TPTP Problem File: SLH0488^1.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 : Dedekind_Real/0000_Dedekind_Real/prob_00936_030900__5704226_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1448 ( 399 unt; 175 typ; 0 def)
% Number of atoms : 4292 (1060 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 12078 ( 469 ~; 58 |; 203 &;9213 @)
% ( 0 <=>;2135 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 1102 (1102 >; 0 *; 0 +; 0 <<)
% Number of symbols : 161 ( 158 usr; 16 con; 0-3 aty)
% Number of variables : 4260 ( 417 ^;3683 !; 160 ?;4260 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:30:07.838
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
set_set_set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Dedekind____Real__Opreal_J_J,type,
set_se4284277844656040150_preal: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Rat__Orat_J_M_Eo_J_J,type,
set_set_rat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Dedekind____Real__Opreal_M_Eo_J_J,type,
set_Dedekind_preal_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
set_set_rat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
set_Dedekind_preal: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Rat__Orat_M_Eo_J_J,type,
set_rat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $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__Dedekind____Real__Opreal,type,
dedekind_preal: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Rat__Orat,type,
rat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (158)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_Eo_M_Eo_J,type,
complete_Sup_Sup_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Dedekind____Real__Opreal_M_Eo_J,type,
comple7848071142344908218real_o: set_Dedekind_preal_o > dedekind_preal > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Rat__Orat_M_Eo_J,type,
comple4580332206425622756_rat_o: set_rat_o > rat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
comple3057484590954014638_rat_o: set_set_rat_o > set_rat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
comple800427960086186211_preal: set_se4284277844656040150_preal > set_Dedekind_preal ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Rat__Orat_J,type,
comple3890839924845867745et_rat: set_set_rat > set_rat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
comple3392050375588816791et_rat: set_set_set_rat > set_set_rat ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001_Eo,type,
condit5488710616941104124bove_o: set_o > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Dedekind____Real__Opreal,type,
condit1512905060068503930_preal: set_Dedekind_preal > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Rat__Orat,type,
condit1579696412822616692ve_rat: set_rat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Rat__Orat_J,type,
condit1969311730731577898et_rat: set_set_rat > $o ).
thf(sy_c_Dedekind__Real_Oadd__set,type,
dedekind_add_set: set_rat > set_rat > set_rat ).
thf(sy_c_Dedekind__Real_Ocut,type,
dedekind_cut: set_rat > $o ).
thf(sy_c_Dedekind__Real_Odiff__set,type,
dedekind_diff_set: set_rat > set_rat > set_rat ).
thf(sy_c_Dedekind__Real_Omult__set,type,
dedekind_mult_set: set_rat > set_rat > set_rat ).
thf(sy_c_Dedekind__Real_Opreal_OAbs__preal,type,
dedekind_Abs_preal: set_rat > dedekind_preal ).
thf(sy_c_Dedekind__Real_Opreal_ORep__preal,type,
dedekind_Rep_preal: dedekind_preal > set_rat ).
thf(sy_c_Dedekind__Real_Opsup,type,
dedekind_psup: set_Dedekind_preal > dedekind_preal ).
thf(sy_c_Finite__Set_OFpow_001t__Dedekind____Real__Opreal,type,
finite6657831709359810172_preal: set_Dedekind_preal > set_se4284277844656040150_preal ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Rat__Orat,type,
finite_Fpow_rat: set_rat > set_set_rat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Rat__Orat_J,type,
finite_Fpow_set_rat: set_set_rat > set_set_set_rat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Dedekind____Real__Opreal,type,
minus_7336623429200594941_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Dedekind____Real__Opreal,type,
plus_p3173629198307831117_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
plus_plus_rat: rat > rat > rat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Dedekind____Real__Opreal,type,
times_3000655703912201937_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
times_times_rat: rat > rat > rat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
zero_zero_rat: rat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Dedekind____Real__Opreal_M_Eo_J,type,
bot_bo4815710026557797371real_o: dedekind_preal > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Rat__Orat_M_Eo_J,type,
bot_bot_rat_o: rat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
bot_bot_set_rat_o: set_rat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
bot_bo4848840305443107682_preal: set_Dedekind_preal ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
bot_bot_set_rat: set_rat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Dedekind____Real__Opreal_J_J,type,
bot_bo5050857909902108994_preal: set_se4284277844656040150_preal ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
bot_bot_set_set_rat: set_set_rat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_M_Eo_J,type,
ord_less_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Dedekind____Real__Opreal_M_Eo_J,type,
ord_le1914443497901267899real_o: ( dedekind_preal > $o ) > ( dedekind_preal > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Rat__Orat_M_Eo_J,type,
ord_less_rat_o: ( rat > $o ) > ( rat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
ord_less_set_rat_o: ( set_rat > $o ) > ( set_rat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Dedekind____Real__Opreal,type,
ord_le5708704896291381698_preal: dedekind_preal > dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__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__Set__Oset_I_Eo_J,type,
ord_less_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
ord_le1802228187270208418_preal: set_Dedekind_preal > set_Dedekind_preal > $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__Set__Oset_It__Rat__Orat_J_J,type,
ord_less_set_set_rat: set_set_rat > set_set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Dedekind____Real__Opreal_M_Eo_J,type,
ord_le5990576202697070023real_o: ( dedekind_preal > $o ) > ( dedekind_preal > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Rat__Orat_M_Eo_J,type,
ord_less_eq_rat_o: ( rat > $o ) > ( rat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
ord_le3214917520859460321_rat_o: ( set_rat > $o ) > ( set_rat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Dedekind____Real__Opreal,type,
ord_le5604041210740703414_preal: dedekind_preal > dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
ord_le7349499860212017814_preal: set_Dedekind_preal > set_Dedekind_preal > $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__Set__Oset_It__Rat__Orat_J_J,type,
ord_le513522071413781156et_rat: set_set_rat > set_set_rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
ord_le8552383839478139994et_rat: set_set_set_rat > set_set_set_rat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
orderi8630438233043249959_preal: ( set_Dedekind_preal > set_Dedekind_preal > $o ) > ( set_Dedekind_preal > set_Dedekind_preal > $o ) > set_Dedekind_preal > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Rat__Orat_J,type,
ordering_top_set_rat: ( set_rat > set_rat > $o ) > ( set_rat > set_rat > $o ) > set_rat > $o ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Dedekind____Real__Opreal,type,
collec1132657498972982273_preal: ( dedekind_preal > $o ) > set_Dedekind_preal ).
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__Set__Oset_It__Rat__Orat_J,type,
collect_set_rat: ( set_rat > $o ) > set_set_rat ).
thf(sy_c_Set_OPow_001t__Dedekind____Real__Opreal,type,
pow_Dedekind_preal: set_Dedekind_preal > set_se4284277844656040150_preal ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Rat__Orat,type,
pow_rat: set_rat > set_set_rat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Rat__Orat_J,type,
pow_set_rat: set_set_rat > set_set_set_rat ).
thf(sy_c_Set_Oimage_001_062_I_Eo_M_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_o_o_set_o: ( ( $o > $o ) > set_o ) > set_o_o > set_set_o ).
thf(sy_c_Set_Oimage_001_062_It__Dedekind____Real__Opreal_M_Eo_J_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
image_8702985839003149832_preal: ( ( dedekind_preal > $o ) > set_Dedekind_preal ) > set_Dedekind_preal_o > set_se4284277844656040150_preal ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Rat__Orat_M_Eo_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_rat_o_set_rat: ( ( rat > $o ) > set_rat ) > set_rat_o > set_set_rat ).
thf(sy_c_Set_Oimage_001_062_It__Set__Oset_It__Rat__Orat_J_M_Eo_J_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
image_3676206661936088128et_rat: ( ( set_rat > $o ) > set_set_rat ) > set_set_rat_o > set_set_set_rat ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Dedekind____Real__Opreal,type,
image_3026614506929738499_preal: ( $o > dedekind_preal ) > set_o > set_Dedekind_preal ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Rat__Orat,type,
image_o_rat: ( $o > rat ) > set_o > set_rat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
image_2856262260070713571_preal: ( $o > set_Dedekind_preal ) > set_o > set_se4284277844656040150_preal ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
image_o_set_nat: ( $o > set_nat ) > set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Rat__Orat_J,type,
image_o_set_rat: ( $o > set_rat ) > set_o > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Dedekind____Real__Opreal_001_062_It__Rat__Orat_M_Eo_J,type,
image_8369558857164962022_rat_o: ( dedekind_preal > rat > $o ) > set_Dedekind_preal > set_rat_o ).
thf(sy_c_Set_Oimage_001t__Dedekind____Real__Opreal_001_Eo,type,
image_248103505173020295real_o: ( dedekind_preal > $o ) > set_Dedekind_preal > set_o ).
thf(sy_c_Set_Oimage_001t__Dedekind____Real__Opreal_001t__Dedekind____Real__Opreal,type,
image_4420143364984939013_preal: ( dedekind_preal > dedekind_preal ) > set_Dedekind_preal > set_Dedekind_preal ).
thf(sy_c_Set_Oimage_001t__Dedekind____Real__Opreal_001t__Nat__Onat,type,
image_7207786835304483489al_nat: ( dedekind_preal > nat ) > set_Dedekind_preal > set_nat ).
thf(sy_c_Set_Oimage_001t__Dedekind____Real__Opreal_001t__Rat__Orat,type,
image_6572656775217987753al_rat: ( dedekind_preal > rat ) > set_Dedekind_preal > set_rat ).
thf(sy_c_Set_Oimage_001t__Dedekind____Real__Opreal_001t__Set__Oset_I_Eo_J,type,
image_8163909360452110695_set_o: ( dedekind_preal > set_o ) > set_Dedekind_preal > set_set_o ).
thf(sy_c_Set_Oimage_001t__Dedekind____Real__Opreal_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
image_85258263552348133_preal: ( dedekind_preal > set_Dedekind_preal ) > set_Dedekind_preal > set_se4284277844656040150_preal ).
thf(sy_c_Set_Oimage_001t__Dedekind____Real__Opreal_001t__Set__Oset_It__Nat__Onat_J,type,
image_5739498456150602327et_nat: ( dedekind_preal > set_nat ) > set_Dedekind_preal > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Dedekind____Real__Opreal_001t__Set__Oset_It__Rat__Orat_J,type,
image_2231269897757205599et_rat: ( dedekind_preal > set_rat ) > set_Dedekind_preal > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Dedekind____Real__Opreal_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
image_8268678932589273365et_rat: ( dedekind_preal > set_set_rat ) > set_Dedekind_preal > set_set_set_rat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Dedekind____Real__Opreal,type,
image_8829711020507855231_preal: ( nat > dedekind_preal ) > set_nat > set_Dedekind_preal ).
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__Set__Oset_I_Eo_J,type,
image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
image_6010050456725241695_preal: ( nat > set_Dedekind_preal ) > set_nat > set_se4284277844656040150_preal ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Rat__Orat_J,type,
image_nat_set_rat: ( nat > set_rat ) > set_nat > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001_Eo,type,
image_rat_o: ( rat > $o ) > set_rat > set_o ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Dedekind____Real__Opreal,type,
image_7112546572676366711_preal: ( rat > dedekind_preal ) > set_rat > set_Dedekind_preal ).
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__Rat__Orat,type,
image_rat_rat: ( rat > rat ) > set_rat > set_rat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Set__Oset_I_Eo_J,type,
image_rat_set_o: ( rat > set_o ) > set_rat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
image_1093189101101737815_preal: ( rat > set_Dedekind_preal ) > set_rat > set_se4284277844656040150_preal ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Set__Oset_It__Nat__Onat_J,type,
image_rat_set_nat: ( rat > set_nat ) > set_rat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Set__Oset_It__Rat__Orat_J,type,
image_rat_set_rat: ( rat > set_rat ) > set_rat > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_062_I_Eo_M_Eo_J,type,
image_set_o_o_o: ( set_o > $o > $o ) > set_set_o > set_o_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_Eo,type,
image_set_o_o: ( set_o > $o ) > set_set_o > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Dedekind____Real__Opreal_J_001_062_It__Dedekind____Real__Opreal_M_Eo_J,type,
image_7211531428116543256real_o: ( set_Dedekind_preal > dedekind_preal > $o ) > set_se4284277844656040150_preal > set_Dedekind_preal_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Dedekind____Real__Opreal_J_001_Eo,type,
image_3678684046732248231real_o: ( set_Dedekind_preal > $o ) > set_se4284277844656040150_preal > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Dedekind____Real__Opreal_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_4042146516684777023et_rat: ( set_Dedekind_preal > set_rat ) > set_se4284277844656040150_preal > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Dedekind____Real__Opreal_J_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
image_1970760655392924405et_rat: ( set_Dedekind_preal > set_set_rat ) > set_se4284277844656040150_preal > set_set_set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001_062_It__Rat__Orat_M_Eo_J,type,
image_set_rat_rat_o: ( set_rat > rat > $o ) > set_set_rat > set_rat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001_Eo,type,
image_set_rat_o: ( set_rat > $o ) > set_set_rat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001t__Dedekind____Real__Opreal,type,
image_6579680541524808769_preal: ( set_rat > dedekind_preal ) > set_set_rat > set_Dedekind_preal ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
image_7647003310076334625_preal: ( set_rat > set_Dedekind_preal ) > set_set_rat > set_se4284277844656040150_preal ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_3939399684171694371et_rat: ( set_rat > set_rat ) > set_set_rat > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_001_062_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
image_5922895781192582838_rat_o: ( set_set_rat > set_rat > $o ) > set_set_set_rat > set_set_rat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_001_Eo,type,
image_set_set_rat_o: ( set_set_rat > $o ) > set_set_set_rat > set_o ).
thf(sy_c_Set_Ois__empty_001t__Dedekind____Real__Opreal,type,
is_emp2142545888348893928_preal: set_Dedekind_preal > $o ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Rat__Orat_J,type,
the_elem_set_rat: set_set_rat > set_rat ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Dedekind____Real__Opreal,type,
member6871284927547481791_preal: dedekind_preal > set_Dedekind_preal > $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__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
member2000626104285586591_preal: set_Dedekind_preal > set_se4284277844656040150_preal > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Rat__Orat_J,type,
member_set_rat: set_rat > set_set_rat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
member_set_set_rat: set_set_rat > set_set_set_rat > $o ).
thf(sy_v_P,type,
p: set_Dedekind_preal ).
thf(sy_v_Y,type,
y: dedekind_preal ).
thf(sy_v_Z,type,
z: dedekind_preal ).
