: nat,M3: nat] :
( ~ ( ord_less_nat @ N3 @ M3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M3 ) )
=> ( M3 = N3 ) ) ) ).
% less_antisym
thf(fact_1198_Suc__less__SucD,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N3 ) )
=> ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_less_SucD
thf(fact_1199_less__trans__Suc,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I2 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1200_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K )
=> ( P @ I3 @ K ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1201_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1202_not__less__less__Suc__eq,axiom,
! [N3: nat,M3: nat] :
( ~ ( ord_less_nat @ N3 @ M3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M3 ) )
= ( N3 = M3 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1203_transitive__stepwise__le,axiom,
! [M3: nat,N3: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ! [X2: nat] : ( R4 @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z2: nat] :
( ( R4 @ X2 @ Y3 )
=> ( ( R4 @ Y3 @ Z2 )
=> ( R4 @ X2 @ Z2 ) ) )
=> ( ! [N4: nat] : ( R4 @ N4 @ ( suc @ N4 ) )
=> ( R4 @ M3 @ N3 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1204_nat__induct__at__least,axiom,
! [M3: nat,N3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( P @ M3 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1205_full__nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N3 ) ) ).
% full_nat_induct
thf(fact_1206_not__less__eq__eq,axiom,
! [M3: nat,N3: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N3 ) )
= ( ord_less_eq_nat @ ( suc @ N3 ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_1207_Suc__n__not__le__n,axiom,
! [N3: nat] :
~ ( ord_less_eq_nat @ ( suc @ N3 ) @ N3 ) ).
% Suc_n_not_le_n
thf(fact_1208_le__Suc__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) )
= ( ( ord_less_eq_nat @ M3 @ N3 )
| ( M3
= ( suc @ N3 ) ) ) ) ).
% le_Suc_eq
thf(fact_1209_Suc__le__D,axiom,
! [N3: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ M7 )
=> ? [M5: nat] :
( M7
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1210_le__SucI,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) ) ) ).
% le_SucI
thf(fact_1211_le__SucE,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N3 )
=> ( M3
= ( suc @ N3 ) ) ) ) ).
% le_SucE
thf(fact_1212_Suc__leD,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% Suc_leD
thf(fact_1213_Suc__mult__cancel1,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M3 )
= ( times_times_nat @ ( suc @ K2 ) @ N3 ) )
= ( M3 = N3 ) ) ).
% Suc_mult_cancel1
thf(fact_1214_Suc__mult__less__cancel1,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M3 ) @ ( times_times_nat @ ( suc @ K2 ) @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1215_Suc__mult__le__cancel1,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M3 ) @ ( times_times_nat @ ( suc @ K2 ) @ N3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1216_Suc__leI,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 ) ) ).
% Suc_leI
thf(fact_1217_Suc__le__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_le_eq
thf(fact_1218_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1219_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1220_Suc__le__lessD,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_le_lessD
thf(fact_1221_le__less__Suc__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M3 ) )
= ( N3 = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_1222_less__Suc__eq__le,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% less_Suc_eq_le
thf(fact_1223_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1224_le__imp__less__Suc,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_nat @ M3 @ ( suc @ N3 ) ) ) ).
% le_imp_less_Suc
thf(fact_1225_less__Suc__eq__0__disj,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
= ( ( M3 = zero_zero_nat )
| ? [J3: nat] :
( ( M3
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N3 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1226_gr0__implies__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ? [M5: nat] :
( N3
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1227_All__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N3 ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1228_gr0__conv__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( ? [M: nat] :
( N3
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1229_Ex__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N3 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1230_ex__least__nat__less,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N3 )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K )
=> ~ ( P @ I ) )
& ( P @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1231_nat__induct__non__zero,axiom,
! [N3: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1232_one__less__mult,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M3 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M3 @ N3 ) ) ) ) ).
% one_less_mult
thf(fact_1233_n__less__m__mult__n,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M3 )
=> ( ord_less_nat @ N3 @ ( times_times_nat @ M3 @ N3 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1234_n__less__n__mult__m,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M3 )
=> ( ord_less_nat @ N3 @ ( times_times_nat @ N3 @ M3 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1235_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_1236_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D5: real,E: real] :
( ( ord_less_real @ D5 @ E )
=> ( ( P @ D5 )
=> ( P @ E ) ) )
=> ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_1237_negative__zless__0,axiom,
! [N3: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1238_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N4: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% negD
thf(fact_1239_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N4: nat] :
( X
!= ( suc @ N4 ) ) ) ).
% list_decode.cases
thf(fact_1240_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_1241_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1242_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ord_less_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1243_bounded__nat__set__is__finite,axiom,
! [N2: set_nat,N3: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ord_less_nat @ X2 @ N3 ) )
=> ( finite_finite_nat @ N2 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1244_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1245_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1246_infinite__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ? [R2: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R2 )
& ! [N5: nat] : ( member_nat @ ( R2 @ N5 ) @ S ) ) ) ).
% infinite_enumerate
thf(fact_1247_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( member_nat @ N @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1248_unbounded__k__infinite,axiom,
! [K2: nat,S: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K2 @ M5 )
=> ? [N5: nat] :
( ( ord_less_nat @ M5 @ N5 )
& ( member_nat @ N5 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1249_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M: nat] :
? [N: nat] :
( ( ord_less_nat @ M @ N )
& ( member_nat @ N @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1250_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_1251_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1252_card__nat,axiom,
( ( finite_card_nat @ top_top_set_nat )
= zero_zero_nat ) ).
% card_nat
thf(fact_1253_card__literal,axiom,
( ( finite_card_literal @ top_top_set_literal )
= zero_zero_nat ) ).
% card_literal
thf(fact_1254_card__num0,axiom,
( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
= zero_zero_nat ) ).
% card_num0
thf(fact_1255_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_1256_add__is__0,axiom,
! [M3: nat,N3: nat] :
( ( ( plus_plus_nat @ M3 @ N3 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N3 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1257_add__Suc__right,axiom,
! [M3: nat,N3: nat] :
( ( plus_plus_nat @ M3 @ ( suc @ N3 ) )
= ( suc @ ( plus_plus_nat @ M3 @ N3 ) ) ) ).
% add_Suc_right
thf(fact_1258_Suc__funpow,axiom,
! [N3: nat] :
( ( compow_nat_nat @ N3 @ suc )
= ( plus_plus_nat @ N3 ) ) ).
% Suc_funpow
thf(fact_1259_nat__add__left__cancel__le,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% nat_add_left_cancel_le
thf(fact_1260_nat__add__left__cancel__less,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% nat_add_left_cancel_less
thf(fact_1261_add__gr__0,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% add_gr_0
thf(fact_1262_mult__Suc__right,axiom,
! [M3: nat,N3: nat] :
( ( times_times_nat @ M3 @ ( suc @ N3 ) )
= ( plus_plus_nat @ M3 @ ( times_times_nat @ M3 @ N3 ) ) ) ).
% mult_Suc_right
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [H: real > real] :
( ( member_real_real @ H @ ( image_nat_real_real @ g @ top_top_set_nat ) )
=> ( ( ord_less_real @ y @ ( H @ x ) )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------