% Relevant facts (1272)
thf(fact_0_preal__less__def,axiom,
( ord_le5708704896291381698_preal
= ( ^ [R: dedekind_preal,S: dedekind_preal] : ( ord_less_set_rat @ ( dedekind_Rep_preal @ R ) @ ( dedekind_Rep_preal @ S ) ) ) ) ).
% preal_less_def
thf(fact_1__092_060open_062_092_060exists_062X_092_060in_062P_O_AZ_A_060_AX_092_060close_062,axiom,
? [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ p )
& ( ord_le5708704896291381698_preal @ z @ X ) ) ).
% \<open>\<exists>X\<in>P. Z < X\<close>
thf(fact_2_assms_I2_J,axiom,
p != bot_bo4848840305443107682_preal ).
% assms(2)
thf(fact_3_le,axiom,
! [X2: dedekind_preal] :
( ( member6871284927547481791_preal @ X2 @ p )
=> ( ord_le5604041210740703414_preal @ X2 @ y ) ) ).
% le
thf(fact_4_minf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Z )
=> ~ ( ord_le5708704896291381698_preal @ T @ X3 ) ) ).
% minf(7)
thf(fact_5_minf_I7_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ~ ( ord_less_rat @ T @ X3 ) ) ).
% minf(7)
thf(fact_6_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_7_minf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Z )
=> ( ord_le5708704896291381698_preal @ X3 @ T ) ) ).
% minf(5)
thf(fact_8_minf_I5_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ( ord_less_rat @ X3 @ T ) ) ).
% minf(5)
thf(fact_9_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_10_minf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_11_minf_I4_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_12_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_13_minf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_14_minf_I3_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_15_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_16_minf_I2_J,axiom,
! [P: dedekind_preal > $o,P2: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Z )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_17_minf_I2_J,axiom,
! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_18_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_19_minf_I1_J,axiom,
! [P: dedekind_preal > $o,P2: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Z )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_20_minf_I1_J,axiom,
! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_21_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_22_pinf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X3 )
=> ( ord_le5708704896291381698_preal @ T @ X3 ) ) ).
% pinf(7)
thf(fact_23_pinf_I7_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( ord_less_rat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_24_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_25_pinf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X3 )
=> ~ ( ord_le5708704896291381698_preal @ X3 @ T ) ) ).
% pinf(5)
thf(fact_26_pinf_I5_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ~ ( ord_less_rat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_27_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_28_pinf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_29_pinf_I4_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_30_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_31_pinf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X3 )
=> ~ ( ord_le5604041210740703414_preal @ X3 @ T ) ) ).
% pinf(6)
thf(fact_32_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_33_pinf_I6_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ~ ( ord_less_eq_rat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_34_pinf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X3 )
=> ( ord_le5604041210740703414_preal @ T @ X3 ) ) ).
% pinf(8)
thf(fact_35_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_36_pinf_I8_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( ord_less_eq_rat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_37_minf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Z )
=> ( ord_le5604041210740703414_preal @ X3 @ T ) ) ).
% minf(6)
thf(fact_38_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_39_minf_I6_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ( ord_less_eq_rat @ X3 @ T ) ) ).
% minf(6)
thf(fact_40_minf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Z )
=> ~ ( ord_le5604041210740703414_preal @ T @ X3 ) ) ).
% minf(8)
thf(fact_41_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_42_minf_I8_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z )
=> ~ ( ord_less_eq_rat @ T @ X3 ) ) ).
% minf(8)
thf(fact_43_psup__le__ub,axiom,
! [P: set_Dedekind_preal,Y: dedekind_preal] :
( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ P )
=> ( ord_le5604041210740703414_preal @ X4 @ Y ) )
=> ( ( P != bot_bo4848840305443107682_preal )
=> ( ord_le5604041210740703414_preal @ ( dedekind_psup @ P ) @ Y ) ) ) ).
% psup_le_ub
thf(fact_44_mem__Rep__preal__Ex,axiom,
! [X2: dedekind_preal] :
? [X: rat] : ( member_rat @ X @ ( dedekind_Rep_preal @ X2 ) ) ).
% mem_Rep_preal_Ex
thf(fact_45_Rep__preal__inject,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ( dedekind_Rep_preal @ X5 )
= ( dedekind_Rep_preal @ Y2 ) )
= ( X5 = Y2 ) ) ).
% Rep_preal_inject
thf(fact_46_preal__psup__le,axiom,
! [P: set_Dedekind_preal,Y: dedekind_preal,X5: dedekind_preal] :
( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ P )
=> ( ord_le5604041210740703414_preal @ X4 @ Y ) )
=> ( ( member6871284927547481791_preal @ X5 @ P )
=> ( ord_le5604041210740703414_preal @ X5 @ ( dedekind_psup @ P ) ) ) ) ).
% preal_psup_le
thf(fact_47_pinf_I1_J,axiom,
! [P: dedekind_preal > $o,P2: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_48_pinf_I1_J,axiom,
! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_49_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_50_pinf_I2_J,axiom,
! [P: dedekind_preal > $o,P2: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_51_pinf_I2_J,axiom,
! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: rat] :
! [X: rat] :
( ( ord_less_rat @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_52_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_53_pinf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z: dedekind_preal] :
! [X3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_54_pinf_I3_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_55_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_56_Rep__preal__inverse,axiom,
! [X5: dedekind_preal] :
( ( dedekind_Abs_preal @ ( dedekind_Rep_preal @ X5 ) )
= X5 ) ).
% Rep_preal_inverse
thf(fact_57_empty__iff,axiom,
! [C: rat] :
~ ( member_rat @ C @ bot_bot_set_rat ) ).
% empty_iff
thf(fact_58_empty__iff,axiom,
! [C: set_rat] :
~ ( member_set_rat @ C @ bot_bot_set_set_rat ) ).
% empty_iff
thf(fact_59_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_60_empty__iff,axiom,
! [C: dedekind_preal] :
~ ( member6871284927547481791_preal @ C @ bot_bo4848840305443107682_preal ) ).
% empty_iff
thf(fact_61_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_62_all__not__in__conv,axiom,
! [A: set_rat] :
( ( ! [X6: rat] :
~ ( member_rat @ X6 @ A ) )
= ( A = bot_bot_set_rat ) ) ).
% all_not_in_conv
thf(fact_63_all__not__in__conv,axiom,
! [A: set_set_rat] :
( ( ! [X6: set_rat] :
~ ( member_set_rat @ X6 @ A ) )
= ( A = bot_bot_set_set_rat ) ) ).
% all_not_in_conv
thf(fact_64_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X6: $o] :
~ ( member_o @ X6 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_65_all__not__in__conv,axiom,
! [A: set_Dedekind_preal] :
( ( ! [X6: dedekind_preal] :
~ ( member6871284927547481791_preal @ X6 @ A ) )
= ( A = bot_bo4848840305443107682_preal ) ) ).
% all_not_in_conv
thf(fact_66_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X6: nat] :
~ ( member_nat @ X6 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_67_Collect__empty__eq,axiom,
! [P: set_rat > $o] :
( ( ( collect_set_rat @ P )
= bot_bot_set_set_rat )
= ( ! [X6: set_rat] :
~ ( P @ X6 ) ) ) ).
% Collect_empty_eq
thf(fact_68_Collect__empty__eq,axiom,
! [P: rat > $o] :
( ( ( collect_rat @ P )
= bot_bot_set_rat )
= ( ! [X6: rat] :
~ ( P @ X6 ) ) ) ).
% Collect_empty_eq
thf(fact_69_Collect__empty__eq,axiom,
! [P: dedekind_preal > $o] :
( ( ( collec1132657498972982273_preal @ P )
= bot_bo4848840305443107682_preal )
= ( ! [X6: dedekind_preal] :
~ ( P @ X6 ) ) ) ).
% Collect_empty_eq
thf(fact_70_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X6: nat] :
~ ( P @ X6 ) ) ) ).
% Collect_empty_eq
thf(fact_71_empty__Collect__eq,axiom,
! [P: set_rat > $o] :
( ( bot_bot_set_set_rat
= ( collect_set_rat @ P ) )
= ( ! [X6: set_rat] :
~ ( P @ X6 ) ) ) ).
% empty_Collect_eq
thf(fact_72_empty__Collect__eq,axiom,
! [P: rat > $o] :
( ( bot_bot_set_rat
= ( collect_rat @ P ) )
= ( ! [X6: rat] :
~ ( P @ X6 ) ) ) ).
% empty_Collect_eq
thf(fact_73_empty__Collect__eq,axiom,
! [P: dedekind_preal > $o] :
( ( bot_bo4848840305443107682_preal
= ( collec1132657498972982273_preal @ P ) )
= ( ! [X6: dedekind_preal] :
~ ( P @ X6 ) ) ) ).
% empty_Collect_eq
thf(fact_74_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X6: nat] :
~ ( P @ X6 ) ) ) ).
% empty_Collect_eq
thf(fact_75_order__refl,axiom,
! [X5: dedekind_preal] : ( ord_le5604041210740703414_preal @ X5 @ X5 ) ).
% order_refl
thf(fact_76_order__refl,axiom,
! [X5: set_rat] : ( ord_less_eq_set_rat @ X5 @ X5 ) ).
% order_refl
thf(fact_77_order__refl,axiom,
! [X5: nat] : ( ord_less_eq_nat @ X5 @ X5 ) ).
% order_refl
thf(fact_78_order__refl,axiom,
! [X5: rat] : ( ord_less_eq_rat @ X5 @ X5 ) ).
% order_refl
thf(fact_79_dual__order_Orefl,axiom,
! [A2: dedekind_preal] : ( ord_le5604041210740703414_preal @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_80_dual__order_Orefl,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_81_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_82_dual__order_Orefl,axiom,
! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_83_not__psubset__empty,axiom,
! [A: set_Dedekind_preal] :
~ ( ord_le1802228187270208418_preal @ A @ bot_bo4848840305443107682_preal ) ).
% not_psubset_empty
thf(fact_84_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_85_not__psubset__empty,axiom,
! [A: set_rat] :
~ ( ord_less_set_rat @ A @ bot_bot_set_rat ) ).
% not_psubset_empty
thf(fact_86_Abs__preal__inverse,axiom,
! [Y2: set_rat] :
( ( member_set_rat @ Y2 @ ( collect_set_rat @ dedekind_cut ) )
=> ( ( dedekind_Rep_preal @ ( dedekind_Abs_preal @ Y2 ) )
= Y2 ) ) ).
% Abs_preal_inverse
thf(fact_87_cut__Rep__preal,axiom,
! [X5: dedekind_preal] : ( dedekind_cut @ ( dedekind_Rep_preal @ X5 ) ) ).
% cut_Rep_preal
thf(fact_88_Rep__preal__induct,axiom,
! [Y2: set_rat,P: set_rat > $o] :
( ( member_set_rat @ Y2 @ ( collect_set_rat @ dedekind_cut ) )
=> ( ! [X: dedekind_preal] : ( P @ ( dedekind_Rep_preal @ X ) )
=> ( P @ Y2 ) ) ) ).
% Rep_preal_induct
thf(fact_89_empty__subsetI,axiom,
! [A: set_Dedekind_preal] : ( ord_le7349499860212017814_preal @ bot_bo4848840305443107682_preal @ A ) ).
% empty_subsetI
thf(fact_90_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_91_empty__subsetI,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ bot_bot_set_rat @ A ) ).
% empty_subsetI
thf(fact_92_subset__empty,axiom,
! [A: set_Dedekind_preal] :
( ( ord_le7349499860212017814_preal @ A @ bot_bo4848840305443107682_preal )
= ( A = bot_bo4848840305443107682_preal ) ) ).
% subset_empty
thf(fact_93_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_94_subset__empty,axiom,
! [A: set_rat] :
( ( ord_less_eq_set_rat @ A @ bot_bot_set_rat )
= ( A = bot_bot_set_rat ) ) ).
% subset_empty
thf(fact_95_psubsetI,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_rat @ A @ B ) ) ) ).
% psubsetI
thf(fact_96_Abs__preal__inject,axiom,
! [X5: set_rat,Y2: set_rat] :
( ( member_set_rat @ X5 @ ( collect_set_rat @ dedekind_cut ) )
=> ( ( member_set_rat @ Y2 @ ( collect_set_rat @ dedekind_cut ) )
=> ( ( ( dedekind_Abs_preal @ X5 )
= ( dedekind_Abs_preal @ Y2 ) )
= ( X5 = Y2 ) ) ) ) ).
% Abs_preal_inject
thf(fact_97_bot__set__def,axiom,
( bot_bot_set_set_rat
= ( collect_set_rat @ bot_bot_set_rat_o ) ) ).
% bot_set_def
thf(fact_98_bot__set__def,axiom,
( bot_bot_set_rat
= ( collect_rat @ bot_bot_rat_o ) ) ).
% bot_set_def
thf(fact_99_bot__set__def,axiom,
( bot_bo4848840305443107682_preal
= ( collec1132657498972982273_preal @ bot_bo4815710026557797371real_o ) ) ).
% bot_set_def
thf(fact_100_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_101_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A3: set_rat,B2: set_rat] :
( ( ord_less_set_rat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_102_subset__psubset__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_103_mem__Collect__eq,axiom,
! [A2: dedekind_preal,P: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ A2 @ ( collec1132657498972982273_preal @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_104_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_105_mem__Collect__eq,axiom,
! [A2: set_rat,P: set_rat > $o] :
( ( member_set_rat @ A2 @ ( collect_set_rat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_106_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_107_mem__Collect__eq,axiom,
! [A2: rat,P: rat > $o] :
( ( member_rat @ A2 @ ( collect_rat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_108_Collect__mem__eq,axiom,
! [A: set_Dedekind_preal] :
( ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] : ( member6871284927547481791_preal @ X6 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_109_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X6: $o] : ( member_o @ X6 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
! [A: set_set_rat] :
( ( collect_set_rat
@ ^ [X6: set_rat] : ( member_set_rat @ X6 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_111_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X6: nat] : ( member_nat @ X6 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_112_Collect__mem__eq,axiom,
! [A: set_rat] :
( ( collect_rat
@ ^ [X6: rat] : ( member_rat @ X6 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_113_Collect__cong,axiom,
! [P: set_rat > $o,Q: set_rat > $o] :
( ! [X: set_rat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_set_rat @ P )
= ( collect_set_rat @ Q ) ) ) ).
% Collect_cong
thf(fact_114_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_115_Collect__cong,axiom,
! [P: rat > $o,Q: rat > $o] :
( ! [X: rat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_rat @ P )
= ( collect_rat @ Q ) ) ) ).
% Collect_cong
thf(fact_116_subset__not__subset__eq,axiom,
( ord_less_set_rat
= ( ^ [A3: set_rat,B2: set_rat] :
( ( ord_less_eq_set_rat @ A3 @ B2 )
& ~ ( ord_less_eq_set_rat @ B2 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_117_psubset__subset__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ord_less_eq_set_rat @ B @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_118_psubset__imp__subset,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ord_less_eq_set_rat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_119_psubset__eq,axiom,
( ord_less_set_rat
= ( ^ [A3: set_rat,B2: set_rat] :
( ( ord_less_eq_set_rat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_120_psubsetE,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ~ ( ( ord_less_eq_set_rat @ A @ B )
=> ( ord_less_eq_set_rat @ B @ A ) ) ) ).
% psubsetE
thf(fact_121_Abs__preal__cases,axiom,
! [X5: dedekind_preal] :
~ ! [Y3: set_rat] :
( ( X5
= ( dedekind_Abs_preal @ Y3 ) )
=> ~ ( member_set_rat @ Y3 @ ( collect_set_rat @ dedekind_cut ) ) ) ).
% Abs_preal_cases
thf(fact_122_preal_OAbs__preal__induct,axiom,
! [P: dedekind_preal > $o,X5: dedekind_preal] :
( ! [Y3: set_rat] :
( ( member_set_rat @ Y3 @ ( collect_set_rat @ dedekind_cut ) )
=> ( P @ ( dedekind_Abs_preal @ Y3 ) ) )
=> ( P @ X5 ) ) ).
% preal.Abs_preal_induct
thf(fact_123_Dedekind__Real_OAbs__preal__induct,axiom,
! [P: dedekind_preal > $o,X5: dedekind_preal] :
( ! [X: set_rat] :
( ( dedekind_cut @ X )
=> ( P @ ( dedekind_Abs_preal @ X ) ) )
=> ( P @ X5 ) ) ).
% Dedekind_Real.Abs_preal_induct
thf(fact_124_preal__Ex__mem,axiom,
! [A: set_rat] :
( ( dedekind_cut @ A )
=> ? [X: rat] : ( member_rat @ X @ A ) ) ).
% preal_Ex_mem
thf(fact_125_preal__exists__greater,axiom,
! [A: set_rat,Y2: rat] :
( ( dedekind_cut @ A )
=> ( ( member_rat @ Y2 @ A )
=> ? [X: rat] :
( ( member_rat @ X @ A )
& ( ord_less_rat @ Y2 @ X ) ) ) ) ).
% preal_exists_greater
thf(fact_126_preal__le__def,axiom,
( ord_le5604041210740703414_preal
= ( ^ [R: dedekind_preal,S: dedekind_preal] : ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ R ) @ ( dedekind_Rep_preal @ S ) ) ) ) ).
% preal_le_def
thf(fact_127_order__antisym__conv,axiom,
! [Y2: dedekind_preal,X5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y2 @ X5 )
=> ( ( ord_le5604041210740703414_preal @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_128_order__antisym__conv,axiom,
! [Y2: set_rat,X5: set_rat] :
( ( ord_less_eq_set_rat @ Y2 @ X5 )
=> ( ( ord_less_eq_set_rat @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_129_order__antisym__conv,axiom,
! [Y2: nat,X5: nat] :
( ( ord_less_eq_nat @ Y2 @ X5 )
=> ( ( ord_less_eq_nat @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_130_order__antisym__conv,axiom,
! [Y2: rat,X5: rat] :
( ( ord_less_eq_rat @ Y2 @ X5 )
=> ( ( ord_less_eq_rat @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_131_linorder__le__cases,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ X5 @ Y2 )
=> ( ord_le5604041210740703414_preal @ Y2 @ X5 ) ) ).
% linorder_le_cases
thf(fact_132_linorder__le__cases,axiom,
! [X5: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X5 ) ) ).
% linorder_le_cases
thf(fact_133_linorder__le__cases,axiom,
! [X5: rat,Y2: rat] :
( ~ ( ord_less_eq_rat @ X5 @ Y2 )
=> ( ord_less_eq_rat @ Y2 @ X5 ) ) ).
% linorder_le_cases
thf(fact_134_ord__le__eq__subst,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_135_ord__le__eq__subst,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_136_ord__le__eq__subst,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > rat,C: rat] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_137_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_138_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_139_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_140_ord__le__eq__subst,axiom,
! [A2: rat,B3: rat,F: rat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_141_ord__le__eq__subst,axiom,
! [A2: rat,B3: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_142_ord__le__eq__subst,axiom,
! [A2: rat,B3: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_143_ord__le__eq__subst,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > set_rat,C: set_rat] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_144_ord__eq__le__subst,axiom,
! [A2: dedekind_preal,F: dedekind_preal > dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_145_ord__eq__le__subst,axiom,
! [A2: nat,F: dedekind_preal > nat,B3: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_146_ord__eq__le__subst,axiom,
! [A2: rat,F: dedekind_preal > rat,B3: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_147_ord__eq__le__subst,axiom,
! [A2: dedekind_preal,F: nat > dedekind_preal,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_148_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_149_ord__eq__le__subst,axiom,
! [A2: rat,F: nat > rat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_150_ord__eq__le__subst,axiom,
! [A2: dedekind_preal,F: rat > dedekind_preal,B3: rat,C: rat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_151_ord__eq__le__subst,axiom,
! [A2: nat,F: rat > nat,B3: rat,C: rat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_152_ord__eq__le__subst,axiom,
! [A2: rat,F: rat > rat,B3: rat,C: rat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_153_ord__eq__le__subst,axiom,
! [A2: set_rat,F: dedekind_preal > set_rat,B3: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_154_linorder__linear,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y2 )
| ( ord_le5604041210740703414_preal @ Y2 @ X5 ) ) ).
% linorder_linear
thf(fact_155_linorder__linear,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X5 ) ) ).
% linorder_linear
thf(fact_156_linorder__linear,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_eq_rat @ X5 @ Y2 )
| ( ord_less_eq_rat @ Y2 @ X5 ) ) ).
% linorder_linear
thf(fact_157_order__eq__refl,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( X5 = Y2 )
=> ( ord_le5604041210740703414_preal @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_158_order__eq__refl,axiom,
! [X5: set_rat,Y2: set_rat] :
( ( X5 = Y2 )
=> ( ord_less_eq_set_rat @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_159_order__eq__refl,axiom,
! [X5: nat,Y2: nat] :
( ( X5 = Y2 )
=> ( ord_less_eq_nat @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_160_order__eq__refl,axiom,
! [X5: rat,Y2: rat] :
( ( X5 = Y2 )
=> ( ord_less_eq_rat @ X5 @ Y2 ) ) ).
% order_eq_refl
thf(fact_161_order__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_162_order__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_163_order__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > rat,C: rat] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_164_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_165_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_166_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_167_order__subst2,axiom,
! [A2: rat,B3: rat,F: rat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B3 ) @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_168_order__subst2,axiom,
! [A2: rat,B3: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_169_order__subst2,axiom,
! [A2: rat,B3: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_170_order__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > set_rat,C: set_rat] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ord_less_eq_set_rat @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_171_order__subst1,axiom,
! [A2: dedekind_preal,F: dedekind_preal > dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_172_order__subst1,axiom,
! [A2: dedekind_preal,F: nat > dedekind_preal,B3: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_173_order__subst1,axiom,
! [A2: dedekind_preal,F: rat > dedekind_preal,B3: rat,C: rat] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_174_order__subst1,axiom,
! [A2: nat,F: dedekind_preal > nat,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_175_order__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_176_order__subst1,axiom,
! [A2: nat,F: rat > nat,B3: rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_177_order__subst1,axiom,
! [A2: rat,F: dedekind_preal > rat,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_178_order__subst1,axiom,
! [A2: rat,F: nat > rat,B3: nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_179_order__subst1,axiom,
! [A2: rat,F: rat > rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_180_order__subst1,axiom,
! [A2: dedekind_preal,F: set_rat > dedekind_preal,B3: set_rat,C: set_rat] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_rat @ B3 @ C )
=> ( ! [X: set_rat,Y3: set_rat] :
( ( ord_less_eq_set_rat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_181_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: dedekind_preal,Z3: dedekind_preal] : ( Y4 = Z3 ) )
= ( ^ [A4: dedekind_preal,B4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A4 @ B4 )
& ( ord_le5604041210740703414_preal @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_182_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_rat,Z3: set_rat] : ( Y4 = Z3 ) )
= ( ^ [A4: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ A4 @ B4 )
& ( ord_less_eq_set_rat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_183_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_184_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: rat,Z3: rat] : ( Y4 = Z3 ) )
= ( ^ [A4: rat,B4: rat] :
( ( ord_less_eq_rat @ A4 @ B4 )
& ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_185_antisym,axiom,
! [A2: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ord_le5604041210740703414_preal @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_186_antisym,axiom,
! [A2: set_rat,B3: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B3 )
=> ( ( ord_less_eq_set_rat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_187_antisym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_188_antisym,axiom,
! [A2: rat,B3: rat] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( ord_less_eq_rat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_189_dual__order_Otrans,axiom,
! [B3: dedekind_preal,A2: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B3 @ A2 )
=> ( ( ord_le5604041210740703414_preal @ C @ B3 )
=> ( ord_le5604041210740703414_preal @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_190_dual__order_Otrans,axiom,
! [B3: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ B3 @ A2 )
=> ( ( ord_less_eq_set_rat @ C @ B3 )
=> ( ord_less_eq_set_rat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_191_dual__order_Otrans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_192_dual__order_Otrans,axiom,
! [B3: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B3 @ A2 )
=> ( ( ord_less_eq_rat @ C @ B3 )
=> ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_193_dual__order_Oantisym,axiom,
! [B3: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B3 @ A2 )
=> ( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_194_dual__order_Oantisym,axiom,
! [B3: set_rat,A2: set_rat] :
( ( ord_less_eq_set_rat @ B3 @ A2 )
=> ( ( ord_less_eq_set_rat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_195_dual__order_Oantisym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_196_dual__order_Oantisym,axiom,
! [B3: rat,A2: rat] :
( ( ord_less_eq_rat @ B3 @ A2 )
=> ( ( ord_less_eq_rat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_197_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: dedekind_preal,Z3: dedekind_preal] : ( Y4 = Z3 ) )
= ( ^ [A4: dedekind_preal,B4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B4 @ A4 )
& ( ord_le5604041210740703414_preal @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_198_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_rat,Z3: set_rat] : ( Y4 = Z3 ) )
= ( ^ [A4: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ B4 @ A4 )
& ( ord_less_eq_set_rat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_199_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_200_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: rat,Z3: rat] : ( Y4 = Z3 ) )
= ( ^ [A4: rat,B4: rat] :
( ( ord_less_eq_rat @ B4 @ A4 )
& ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_201_linorder__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A2: dedekind_preal,B3: dedekind_preal] :
( ! [A5: dedekind_preal,B5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: dedekind_preal,B5: dedekind_preal] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_202_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_203_linorder__wlog,axiom,
! [P: rat > rat > $o,A2: rat,B3: rat] :
( ! [A5: rat,B5: rat] :
( ( ord_less_eq_rat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: rat,B5: rat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_204_order__trans,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal,Z4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y2 )
=> ( ( ord_le5604041210740703414_preal @ Y2 @ Z4 )
=> ( ord_le5604041210740703414_preal @ X5 @ Z4 ) ) ) ).
% order_trans
thf(fact_205_order__trans,axiom,
! [X5: set_rat,Y2: set_rat,Z4: set_rat] :
( ( ord_less_eq_set_rat @ X5 @ Y2 )
=> ( ( ord_less_eq_set_rat @ Y2 @ Z4 )
=> ( ord_less_eq_set_rat @ X5 @ Z4 ) ) ) ).
% order_trans
thf(fact_206_order__trans,axiom,
! [X5: nat,Y2: nat,Z4: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z4 )
=> ( ord_less_eq_nat @ X5 @ Z4 ) ) ) ).
% order_trans
thf(fact_207_order__trans,axiom,
! [X5: rat,Y2: rat,Z4: rat] :
( ( ord_less_eq_rat @ X5 @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ Z4 )
=> ( ord_less_eq_rat @ X5 @ Z4 ) ) ) ).
% order_trans
thf(fact_208_order_Otrans,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ord_le5604041210740703414_preal @ A2 @ C ) ) ) ).
% order.trans
thf(fact_209_order_Otrans,axiom,
! [A2: set_rat,B3: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B3 )
=> ( ( ord_less_eq_set_rat @ B3 @ C )
=> ( ord_less_eq_set_rat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_210_order_Otrans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_211_order_Otrans,axiom,
! [A2: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_212_order__antisym,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y2 )
=> ( ( ord_le5604041210740703414_preal @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_213_order__antisym,axiom,
! [X5: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X5 @ Y2 )
=> ( ( ord_less_eq_set_rat @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_214_order__antisym,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_215_order__antisym,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_eq_rat @ X5 @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ X5 )
=> ( X5 = Y2 ) ) ) ).
% order_antisym
thf(fact_216_ord__le__eq__trans,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le5604041210740703414_preal @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_217_ord__le__eq__trans,axiom,
! [A2: set_rat,B3: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_rat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_218_ord__le__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_219_ord__le__eq__trans,axiom,
! [A2: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_220_ord__eq__le__trans,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( A2 = B3 )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ord_le5604041210740703414_preal @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_221_ord__eq__le__trans,axiom,
! [A2: set_rat,B3: set_rat,C: set_rat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_rat @ B3 @ C )
=> ( ord_less_eq_set_rat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_222_ord__eq__le__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_223_ord__eq__le__trans,axiom,
! [A2: rat,B3: rat,C: rat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_224_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: dedekind_preal,Z3: dedekind_preal] : ( Y4 = Z3 ) )
= ( ^ [X6: dedekind_preal,Y5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X6 @ Y5 )
& ( ord_le5604041210740703414_preal @ Y5 @ X6 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_225_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_rat,Z3: set_rat] : ( Y4 = Z3 ) )
= ( ^ [X6: set_rat,Y5: set_rat] :
( ( ord_less_eq_set_rat @ X6 @ Y5 )
& ( ord_less_eq_set_rat @ Y5 @ X6 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_226_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X6 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_227_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: rat,Z3: rat] : ( Y4 = Z3 ) )
= ( ^ [X6: rat,Y5: rat] :
( ( ord_less_eq_rat @ X6 @ Y5 )
& ( ord_less_eq_rat @ Y5 @ X6 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_228_le__cases3,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal,Z4: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ X5 @ Y2 )
=> ~ ( ord_le5604041210740703414_preal @ Y2 @ Z4 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y2 @ X5 )
=> ~ ( ord_le5604041210740703414_preal @ X5 @ Z4 ) )
=> ( ( ( ord_le5604041210740703414_preal @ X5 @ Z4 )
=> ~ ( ord_le5604041210740703414_preal @ Z4 @ Y2 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Z4 @ Y2 )
=> ~ ( ord_le5604041210740703414_preal @ Y2 @ X5 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y2 @ Z4 )
=> ~ ( ord_le5604041210740703414_preal @ Z4 @ X5 ) )
=> ~ ( ( ord_le5604041210740703414_preal @ Z4 @ X5 )
=> ~ ( ord_le5604041210740703414_preal @ X5 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_229_le__cases3,axiom,
! [X5: nat,Y2: nat,Z4: nat] :
( ( ( ord_less_eq_nat @ X5 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ X5 @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z4 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X5 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ X5 ) )
=> ~ ( ( ord_less_eq_nat @ Z4 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_230_le__cases3,axiom,
! [X5: rat,Y2: rat,Z4: rat] :
( ( ( ord_less_eq_rat @ X5 @ Y2 )
=> ~ ( ord_less_eq_rat @ Y2 @ Z4 ) )
=> ( ( ( ord_less_eq_rat @ Y2 @ X5 )
=> ~ ( ord_less_eq_rat @ X5 @ Z4 ) )
=> ( ( ( ord_less_eq_rat @ X5 @ Z4 )
=> ~ ( ord_less_eq_rat @ Z4 @ Y2 ) )
=> ( ( ( ord_less_eq_rat @ Z4 @ Y2 )
=> ~ ( ord_less_eq_rat @ Y2 @ X5 ) )
=> ( ( ( ord_less_eq_rat @ Y2 @ Z4 )
=> ~ ( ord_less_eq_rat @ Z4 @ X5 ) )
=> ~ ( ( ord_less_eq_rat @ Z4 @ X5 )
=> ~ ( ord_less_eq_rat @ X5 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_231_nle__le,axiom,
! [A2: dedekind_preal,B3: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ A2 @ B3 ) )
= ( ( ord_le5604041210740703414_preal @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_232_nle__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_233_nle__le,axiom,
! [A2: rat,B3: rat] :
( ( ~ ( ord_less_eq_rat @ A2 @ B3 ) )
= ( ( ord_less_eq_rat @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_234_lt__ex,axiom,
! [X5: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X5 ) ).
% lt_ex
thf(fact_235_gt__ex,axiom,
! [X5: rat] :
? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% gt_ex
thf(fact_236_gt__ex,axiom,
! [X5: nat] :
? [X_1: nat] : ( ord_less_nat @ X5 @ X_1 ) ).
% gt_ex
thf(fact_237_dense,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_rat @ X5 @ Y2 )
=> ? [Z: rat] :
( ( ord_less_rat @ X5 @ Z )
& ( ord_less_rat @ Z @ Y2 ) ) ) ).
% dense
thf(fact_238_less__imp__neq,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% less_imp_neq
thf(fact_239_less__imp__neq,axiom,
! [X5: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% less_imp_neq
thf(fact_240_less__imp__neq,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_rat @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% less_imp_neq
thf(fact_241_less__imp__neq,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% less_imp_neq
thf(fact_242_order_Oasym,axiom,
! [A2: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ~ ( ord_le5708704896291381698_preal @ B3 @ A2 ) ) ).
% order.asym
thf(fact_243_order_Oasym,axiom,
! [A2: set_rat,B3: set_rat] :
( ( ord_less_set_rat @ A2 @ B3 )
=> ~ ( ord_less_set_rat @ B3 @ A2 ) ) ).
% order.asym
thf(fact_244_order_Oasym,axiom,
! [A2: rat,B3: rat] :
( ( ord_less_rat @ A2 @ B3 )
=> ~ ( ord_less_rat @ B3 @ A2 ) ) ).
% order.asym
thf(fact_245_order_Oasym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order.asym
thf(fact_246_ord__eq__less__trans,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( A2 = B3 )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ord_le5708704896291381698_preal @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_247_ord__eq__less__trans,axiom,
! [A2: set_rat,B3: set_rat,C: set_rat] :
( ( A2 = B3 )
=> ( ( ord_less_set_rat @ B3 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_248_ord__eq__less__trans,axiom,
! [A2: rat,B3: rat,C: rat] :
( ( A2 = B3 )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_249_ord__eq__less__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_250_ord__less__eq__trans,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le5708704896291381698_preal @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_251_ord__less__eq__trans,axiom,
! [A2: set_rat,B3: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_252_ord__less__eq__trans,axiom,
! [A2: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_253_ord__less__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_254_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X )
=> ( P @ Y6 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_255_antisym__conv3,axiom,
! [Y2: dedekind_preal,X5: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ Y2 @ X5 )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X5 @ Y2 ) )
= ( X5 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_256_antisym__conv3,axiom,
! [Y2: rat,X5: rat] :
( ~ ( ord_less_rat @ Y2 @ X5 )
=> ( ( ~ ( ord_less_rat @ X5 @ Y2 ) )
= ( X5 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_257_antisym__conv3,axiom,
! [Y2: nat,X5: nat] :
( ~ ( ord_less_nat @ Y2 @ X5 )
=> ( ( ~ ( ord_less_nat @ X5 @ Y2 ) )
= ( X5 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_258_linorder__cases,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ( ( X5 != Y2 )
=> ( ord_le5708704896291381698_preal @ Y2 @ X5 ) ) ) ).
% linorder_cases
thf(fact_259_linorder__cases,axiom,
! [X5: rat,Y2: rat] :
( ~ ( ord_less_rat @ X5 @ Y2 )
=> ( ( X5 != Y2 )
=> ( ord_less_rat @ Y2 @ X5 ) ) ) ).
% linorder_cases
thf(fact_260_linorder__cases,axiom,
! [X5: nat,Y2: nat] :
( ~ ( ord_less_nat @ X5 @ Y2 )
=> ( ( X5 != Y2 )
=> ( ord_less_nat @ Y2 @ X5 ) ) ) ).
% linorder_cases
thf(fact_261_dual__order_Oasym,axiom,
! [B3: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B3 @ A2 )
=> ~ ( ord_le5708704896291381698_preal @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_262_dual__order_Oasym,axiom,
! [B3: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B3 @ A2 )
=> ~ ( ord_less_set_rat @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_263_dual__order_Oasym,axiom,
! [B3: rat,A2: rat] :
( ( ord_less_rat @ B3 @ A2 )
=> ~ ( ord_less_rat @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_264_dual__order_Oasym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_265_dual__order_Oirrefl,axiom,
! [A2: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_266_dual__order_Oirrefl,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_267_dual__order_Oirrefl,axiom,
! [A2: rat] :
~ ( ord_less_rat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_268_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_269_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X7: nat] : ( P3 @ X7 ) )
= ( ^ [P4: nat > $o] :
? [N: nat] :
( ( P4 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P4 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_270_linorder__less__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A2: dedekind_preal,B3: dedekind_preal] :
( ! [A5: dedekind_preal,B5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: dedekind_preal] : ( P @ A5 @ A5 )
=> ( ! [A5: dedekind_preal,B5: dedekind_preal] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_271_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A2: rat,B3: rat] :
( ! [A5: rat,B5: rat] :
( ( ord_less_rat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: rat] : ( P @ A5 @ A5 )
=> ( ! [A5: rat,B5: rat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_272_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_273_order_Ostrict__trans,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ord_le5708704896291381698_preal @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_274_order_Ostrict__trans,axiom,
! [A2: set_rat,B3: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B3 )
=> ( ( ord_less_set_rat @ B3 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_275_order_Ostrict__trans,axiom,
! [A2: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_276_order_Ostrict__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_277_not__less__iff__gr__or__eq,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X5 @ Y2 ) )
= ( ( ord_le5708704896291381698_preal @ Y2 @ X5 )
| ( X5 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_278_not__less__iff__gr__or__eq,axiom,
! [X5: rat,Y2: rat] :
( ( ~ ( ord_less_rat @ X5 @ Y2 ) )
= ( ( ord_less_rat @ Y2 @ X5 )
| ( X5 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_279_not__less__iff__gr__or__eq,axiom,
! [X5: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X5 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X5 )
| ( X5 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_280_dual__order_Ostrict__trans,axiom,
! [B3: dedekind_preal,A2: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B3 @ A2 )
=> ( ( ord_le5708704896291381698_preal @ C @ B3 )
=> ( ord_le5708704896291381698_preal @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_281_dual__order_Ostrict__trans,axiom,
! [B3: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ B3 @ A2 )
=> ( ( ord_less_set_rat @ C @ B3 )
=> ( ord_less_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_282_dual__order_Ostrict__trans,axiom,
! [B3: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B3 @ A2 )
=> ( ( ord_less_rat @ C @ B3 )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_283_dual__order_Ostrict__trans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_284_order_Ostrict__implies__not__eq,axiom,
! [A2: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_285_order_Ostrict__implies__not__eq,axiom,
! [A2: set_rat,B3: set_rat] :
( ( ord_less_set_rat @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_286_order_Ostrict__implies__not__eq,axiom,
! [A2: rat,B3: rat] :
( ( ord_less_rat @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_287_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_288_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_289_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_290_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: rat,A2: rat] :
( ( ord_less_rat @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_291_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_292_linorder__neqE,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( X5 != Y2 )
=> ( ~ ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ( ord_le5708704896291381698_preal @ Y2 @ X5 ) ) ) ).
% linorder_neqE
thf(fact_293_linorder__neqE,axiom,
! [X5: rat,Y2: rat] :
( ( X5 != Y2 )
=> ( ~ ( ord_less_rat @ X5 @ Y2 )
=> ( ord_less_rat @ Y2 @ X5 ) ) ) ).
% linorder_neqE
thf(fact_294_linorder__neqE,axiom,
! [X5: nat,Y2: nat] :
( ( X5 != Y2 )
=> ( ~ ( ord_less_nat @ X5 @ Y2 )
=> ( ord_less_nat @ Y2 @ X5 ) ) ) ).
% linorder_neqE
thf(fact_295_order__less__asym,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ~ ( ord_le5708704896291381698_preal @ Y2 @ X5 ) ) ).
% order_less_asym
thf(fact_296_order__less__asym,axiom,
! [X5: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X5 @ Y2 )
=> ~ ( ord_less_set_rat @ Y2 @ X5 ) ) ).
% order_less_asym
thf(fact_297_order__less__asym,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_rat @ X5 @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X5 ) ) ).
% order_less_asym
thf(fact_298_order__less__asym,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X5 ) ) ).
% order_less_asym
thf(fact_299_linorder__neq__iff,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( X5 != Y2 )
= ( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
| ( ord_le5708704896291381698_preal @ Y2 @ X5 ) ) ) ).
% linorder_neq_iff
thf(fact_300_linorder__neq__iff,axiom,
! [X5: rat,Y2: rat] :
( ( X5 != Y2 )
= ( ( ord_less_rat @ X5 @ Y2 )
| ( ord_less_rat @ Y2 @ X5 ) ) ) ).
% linorder_neq_iff
thf(fact_301_linorder__neq__iff,axiom,
! [X5: nat,Y2: nat] :
( ( X5 != Y2 )
= ( ( ord_less_nat @ X5 @ Y2 )
| ( ord_less_nat @ Y2 @ X5 ) ) ) ).
% linorder_neq_iff
thf(fact_302_order__less__asym_H,axiom,
! [A2: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ~ ( ord_le5708704896291381698_preal @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_303_order__less__asym_H,axiom,
! [A2: set_rat,B3: set_rat] :
( ( ord_less_set_rat @ A2 @ B3 )
=> ~ ( ord_less_set_rat @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_304_order__less__asym_H,axiom,
! [A2: rat,B3: rat] :
( ( ord_less_rat @ A2 @ B3 )
=> ~ ( ord_less_rat @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_305_order__less__asym_H,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_306_order__less__trans,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal,Z4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ( ( ord_le5708704896291381698_preal @ Y2 @ Z4 )
=> ( ord_le5708704896291381698_preal @ X5 @ Z4 ) ) ) ).
% order_less_trans
thf(fact_307_order__less__trans,axiom,
! [X5: set_rat,Y2: set_rat,Z4: set_rat] :
( ( ord_less_set_rat @ X5 @ Y2 )
=> ( ( ord_less_set_rat @ Y2 @ Z4 )
=> ( ord_less_set_rat @ X5 @ Z4 ) ) ) ).
% order_less_trans
thf(fact_308_order__less__trans,axiom,
! [X5: rat,Y2: rat,Z4: rat] :
( ( ord_less_rat @ X5 @ Y2 )
=> ( ( ord_less_rat @ Y2 @ Z4 )
=> ( ord_less_rat @ X5 @ Z4 ) ) ) ).
% order_less_trans
thf(fact_309_order__less__trans,axiom,
! [X5: nat,Y2: nat,Z4: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z4 )
=> ( ord_less_nat @ X5 @ Z4 ) ) ) ).
% order_less_trans
thf(fact_310_ord__eq__less__subst,axiom,
! [A2: dedekind_preal,F: dedekind_preal > dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_311_ord__eq__less__subst,axiom,
! [A2: rat,F: dedekind_preal > rat,B3: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_312_ord__eq__less__subst,axiom,
! [A2: nat,F: dedekind_preal > nat,B3: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_313_ord__eq__less__subst,axiom,
! [A2: dedekind_preal,F: rat > dedekind_preal,B3: rat,C: rat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_314_ord__eq__less__subst,axiom,
! [A2: rat,F: rat > rat,B3: rat,C: rat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_315_ord__eq__less__subst,axiom,
! [A2: nat,F: rat > nat,B3: rat,C: rat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_316_ord__eq__less__subst,axiom,
! [A2: dedekind_preal,F: nat > dedekind_preal,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_317_ord__eq__less__subst,axiom,
! [A2: rat,F: nat > rat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_318_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_319_ord__eq__less__subst,axiom,
! [A2: set_rat,F: dedekind_preal > set_rat,B3: dedekind_preal,C: dedekind_preal] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_320_ord__less__eq__subst,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_321_ord__less__eq__subst,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > rat,C: rat] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_322_ord__less__eq__subst,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_323_ord__less__eq__subst,axiom,
! [A2: rat,B3: rat,F: rat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_324_ord__less__eq__subst,axiom,
! [A2: rat,B3: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_325_ord__less__eq__subst,axiom,
! [A2: rat,B3: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_326_ord__less__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_327_ord__less__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_328_ord__less__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_329_ord__less__eq__subst,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > set_rat,C: set_rat] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_330_order__less__irrefl,axiom,
! [X5: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_331_order__less__irrefl,axiom,
! [X5: set_rat] :
~ ( ord_less_set_rat @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_332_order__less__irrefl,axiom,
! [X5: rat] :
~ ( ord_less_rat @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_333_order__less__irrefl,axiom,
! [X5: nat] :
~ ( ord_less_nat @ X5 @ X5 ) ).
% order_less_irrefl
thf(fact_334_order__less__subst1,axiom,
! [A2: dedekind_preal,F: dedekind_preal > dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_335_order__less__subst1,axiom,
! [A2: dedekind_preal,F: rat > dedekind_preal,B3: rat,C: rat] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_336_order__less__subst1,axiom,
! [A2: dedekind_preal,F: nat > dedekind_preal,B3: nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_337_order__less__subst1,axiom,
! [A2: rat,F: dedekind_preal > rat,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_less_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_338_order__less__subst1,axiom,
! [A2: rat,F: rat > rat,B3: rat,C: rat] :
( ( ord_less_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_339_order__less__subst1,axiom,
! [A2: rat,F: nat > rat,B3: nat,C: nat] :
( ( ord_less_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_340_order__less__subst1,axiom,
! [A2: nat,F: dedekind_preal > nat,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_341_order__less__subst1,axiom,
! [A2: nat,F: rat > nat,B3: rat,C: rat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_342_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_343_order__less__subst1,axiom,
! [A2: dedekind_preal,F: set_rat > dedekind_preal,B3: set_rat,C: set_rat] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_set_rat @ B3 @ C )
=> ( ! [X: set_rat,Y3: set_rat] :
( ( ord_less_set_rat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_344_order__less__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_345_order__less__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > rat,C: rat] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ord_less_rat @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_346_order__less__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_347_order__less__subst2,axiom,
! [A2: rat,B3: rat,F: rat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B3 ) @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_348_order__less__subst2,axiom,
! [A2: rat,B3: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ( ord_less_rat @ ( F @ B3 ) @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_349_order__less__subst2,axiom,
! [A2: rat,B3: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_350_order__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_351_order__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_rat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_352_order__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_353_order__less__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > set_rat,C: set_rat] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ord_less_set_rat @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_set_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_354_order__less__not__sym,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ~ ( ord_le5708704896291381698_preal @ Y2 @ X5 ) ) ).
% order_less_not_sym
thf(fact_355_order__less__not__sym,axiom,
! [X5: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X5 @ Y2 )
=> ~ ( ord_less_set_rat @ Y2 @ X5 ) ) ).
% order_less_not_sym
thf(fact_356_order__less__not__sym,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_rat @ X5 @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X5 ) ) ).
% order_less_not_sym
thf(fact_357_order__less__not__sym,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X5 ) ) ).
% order_less_not_sym
thf(fact_358_order__less__imp__triv,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal,P: $o] :
( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ( ( ord_le5708704896291381698_preal @ Y2 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_359_order__less__imp__triv,axiom,
! [X5: set_rat,Y2: set_rat,P: $o] :
( ( ord_less_set_rat @ X5 @ Y2 )
=> ( ( ord_less_set_rat @ Y2 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_360_order__less__imp__triv,axiom,
! [X5: rat,Y2: rat,P: $o] :
( ( ord_less_rat @ X5 @ Y2 )
=> ( ( ord_less_rat @ Y2 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_361_order__less__imp__triv,axiom,
! [X5: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X5 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_362_linorder__less__linear,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
| ( X5 = Y2 )
| ( ord_le5708704896291381698_preal @ Y2 @ X5 ) ) ).
% linorder_less_linear
thf(fact_363_linorder__less__linear,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_rat @ X5 @ Y2 )
| ( X5 = Y2 )
| ( ord_less_rat @ Y2 @ X5 ) ) ).
% linorder_less_linear
thf(fact_364_linorder__less__linear,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
| ( X5 = Y2 )
| ( ord_less_nat @ Y2 @ X5 ) ) ).
% linorder_less_linear
thf(fact_365_order__less__imp__not__eq,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_366_order__less__imp__not__eq,axiom,
! [X5: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_367_order__less__imp__not__eq,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_rat @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_368_order__less__imp__not__eq,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( X5 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_369_order__less__imp__not__eq2,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ( Y2 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_370_order__less__imp__not__eq2,axiom,
! [X5: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X5 @ Y2 )
=> ( Y2 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_371_order__less__imp__not__eq2,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_rat @ X5 @ Y2 )
=> ( Y2 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_372_order__less__imp__not__eq2,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( Y2 != X5 ) ) ).
% order_less_imp_not_eq2
thf(fact_373_order__less__imp__not__less,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ~ ( ord_le5708704896291381698_preal @ Y2 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_374_order__less__imp__not__less,axiom,
! [X5: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X5 @ Y2 )
=> ~ ( ord_less_set_rat @ Y2 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_375_order__less__imp__not__less,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_rat @ X5 @ Y2 )
=> ~ ( ord_less_rat @ Y2 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_376_order__less__imp__not__less,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X5 ) ) ).
% order_less_imp_not_less
thf(fact_377_ex__in__conv,axiom,
! [A: set_rat] :
( ( ? [X6: rat] : ( member_rat @ X6 @ A ) )
= ( A != bot_bot_set_rat ) ) ).
% ex_in_conv
thf(fact_378_ex__in__conv,axiom,
! [A: set_set_rat] :
( ( ? [X6: set_rat] : ( member_set_rat @ X6 @ A ) )
= ( A != bot_bot_set_set_rat ) ) ).
% ex_in_conv
thf(fact_379_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X6: $o] : ( member_o @ X6 @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_380_ex__in__conv,axiom,
! [A: set_Dedekind_preal] :
( ( ? [X6: dedekind_preal] : ( member6871284927547481791_preal @ X6 @ A ) )
= ( A != bot_bo4848840305443107682_preal ) ) ).
% ex_in_conv
thf(fact_381_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X6: nat] : ( member_nat @ X6 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_382_equals0I,axiom,
! [A: set_rat] :
( ! [Y3: rat] :
~ ( member_rat @ Y3 @ A )
=> ( A = bot_bot_set_rat ) ) ).
% equals0I
thf(fact_383_equals0I,axiom,
! [A: set_set_rat] :
( ! [Y3: set_rat] :
~ ( member_set_rat @ Y3 @ A )
=> ( A = bot_bot_set_set_rat ) ) ).
% equals0I
thf(fact_384_equals0I,axiom,
! [A: set_o] :
( ! [Y3: $o] :
~ ( member_o @ Y3 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_385_equals0I,axiom,
! [A: set_Dedekind_preal] :
( ! [Y3: dedekind_preal] :
~ ( member6871284927547481791_preal @ Y3 @ A )
=> ( A = bot_bo4848840305443107682_preal ) ) ).
% equals0I
thf(fact_386_equals0I,axiom,
! [A: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_387_equals0D,axiom,
! [A: set_rat,A2: rat] :
( ( A = bot_bot_set_rat )
=> ~ ( member_rat @ A2 @ A ) ) ).
% equals0D
thf(fact_388_equals0D,axiom,
! [A: set_set_rat,A2: set_rat] :
( ( A = bot_bot_set_set_rat )
=> ~ ( member_set_rat @ A2 @ A ) ) ).
% equals0D
thf(fact_389_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_390_equals0D,axiom,
! [A: set_Dedekind_preal,A2: dedekind_preal] :
( ( A = bot_bo4848840305443107682_preal )
=> ~ ( member6871284927547481791_preal @ A2 @ A ) ) ).
% equals0D
thf(fact_391_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_392_emptyE,axiom,
! [A2: rat] :
~ ( member_rat @ A2 @ bot_bot_set_rat ) ).
% emptyE
thf(fact_393_emptyE,axiom,
! [A2: set_rat] :
~ ( member_set_rat @ A2 @ bot_bot_set_set_rat ) ).
% emptyE
thf(fact_394_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_395_emptyE,axiom,
! [A2: dedekind_preal] :
~ ( member6871284927547481791_preal @ A2 @ bot_bo4848840305443107682_preal ) ).
% emptyE
thf(fact_396_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_397_psubset__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_398_psubsetD,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,C: dedekind_preal] :
( ( ord_le1802228187270208418_preal @ A @ B )
=> ( ( member6871284927547481791_preal @ C @ A )
=> ( member6871284927547481791_preal @ C @ B ) ) ) ).
% psubsetD
thf(fact_399_psubsetD,axiom,
! [A: set_set_rat,B: set_set_rat,C: set_rat] :
( ( ord_less_set_set_rat @ A @ B )
=> ( ( member_set_rat @ C @ A )
=> ( member_set_rat @ C @ B ) ) ) ).
% psubsetD
thf(fact_400_psubsetD,axiom,
! [A: set_o,B: set_o,C: $o] :
( ( ord_less_set_o @ A @ B )
=> ( ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% psubsetD
thf(fact_401_psubsetD,axiom,
! [A: set_rat,B: set_rat,C: rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( member_rat @ C @ A )
=> ( member_rat @ C @ B ) ) ) ).
% psubsetD
thf(fact_402_leD,axiom,
! [Y2: dedekind_preal,X5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y2 @ X5 )
=> ~ ( ord_le5708704896291381698_preal @ X5 @ Y2 ) ) ).
% leD
thf(fact_403_leD,axiom,
! [Y2: set_rat,X5: set_rat] :
( ( ord_less_eq_set_rat @ Y2 @ X5 )
=> ~ ( ord_less_set_rat @ X5 @ Y2 ) ) ).
% leD
thf(fact_404_leD,axiom,
! [Y2: nat,X5: nat] :
( ( ord_less_eq_nat @ Y2 @ X5 )
=> ~ ( ord_less_nat @ X5 @ Y2 ) ) ).
% leD
thf(fact_405_leD,axiom,
! [Y2: rat,X5: rat] :
( ( ord_less_eq_rat @ Y2 @ X5 )
=> ~ ( ord_less_rat @ X5 @ Y2 ) ) ).
% leD
thf(fact_406_leI,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ( ord_le5604041210740703414_preal @ Y2 @ X5 ) ) ).
% leI
thf(fact_407_leI,axiom,
! [X5: nat,Y2: nat] :
( ~ ( ord_less_nat @ X5 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X5 ) ) ).
% leI
thf(fact_408_leI,axiom,
! [X5: rat,Y2: rat] :
( ~ ( ord_less_rat @ X5 @ Y2 )
=> ( ord_less_eq_rat @ Y2 @ X5 ) ) ).
% leI
thf(fact_409_nless__le,axiom,
! [A2: dedekind_preal,B3: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ A2 @ B3 ) )
= ( ~ ( ord_le5604041210740703414_preal @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_410_nless__le,axiom,
! [A2: set_rat,B3: set_rat] :
( ( ~ ( ord_less_set_rat @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_set_rat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_411_nless__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_412_nless__le,axiom,
! [A2: rat,B3: rat] :
( ( ~ ( ord_less_rat @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_rat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_413_antisym__conv1,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ( ( ord_le5604041210740703414_preal @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_414_antisym__conv1,axiom,
! [X5: set_rat,Y2: set_rat] :
( ~ ( ord_less_set_rat @ X5 @ Y2 )
=> ( ( ord_less_eq_set_rat @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_415_antisym__conv1,axiom,
! [X5: nat,Y2: nat] :
( ~ ( ord_less_nat @ X5 @ Y2 )
=> ( ( ord_less_eq_nat @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_416_antisym__conv1,axiom,
! [X5: rat,Y2: rat] :
( ~ ( ord_less_rat @ X5 @ Y2 )
=> ( ( ord_less_eq_rat @ X5 @ Y2 )
= ( X5 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_417_antisym__conv2,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y2 )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X5 @ Y2 ) )
= ( X5 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_418_antisym__conv2,axiom,
! [X5: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X5 @ Y2 )
=> ( ( ~ ( ord_less_set_rat @ X5 @ Y2 ) )
= ( X5 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_419_antisym__conv2,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X5 @ Y2 ) )
= ( X5 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_420_antisym__conv2,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_eq_rat @ X5 @ Y2 )
=> ( ( ~ ( ord_less_rat @ X5 @ Y2 ) )
= ( X5 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_421_dense__ge,axiom,
! [Z4: rat,Y2: rat] :
( ! [X: rat] :
( ( ord_less_rat @ Z4 @ X )
=> ( ord_less_eq_rat @ Y2 @ X ) )
=> ( ord_less_eq_rat @ Y2 @ Z4 ) ) ).
% dense_ge
thf(fact_422_dense__le,axiom,
! [Y2: rat,Z4: rat] :
( ! [X: rat] :
( ( ord_less_rat @ X @ Y2 )
=> ( ord_less_eq_rat @ X @ Z4 ) )
=> ( ord_less_eq_rat @ Y2 @ Z4 ) ) ).
% dense_le
thf(fact_423_less__le__not__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X6: dedekind_preal,Y5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X6 @ Y5 )
& ~ ( ord_le5604041210740703414_preal @ Y5 @ X6 ) ) ) ) ).
% less_le_not_le
thf(fact_424_less__le__not__le,axiom,
( ord_less_set_rat
= ( ^ [X6: set_rat,Y5: set_rat] :
( ( ord_less_eq_set_rat @ X6 @ Y5 )
& ~ ( ord_less_eq_set_rat @ Y5 @ X6 ) ) ) ) ).
% less_le_not_le
thf(fact_425_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X6 ) ) ) ) ).
% less_le_not_le
thf(fact_426_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X6: rat,Y5: rat] :
( ( ord_less_eq_rat @ X6 @ Y5 )
& ~ ( ord_less_eq_rat @ Y5 @ X6 ) ) ) ) ).
% less_le_not_le
thf(fact_427_not__le__imp__less,axiom,
! [Y2: dedekind_preal,X5: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ Y2 @ X5 )
=> ( ord_le5708704896291381698_preal @ X5 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_428_not__le__imp__less,axiom,
! [Y2: nat,X5: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X5 )
=> ( ord_less_nat @ X5 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_429_not__le__imp__less,axiom,
! [Y2: rat,X5: rat] :
( ~ ( ord_less_eq_rat @ Y2 @ X5 )
=> ( ord_less_rat @ X5 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_430_order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [A4: dedekind_preal,B4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_431_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_rat
= ( ^ [A4: set_rat,B4: set_rat] :
( ( ord_less_set_rat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_432_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_433_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A4: rat,B4: rat] :
( ( ord_less_rat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_434_order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A4: dedekind_preal,B4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_435_order_Ostrict__iff__order,axiom,
( ord_less_set_rat
= ( ^ [A4: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_436_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_437_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A4: rat,B4: rat] :
( ( ord_less_eq_rat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_438_order_Ostrict__trans1,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ord_le5708704896291381698_preal @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_439_order_Ostrict__trans1,axiom,
! [A2: set_rat,B3: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B3 )
=> ( ( ord_less_set_rat @ B3 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_440_order_Ostrict__trans1,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_441_order_Ostrict__trans1,axiom,
! [A2: rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_442_order_Ostrict__trans2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ord_le5708704896291381698_preal @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_443_order_Ostrict__trans2,axiom,
! [A2: set_rat,B3: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B3 )
=> ( ( ord_less_eq_set_rat @ B3 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_444_order_Ostrict__trans2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_445_order_Ostrict__trans2,axiom,
! [A2: rat,B3: rat,C: rat] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_446_order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A4: dedekind_preal,B4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A4 @ B4 )
& ~ ( ord_le5604041210740703414_preal @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_447_order_Ostrict__iff__not,axiom,
( ord_less_set_rat
= ( ^ [A4: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ A4 @ B4 )
& ~ ( ord_less_eq_set_rat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_448_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_449_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A4: rat,B4: rat] :
( ( ord_less_eq_rat @ A4 @ B4 )
& ~ ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_450_dense__ge__bounded,axiom,
! [Z4: rat,X5: rat,Y2: rat] :
( ( ord_less_rat @ Z4 @ X5 )
=> ( ! [W: rat] :
( ( ord_less_rat @ Z4 @ W )
=> ( ( ord_less_rat @ W @ X5 )
=> ( ord_less_eq_rat @ Y2 @ W ) ) )
=> ( ord_less_eq_rat @ Y2 @ Z4 ) ) ) ).
% dense_ge_bounded
thf(fact_451_dense__le__bounded,axiom,
! [X5: rat,Y2: rat,Z4: rat] :
( ( ord_less_rat @ X5 @ Y2 )
=> ( ! [W: rat] :
( ( ord_less_rat @ X5 @ W )
=> ( ( ord_less_rat @ W @ Y2 )
=> ( ord_less_eq_rat @ W @ Z4 ) ) )
=> ( ord_less_eq_rat @ Y2 @ Z4 ) ) ) ).
% dense_le_bounded
thf(fact_452_dual__order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [B4: dedekind_preal,A4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_453_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_rat
= ( ^ [B4: set_rat,A4: set_rat] :
( ( ord_less_set_rat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_454_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_455_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B4: rat,A4: rat] :
( ( ord_less_rat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_456_dual__order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B4: dedekind_preal,A4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_457_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_rat
= ( ^ [B4: set_rat,A4: set_rat] :
( ( ord_less_eq_set_rat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_458_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_459_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B4: rat,A4: rat] :
( ( ord_less_eq_rat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_460_dual__order_Ostrict__trans1,axiom,
! [B3: dedekind_preal,A2: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B3 @ A2 )
=> ( ( ord_le5708704896291381698_preal @ C @ B3 )
=> ( ord_le5708704896291381698_preal @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_461_dual__order_Ostrict__trans1,axiom,
! [B3: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_eq_set_rat @ B3 @ A2 )
=> ( ( ord_less_set_rat @ C @ B3 )
=> ( ord_less_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_462_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_463_dual__order_Ostrict__trans1,axiom,
! [B3: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B3 @ A2 )
=> ( ( ord_less_rat @ C @ B3 )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_464_dual__order_Ostrict__trans2,axiom,
! [B3: dedekind_preal,A2: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B3 @ A2 )
=> ( ( ord_le5604041210740703414_preal @ C @ B3 )
=> ( ord_le5708704896291381698_preal @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_465_dual__order_Ostrict__trans2,axiom,
! [B3: set_rat,A2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ B3 @ A2 )
=> ( ( ord_less_eq_set_rat @ C @ B3 )
=> ( ord_less_set_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_466_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_467_dual__order_Ostrict__trans2,axiom,
! [B3: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B3 @ A2 )
=> ( ( ord_less_eq_rat @ C @ B3 )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_468_dual__order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B4: dedekind_preal,A4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B4 @ A4 )
& ~ ( ord_le5604041210740703414_preal @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_469_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_rat
= ( ^ [B4: set_rat,A4: set_rat] :
( ( ord_less_eq_set_rat @ B4 @ A4 )
& ~ ( ord_less_eq_set_rat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_470_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_471_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B4: rat,A4: rat] :
( ( ord_less_eq_rat @ B4 @ A4 )
& ~ ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_472_order_Ostrict__implies__order,axiom,
! [A2: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ord_le5604041210740703414_preal @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_473_order_Ostrict__implies__order,axiom,
! [A2: set_rat,B3: set_rat] :
( ( ord_less_set_rat @ A2 @ B3 )
=> ( ord_less_eq_set_rat @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_474_order_Ostrict__implies__order,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_475_order_Ostrict__implies__order,axiom,
! [A2: rat,B3: rat] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_476_dual__order_Ostrict__implies__order,axiom,
! [B3: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B3 @ A2 )
=> ( ord_le5604041210740703414_preal @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_477_dual__order_Ostrict__implies__order,axiom,
! [B3: set_rat,A2: set_rat] :
( ( ord_less_set_rat @ B3 @ A2 )
=> ( ord_less_eq_set_rat @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_478_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_479_dual__order_Ostrict__implies__order,axiom,
! [B3: rat,A2: rat] :
( ( ord_less_rat @ B3 @ A2 )
=> ( ord_less_eq_rat @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_480_order__le__less,axiom,
( ord_le5604041210740703414_preal
= ( ^ [X6: dedekind_preal,Y5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X6 @ Y5 )
| ( X6 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_481_order__le__less,axiom,
( ord_less_eq_set_rat
= ( ^ [X6: set_rat,Y5: set_rat] :
( ( ord_less_set_rat @ X6 @ Y5 )
| ( X6 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_482_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
| ( X6 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_483_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X6: rat,Y5: rat] :
( ( ord_less_rat @ X6 @ Y5 )
| ( X6 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_484_order__less__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X6: dedekind_preal,Y5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X6 @ Y5 )
& ( X6 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_485_order__less__le,axiom,
( ord_less_set_rat
= ( ^ [X6: set_rat,Y5: set_rat] :
( ( ord_less_eq_set_rat @ X6 @ Y5 )
& ( X6 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_486_order__less__le,axiom,
( ord_less_nat
= ( ^ [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
& ( X6 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_487_order__less__le,axiom,
( ord_less_rat
= ( ^ [X6: rat,Y5: rat] :
( ( ord_less_eq_rat @ X6 @ Y5 )
& ( X6 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_488_linorder__not__le,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ X5 @ Y2 ) )
= ( ord_le5708704896291381698_preal @ Y2 @ X5 ) ) ).
% linorder_not_le
thf(fact_489_linorder__not__le,axiom,
! [X5: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X5 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X5 ) ) ).
% linorder_not_le
thf(fact_490_linorder__not__le,axiom,
! [X5: rat,Y2: rat] :
( ( ~ ( ord_less_eq_rat @ X5 @ Y2 ) )
= ( ord_less_rat @ Y2 @ X5 ) ) ).
% linorder_not_le
thf(fact_491_linorder__not__less,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X5 @ Y2 ) )
= ( ord_le5604041210740703414_preal @ Y2 @ X5 ) ) ).
% linorder_not_less
thf(fact_492_linorder__not__less,axiom,
! [X5: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X5 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X5 ) ) ).
% linorder_not_less
thf(fact_493_linorder__not__less,axiom,
! [X5: rat,Y2: rat] :
( ( ~ ( ord_less_rat @ X5 @ Y2 ) )
= ( ord_less_eq_rat @ Y2 @ X5 ) ) ).
% linorder_not_less
thf(fact_494_order__less__imp__le,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ( ord_le5604041210740703414_preal @ X5 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_495_order__less__imp__le,axiom,
! [X5: set_rat,Y2: set_rat] :
( ( ord_less_set_rat @ X5 @ Y2 )
=> ( ord_less_eq_set_rat @ X5 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_496_order__less__imp__le,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ord_less_eq_nat @ X5 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_497_order__less__imp__le,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_rat @ X5 @ Y2 )
=> ( ord_less_eq_rat @ X5 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_498_order__le__neq__trans,axiom,
! [A2: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_le5708704896291381698_preal @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_499_order__le__neq__trans,axiom,
! [A2: set_rat,B3: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_rat @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_500_order__le__neq__trans,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_501_order__le__neq__trans,axiom,
! [A2: rat,B3: rat] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_rat @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_502_order__neq__le__trans,axiom,
! [A2: dedekind_preal,B3: dedekind_preal] :
( ( A2 != B3 )
=> ( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ord_le5708704896291381698_preal @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_503_order__neq__le__trans,axiom,
! [A2: set_rat,B3: set_rat] :
( ( A2 != B3 )
=> ( ( ord_less_eq_set_rat @ A2 @ B3 )
=> ( ord_less_set_rat @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_504_order__neq__le__trans,axiom,
! [A2: nat,B3: nat] :
( ( A2 != B3 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_505_order__neq__le__trans,axiom,
! [A2: rat,B3: rat] :
( ( A2 != B3 )
=> ( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ord_less_rat @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_506_order__le__less__trans,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal,Z4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y2 )
=> ( ( ord_le5708704896291381698_preal @ Y2 @ Z4 )
=> ( ord_le5708704896291381698_preal @ X5 @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_507_order__le__less__trans,axiom,
! [X5: set_rat,Y2: set_rat,Z4: set_rat] :
( ( ord_less_eq_set_rat @ X5 @ Y2 )
=> ( ( ord_less_set_rat @ Y2 @ Z4 )
=> ( ord_less_set_rat @ X5 @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_508_order__le__less__trans,axiom,
! [X5: nat,Y2: nat,Z4: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z4 )
=> ( ord_less_nat @ X5 @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_509_order__le__less__trans,axiom,
! [X5: rat,Y2: rat,Z4: rat] :
( ( ord_less_eq_rat @ X5 @ Y2 )
=> ( ( ord_less_rat @ Y2 @ Z4 )
=> ( ord_less_rat @ X5 @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_510_order__less__le__trans,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal,Z4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
=> ( ( ord_le5604041210740703414_preal @ Y2 @ Z4 )
=> ( ord_le5708704896291381698_preal @ X5 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_511_order__less__le__trans,axiom,
! [X5: set_rat,Y2: set_rat,Z4: set_rat] :
( ( ord_less_set_rat @ X5 @ Y2 )
=> ( ( ord_less_eq_set_rat @ Y2 @ Z4 )
=> ( ord_less_set_rat @ X5 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_512_order__less__le__trans,axiom,
! [X5: nat,Y2: nat,Z4: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z4 )
=> ( ord_less_nat @ X5 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_513_order__less__le__trans,axiom,
! [X5: rat,Y2: rat,Z4: rat] :
( ( ord_less_rat @ X5 @ Y2 )
=> ( ( ord_less_eq_rat @ Y2 @ Z4 )
=> ( ord_less_rat @ X5 @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_514_order__le__less__subst1,axiom,
! [A2: dedekind_preal,F: dedekind_preal > dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_515_order__le__less__subst1,axiom,
! [A2: dedekind_preal,F: rat > dedekind_preal,B3: rat,C: rat] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_516_order__le__less__subst1,axiom,
! [A2: dedekind_preal,F: nat > dedekind_preal,B3: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_517_order__le__less__subst1,axiom,
! [A2: nat,F: dedekind_preal > nat,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_518_order__le__less__subst1,axiom,
! [A2: nat,F: rat > nat,B3: rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_519_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_520_order__le__less__subst1,axiom,
! [A2: rat,F: dedekind_preal > rat,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5708704896291381698_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_521_order__le__less__subst1,axiom,
! [A2: rat,F: rat > rat,B3: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_522_order__le__less__subst1,axiom,
! [A2: rat,F: nat > rat,B3: nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_523_order__le__less__subst1,axiom,
! [A2: dedekind_preal,F: set_rat > dedekind_preal,B3: set_rat,C: set_rat] :
( ( ord_le5604041210740703414_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_set_rat @ B3 @ C )
=> ( ! [X: set_rat,Y3: set_rat] :
( ( ord_less_set_rat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_524_order__le__less__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_525_order__le__less__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_526_order__le__less__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > rat,C: rat] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ord_less_rat @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_527_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_528_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_529_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_rat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_530_order__le__less__subst2,axiom,
! [A2: rat,B3: rat,F: rat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B3 ) @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_531_order__le__less__subst2,axiom,
! [A2: rat,B3: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_532_order__le__less__subst2,axiom,
! [A2: rat,B3: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B3 )
=> ( ( ord_less_rat @ ( F @ B3 ) @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_533_order__le__less__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > set_rat,C: set_rat] :
( ( ord_le5604041210740703414_preal @ A2 @ B3 )
=> ( ( ord_less_set_rat @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_534_order__less__le__subst1,axiom,
! [A2: dedekind_preal,F: dedekind_preal > dedekind_preal,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_535_order__less__le__subst1,axiom,
! [A2: nat,F: dedekind_preal > nat,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_536_order__less__le__subst1,axiom,
! [A2: rat,F: dedekind_preal > rat,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_less_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_537_order__less__le__subst1,axiom,
! [A2: dedekind_preal,F: nat > dedekind_preal,B3: nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_538_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_539_order__less__le__subst1,axiom,
! [A2: rat,F: nat > rat,B3: nat,C: nat] :
( ( ord_less_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_540_order__less__le__subst1,axiom,
! [A2: dedekind_preal,F: rat > dedekind_preal,B3: rat,C: rat] :
( ( ord_le5708704896291381698_preal @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_541_order__less__le__subst1,axiom,
! [A2: nat,F: rat > nat,B3: rat,C: rat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_542_order__less__le__subst1,axiom,
! [A2: rat,F: rat > rat,B3: rat,C: rat] :
( ( ord_less_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_rat @ B3 @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_eq_rat @ X @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_543_order__less__le__subst1,axiom,
! [A2: set_rat,F: dedekind_preal > set_rat,B3: dedekind_preal,C: dedekind_preal] :
( ( ord_less_set_rat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le5604041210740703414_preal @ B3 @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_544_order__less__le__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_545_order__less__le__subst2,axiom,
! [A2: rat,B3: rat,F: rat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B3 ) @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_546_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_547_order__less__le__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_548_order__less__le__subst2,axiom,
! [A2: rat,B3: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_549_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_550_order__less__le__subst2,axiom,
! [A2: dedekind_preal,B3: dedekind_preal,F: dedekind_preal > rat,C: rat] :
( ( ord_le5708704896291381698_preal @ A2 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_551_order__less__le__subst2,axiom,
! [A2: rat,B3: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A2 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X: rat,Y3: rat] :
( ( ord_less_rat @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_552_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_rat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_553_order__less__le__subst2,axiom,
! [A2: set_rat,B3: set_rat,F: set_rat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_set_rat @ A2 @ B3 )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B3 ) @ C )
=> ( ! [X: set_rat,Y3: set_rat] :
( ( ord_less_set_rat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_554_linorder__le__less__linear,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y2 )
| ( ord_le5708704896291381698_preal @ Y2 @ X5 ) ) ).
% linorder_le_less_linear
thf(fact_555_linorder__le__less__linear,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
| ( ord_less_nat @ Y2 @ X5 ) ) ).
% linorder_le_less_linear
thf(fact_556_linorder__le__less__linear,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_eq_rat @ X5 @ Y2 )
| ( ord_less_rat @ Y2 @ X5 ) ) ).
% linorder_le_less_linear
thf(fact_557_order__le__imp__less__or__eq,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y2 )
=> ( ( ord_le5708704896291381698_preal @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_558_order__le__imp__less__or__eq,axiom,
! [X5: set_rat,Y2: set_rat] :
( ( ord_less_eq_set_rat @ X5 @ Y2 )
=> ( ( ord_less_set_rat @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_559_order__le__imp__less__or__eq,axiom,
! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ( ord_less_nat @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_560_order__le__imp__less__or__eq,axiom,
! [X5: rat,Y2: rat] :
( ( ord_less_eq_rat @ X5 @ Y2 )
=> ( ( ord_less_rat @ X5 @ Y2 )
| ( X5 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_561_bot_Oextremum__uniqueI,axiom,
! [A2: set_Dedekind_preal] :
( ( ord_le7349499860212017814_preal @ A2 @ bot_bo4848840305443107682_preal )
=> ( A2 = bot_bo4848840305443107682_preal ) ) ).
% bot.extremum_uniqueI
thf(fact_562_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_563_bot_Oextremum__uniqueI,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ bot_bot_set_rat )
=> ( A2 = bot_bot_set_rat ) ) ).
% bot.extremum_uniqueI
thf(fact_564_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_565_bot_Oextremum__unique,axiom,
! [A2: set_Dedekind_preal] :
( ( ord_le7349499860212017814_preal @ A2 @ bot_bo4848840305443107682_preal )
= ( A2 = bot_bo4848840305443107682_preal ) ) ).
% bot.extremum_unique
thf(fact_566_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_567_bot_Oextremum__unique,axiom,
! [A2: set_rat] :
( ( ord_less_eq_set_rat @ A2 @ bot_bot_set_rat )
= ( A2 = bot_bot_set_rat ) ) ).
% bot.extremum_unique
thf(fact_568_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_569_bot_Oextremum,axiom,
! [A2: set_Dedekind_preal] : ( ord_le7349499860212017814_preal @ bot_bo4848840305443107682_preal @ A2 ) ).
% bot.extremum
thf(fact_570_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_571_bot_Oextremum,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ bot_bot_set_rat @ A2 ) ).
% bot.extremum
thf(fact_572_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_573_bot_Oextremum__strict,axiom,
! [A2: set_Dedekind_preal] :
~ ( ord_le1802228187270208418_preal @ A2 @ bot_bo4848840305443107682_preal ) ).
% bot.extremum_strict
thf(fact_574_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_575_bot_Oextremum__strict,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ bot_bot_set_rat ) ).
% bot.extremum_strict
thf(fact_576_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_577_bot_Onot__eq__extremum,axiom,
! [A2: set_Dedekind_preal] :
( ( A2 != bot_bo4848840305443107682_preal )
= ( ord_le1802228187270208418_preal @ bot_bo4848840305443107682_preal @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_578_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_579_bot_Onot__eq__extremum,axiom,
! [A2: set_rat] :
( ( A2 != bot_bot_set_rat )
= ( ord_less_set_rat @ bot_bot_set_rat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_580_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_581_Rep__preal,axiom,
! [X5: dedekind_preal] : ( member_set_rat @ ( dedekind_Rep_preal @ X5 ) @ ( collect_set_rat @ dedekind_cut ) ) ).
% Rep_preal
thf(fact_582_Rep__preal__cases,axiom,
! [Y2: set_rat] :
( ( member_set_rat @ Y2 @ ( collect_set_rat @ dedekind_cut ) )
=> ~ ! [X: dedekind_preal] :
( Y2
!= ( dedekind_Rep_preal @ X ) ) ) ).
% Rep_preal_cases
thf(fact_583_image__is__empty,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( ( image_2231269897757205599et_rat @ F @ A )
= bot_bot_set_set_rat )
= ( A = bot_bo4848840305443107682_preal ) ) ).
% image_is_empty
thf(fact_584_image__is__empty,axiom,
! [F: dedekind_preal > dedekind_preal,A: set_Dedekind_preal] :
( ( ( image_4420143364984939013_preal @ F @ A )
= bot_bo4848840305443107682_preal )
= ( A = bot_bo4848840305443107682_preal ) ) ).
% image_is_empty
thf(fact_585_image__is__empty,axiom,
! [F: nat > dedekind_preal,A: set_nat] :
( ( ( image_8829711020507855231_preal @ F @ A )
= bot_bo4848840305443107682_preal )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_586_image__is__empty,axiom,
! [F: dedekind_preal > nat,A: set_Dedekind_preal] :
( ( ( image_7207786835304483489al_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bo4848840305443107682_preal ) ) ).
% image_is_empty
thf(fact_587_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_588_empty__is__image,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( bot_bot_set_set_rat
= ( image_2231269897757205599et_rat @ F @ A ) )
= ( A = bot_bo4848840305443107682_preal ) ) ).
% empty_is_image
thf(fact_589_empty__is__image,axiom,
! [F: dedekind_preal > dedekind_preal,A: set_Dedekind_preal] :
( ( bot_bo4848840305443107682_preal
= ( image_4420143364984939013_preal @ F @ A ) )
= ( A = bot_bo4848840305443107682_preal ) ) ).
% empty_is_image
thf(fact_590_empty__is__image,axiom,
! [F: nat > dedekind_preal,A: set_nat] :
( ( bot_bo4848840305443107682_preal
= ( image_8829711020507855231_preal @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_591_empty__is__image,axiom,
! [F: dedekind_preal > nat,A: set_Dedekind_preal] :
( ( bot_bot_set_nat
= ( image_7207786835304483489al_nat @ F @ A ) )
= ( A = bot_bo4848840305443107682_preal ) ) ).
% empty_is_image
thf(fact_592_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_593_image__empty,axiom,
! [F: dedekind_preal > set_rat] :
( ( image_2231269897757205599et_rat @ F @ bot_bo4848840305443107682_preal )
= bot_bot_set_set_rat ) ).
% image_empty
thf(fact_594_image__empty,axiom,
! [F: dedekind_preal > dedekind_preal] :
( ( image_4420143364984939013_preal @ F @ bot_bo4848840305443107682_preal )
= bot_bo4848840305443107682_preal ) ).
% image_empty
thf(fact_595_image__empty,axiom,
! [F: dedekind_preal > nat] :
( ( image_7207786835304483489al_nat @ F @ bot_bo4848840305443107682_preal )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_596_image__empty,axiom,
! [F: nat > dedekind_preal] :
( ( image_8829711020507855231_preal @ F @ bot_bot_set_nat )
= bot_bo4848840305443107682_preal ) ).
% image_empty
thf(fact_597_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_598_mem__diff__set,axiom,
! [R2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S2 )
=> ( dedekind_cut @ ( dedekind_diff_set @ ( dedekind_Rep_preal @ S2 ) @ ( dedekind_Rep_preal @ R2 ) ) ) ) ).
% mem_diff_set
thf(fact_599_verit__comp__simplify1_I3_J,axiom,
! [B6: dedekind_preal,A6: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ B6 @ A6 ) )
= ( ord_le5708704896291381698_preal @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_600_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_601_verit__comp__simplify1_I3_J,axiom,
! [B6: rat,A6: rat] :
( ( ~ ( ord_less_eq_rat @ B6 @ A6 ) )
= ( ord_less_rat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_602_complete__interval,axiom,
! [A2: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B3 )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A2 @ X )
& ( ord_less_nat @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_603_Sup__empty,axiom,
( ( comple800427960086186211_preal @ bot_bo5050857909902108994_preal )
= bot_bo4848840305443107682_preal ) ).
% Sup_empty
thf(fact_604_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_605_Sup__empty,axiom,
( ( comple3890839924845867745et_rat @ bot_bot_set_set_rat )
= bot_bot_set_rat ) ).
% Sup_empty
thf(fact_606_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_607_Set_Ois__empty__def,axiom,
( is_emp2142545888348893928_preal
= ( ^ [A3: set_Dedekind_preal] : ( A3 = bot_bo4848840305443107682_preal ) ) ) ).
% Set.is_empty_def
thf(fact_608_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_609__092_060open_062_I_092_060And_062X_O_AX_A_092_060in_062_AP_A_092_060Longrightarrow_062_AX_A_092_060in_062_AP_J_A_092_060Longrightarrow_062_ADedekind__Real_Ocut_A_I_092_060Union_062_A_IRep__preal_A_096_AP_J_J_092_060close_062,axiom,
( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ p )
=> ( member6871284927547481791_preal @ X4 @ p ) )
=> ( dedekind_cut @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ dedekind_Rep_preal @ p ) ) ) ) ).
% \<open>(\<And>X. X \<in> P \<Longrightarrow> X \<in> P) \<Longrightarrow> Dedekind_Real.cut (\<Union> (Rep_preal ` P))\<close>
thf(fact_610_mem__add__set,axiom,
! [A: set_rat,B: set_rat] :
( ( dedekind_cut @ A )
=> ( ( dedekind_cut @ B )
=> ( dedekind_cut @ ( dedekind_add_set @ A @ B ) ) ) ) ).
% mem_add_set
thf(fact_611_mem__mult__set,axiom,
! [A: set_rat,B: set_rat] :
( ( dedekind_cut @ A )
=> ( ( dedekind_cut @ B )
=> ( dedekind_cut @ ( dedekind_mult_set @ A @ B ) ) ) ) ).
% mem_mult_set
thf(fact_612_psup__def,axiom,
( dedekind_psup
= ( ^ [P4: set_Dedekind_preal] : ( dedekind_Abs_preal @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ dedekind_Rep_preal @ P4 ) ) ) ) ) ).
% psup_def
thf(fact_613_image__eqI,axiom,
! [B3: dedekind_preal,F: dedekind_preal > dedekind_preal,X5: dedekind_preal,A: set_Dedekind_preal] :
( ( B3
= ( F @ X5 ) )
=> ( ( member6871284927547481791_preal @ X5 @ A )
=> ( member6871284927547481791_preal @ B3 @ ( image_4420143364984939013_preal @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_614_image__eqI,axiom,
! [B3: rat,F: dedekind_preal > rat,X5: dedekind_preal,A: set_Dedekind_preal] :
( ( B3
= ( F @ X5 ) )
=> ( ( member6871284927547481791_preal @ X5 @ A )
=> ( member_rat @ B3 @ ( image_6572656775217987753al_rat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_615_image__eqI,axiom,
! [B3: $o,F: dedekind_preal > $o,X5: dedekind_preal,A: set_Dedekind_preal] :
( ( B3
= ( F @ X5 ) )
=> ( ( member6871284927547481791_preal @ X5 @ A )
=> ( member_o @ B3 @ ( image_248103505173020295real_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_616_image__eqI,axiom,
! [B3: dedekind_preal,F: rat > dedekind_preal,X5: rat,A: set_rat] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_rat @ X5 @ A )
=> ( member6871284927547481791_preal @ B3 @ ( image_7112546572676366711_preal @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_617_image__eqI,axiom,
! [B3: rat,F: rat > rat,X5: rat,A: set_rat] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_rat @ X5 @ A )
=> ( member_rat @ B3 @ ( image_rat_rat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_618_image__eqI,axiom,
! [B3: $o,F: rat > $o,X5: rat,A: set_rat] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_rat @ X5 @ A )
=> ( member_o @ B3 @ ( image_rat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_619_image__eqI,axiom,
! [B3: dedekind_preal,F: $o > dedekind_preal,X5: $o,A: set_o] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_o @ X5 @ A )
=> ( member6871284927547481791_preal @ B3 @ ( image_3026614506929738499_preal @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_620_image__eqI,axiom,
! [B3: rat,F: $o > rat,X5: $o,A: set_o] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_o @ X5 @ A )
=> ( member_rat @ B3 @ ( image_o_rat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_621_image__eqI,axiom,
! [B3: $o,F: $o > $o,X5: $o,A: set_o] :
( ( B3
= ( F @ X5 ) )
=> ( ( member_o @ X5 @ A )
=> ( member_o @ B3 @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_622_image__eqI,axiom,
! [B3: set_rat,F: dedekind_preal > set_rat,X5: dedekind_preal,A: set_Dedekind_preal] :
( ( B3
= ( F @ X5 ) )
=> ( ( member6871284927547481791_preal @ X5 @ A )
=> ( member_set_rat @ B3 @ ( image_2231269897757205599et_rat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_623_subsetI,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal] :
( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( member6871284927547481791_preal @ X @ B ) )
=> ( ord_le7349499860212017814_preal @ A @ B ) ) ).
% subsetI
thf(fact_624_subsetI,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ( member_set_rat @ X @ B ) )
=> ( ord_le513522071413781156et_rat @ A @ B ) ) ).
% subsetI
thf(fact_625_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ X @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_626_subsetI,axiom,
! [A: set_rat,B: set_rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( member_rat @ X @ B ) )
=> ( ord_less_eq_set_rat @ A @ B ) ) ).
% subsetI
thf(fact_627_subset__antisym,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( ord_less_eq_set_rat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_628_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_rat,P: rat > $o] :
( ( ? [X6: rat] :
( ( member_rat @ X6 @ ( comple3890839924845867745et_rat @ A ) )
& ( P @ X6 ) ) )
= ( ? [X6: set_rat] :
( ( member_set_rat @ X6 @ A )
& ? [Y5: rat] :
( ( member_rat @ Y5 @ X6 )
& ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_629_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_rat,P: rat > $o] :
( ( ! [X6: rat] :
( ( member_rat @ X6 @ ( comple3890839924845867745et_rat @ A ) )
=> ( P @ X6 ) ) )
= ( ! [X6: set_rat] :
( ( member_set_rat @ X6 @ A )
=> ! [Y5: rat] :
( ( member_rat @ Y5 @ X6 )
=> ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_630_UnionI,axiom,
! [X2: set_Dedekind_preal,C2: set_se4284277844656040150_preal,A: dedekind_preal] :
( ( member2000626104285586591_preal @ X2 @ C2 )
=> ( ( member6871284927547481791_preal @ A @ X2 )
=> ( member6871284927547481791_preal @ A @ ( comple800427960086186211_preal @ C2 ) ) ) ) ).
% UnionI
thf(fact_631_UnionI,axiom,
! [X2: set_set_rat,C2: set_set_set_rat,A: set_rat] :
( ( member_set_set_rat @ X2 @ C2 )
=> ( ( member_set_rat @ A @ X2 )
=> ( member_set_rat @ A @ ( comple3392050375588816791et_rat @ C2 ) ) ) ) ).
% UnionI
thf(fact_632_UnionI,axiom,
! [X2: set_o,C2: set_set_o,A: $o] :
( ( member_set_o @ X2 @ C2 )
=> ( ( member_o @ A @ X2 )
=> ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) ) ) ) ).
% UnionI
thf(fact_633_UnionI,axiom,
! [X2: set_rat,C2: set_set_rat,A: rat] :
( ( member_set_rat @ X2 @ C2 )
=> ( ( member_rat @ A @ X2 )
=> ( member_rat @ A @ ( comple3890839924845867745et_rat @ C2 ) ) ) ) ).
% UnionI
thf(fact_634_Union__iff,axiom,
! [A: dedekind_preal,C2: set_se4284277844656040150_preal] :
( ( member6871284927547481791_preal @ A @ ( comple800427960086186211_preal @ C2 ) )
= ( ? [X6: set_Dedekind_preal] :
( ( member2000626104285586591_preal @ X6 @ C2 )
& ( member6871284927547481791_preal @ A @ X6 ) ) ) ) ).
% Union_iff
thf(fact_635_Union__iff,axiom,
! [A: set_rat,C2: set_set_set_rat] :
( ( member_set_rat @ A @ ( comple3392050375588816791et_rat @ C2 ) )
= ( ? [X6: set_set_rat] :
( ( member_set_set_rat @ X6 @ C2 )
& ( member_set_rat @ A @ X6 ) ) ) ) ).
% Union_iff
thf(fact_636_Union__iff,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) )
= ( ? [X6: set_o] :
( ( member_set_o @ X6 @ C2 )
& ( member_o @ A @ X6 ) ) ) ) ).
% Union_iff
thf(fact_637_Union__iff,axiom,
! [A: rat,C2: set_set_rat] :
( ( member_rat @ A @ ( comple3890839924845867745et_rat @ C2 ) )
= ( ? [X6: set_rat] :
( ( member_set_rat @ X6 @ C2 )
& ( member_rat @ A @ X6 ) ) ) ) ).
% Union_iff
thf(fact_638_Sup__bot__conv_I2_J,axiom,
! [A: set_se4284277844656040150_preal] :
( ( bot_bo4848840305443107682_preal
= ( comple800427960086186211_preal @ A ) )
= ( ! [X6: set_Dedekind_preal] :
( ( member2000626104285586591_preal @ X6 @ A )
=> ( X6 = bot_bo4848840305443107682_preal ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_639_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A )
=> ( X6 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_640_Sup__bot__conv_I2_J,axiom,
! [A: set_set_rat] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ A ) )
= ( ! [X6: set_rat] :
( ( member_set_rat @ X6 @ A )
=> ( X6 = bot_bot_set_rat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_641_Sup__bot__conv_I2_J,axiom,
! [A: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A ) )
= ( ! [X6: $o] :
( ( member_o @ X6 @ A )
=> ( X6 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_642_Sup__bot__conv_I1_J,axiom,
! [A: set_se4284277844656040150_preal] :
( ( ( comple800427960086186211_preal @ A )
= bot_bo4848840305443107682_preal )
= ( ! [X6: set_Dedekind_preal] :
( ( member2000626104285586591_preal @ X6 @ A )
=> ( X6 = bot_bo4848840305443107682_preal ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_643_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A )
=> ( X6 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_644_Sup__bot__conv_I1_J,axiom,
! [A: set_set_rat] :
( ( ( comple3890839924845867745et_rat @ A )
= bot_bot_set_rat )
= ( ! [X6: set_rat] :
( ( member_set_rat @ X6 @ A )
=> ( X6 = bot_bot_set_rat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_645_Sup__bot__conv_I1_J,axiom,
! [A: set_o] :
( ( ( complete_Sup_Sup_o @ A )
= bot_bot_o )
= ( ! [X6: $o] :
( ( member_o @ X6 @ A )
=> ( X6 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_646_UN__ball__bex__simps_I4_J,axiom,
! [B: dedekind_preal > set_rat,A: set_Dedekind_preal,P: rat > $o] :
( ( ? [X6: rat] :
( ( member_rat @ X6 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) )
& ( P @ X6 ) ) )
= ( ? [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
& ? [Y5: rat] :
( ( member_rat @ Y5 @ ( B @ X6 ) )
& ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_647_UN__ball__bex__simps_I2_J,axiom,
! [B: dedekind_preal > set_rat,A: set_Dedekind_preal,P: rat > $o] :
( ( ! [X6: rat] :
( ( member_rat @ X6 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) )
=> ( P @ X6 ) ) )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
=> ! [Y5: rat] :
( ( member_rat @ Y5 @ ( B @ X6 ) )
=> ( P @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_648_bex__UN,axiom,
! [B: dedekind_preal > set_rat,A: set_Dedekind_preal,P: rat > $o] :
( ( ? [X6: rat] :
( ( member_rat @ X6 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) )
& ( P @ X6 ) ) )
= ( ? [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
& ? [Y5: rat] :
( ( member_rat @ Y5 @ ( B @ X6 ) )
& ( P @ Y5 ) ) ) ) ) ).
% bex_UN
thf(fact_649_ball__UN,axiom,
! [B: dedekind_preal > set_rat,A: set_Dedekind_preal,P: rat > $o] :
( ( ! [X6: rat] :
( ( member_rat @ X6 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) )
=> ( P @ X6 ) ) )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
=> ! [Y5: rat] :
( ( member_rat @ Y5 @ ( B @ X6 ) )
=> ( P @ Y5 ) ) ) ) ) ).
% ball_UN
thf(fact_650_SUP__identity__eq,axiom,
! [A: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X6: set_rat] : X6
@ A ) )
= ( comple3890839924845867745et_rat @ A ) ) ).
% SUP_identity_eq
thf(fact_651_SUP__identity__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X6: $o] : X6
@ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_identity_eq
thf(fact_652_SUP__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X6: nat] : X6
@ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_653_UN__I,axiom,
! [A2: dedekind_preal,A: set_Dedekind_preal,B3: dedekind_preal,B: dedekind_preal > set_Dedekind_preal] :
( ( member6871284927547481791_preal @ A2 @ A )
=> ( ( member6871284927547481791_preal @ B3 @ ( B @ A2 ) )
=> ( member6871284927547481791_preal @ B3 @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_654_UN__I,axiom,
! [A2: dedekind_preal,A: set_Dedekind_preal,B3: $o,B: dedekind_preal > set_o] :
( ( member6871284927547481791_preal @ A2 @ A )
=> ( ( member_o @ B3 @ ( B @ A2 ) )
=> ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_655_UN__I,axiom,
! [A2: rat,A: set_rat,B3: dedekind_preal,B: rat > set_Dedekind_preal] :
( ( member_rat @ A2 @ A )
=> ( ( member6871284927547481791_preal @ B3 @ ( B @ A2 ) )
=> ( member6871284927547481791_preal @ B3 @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_656_UN__I,axiom,
! [A2: rat,A: set_rat,B3: $o,B: rat > set_o] :
( ( member_rat @ A2 @ A )
=> ( ( member_o @ B3 @ ( B @ A2 ) )
=> ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_657_UN__I,axiom,
! [A2: $o,A: set_o,B3: dedekind_preal,B: $o > set_Dedekind_preal] :
( ( member_o @ A2 @ A )
=> ( ( member6871284927547481791_preal @ B3 @ ( B @ A2 ) )
=> ( member6871284927547481791_preal @ B3 @ ( comple800427960086186211_preal @ ( image_2856262260070713571_preal @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_658_UN__I,axiom,
! [A2: $o,A: set_o,B3: $o,B: $o > set_o] :
( ( member_o @ A2 @ A )
=> ( ( member_o @ B3 @ ( B @ A2 ) )
=> ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_659_UN__I,axiom,
! [A2: dedekind_preal,A: set_Dedekind_preal,B3: rat,B: dedekind_preal > set_rat] :
( ( member6871284927547481791_preal @ A2 @ A )
=> ( ( member_rat @ B3 @ ( B @ A2 ) )
=> ( member_rat @ B3 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_660_UN__I,axiom,
! [A2: rat,A: set_rat,B3: rat,B: rat > set_rat] :
( ( member_rat @ A2 @ A )
=> ( ( member_rat @ B3 @ ( B @ A2 ) )
=> ( member_rat @ B3 @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_661_UN__I,axiom,
! [A2: $o,A: set_o,B3: rat,B: $o > set_rat] :
( ( member_o @ A2 @ A )
=> ( ( member_rat @ B3 @ ( B @ A2 ) )
=> ( member_rat @ B3 @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_662_UN__I,axiom,
! [A2: dedekind_preal,A: set_Dedekind_preal,B3: set_rat,B: dedekind_preal > set_set_rat] :
( ( member6871284927547481791_preal @ A2 @ A )
=> ( ( member_set_rat @ B3 @ ( B @ A2 ) )
=> ( member_set_rat @ B3 @ ( comple3392050375588816791et_rat @ ( image_8268678932589273365et_rat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_663_UN__iff,axiom,
! [B3: rat,B: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( member_rat @ B3 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) )
= ( ? [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
& ( member_rat @ B3 @ ( B @ X6 ) ) ) ) ) ).
% UN_iff
thf(fact_664_SUP__bot__conv_I2_J,axiom,
! [B: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
=> ( ( B @ X6 )
= bot_bot_set_rat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_665_SUP__bot__conv_I1_J,axiom,
! [B: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) )
= bot_bot_set_rat )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
=> ( ( B @ X6 )
= bot_bot_set_rat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_666_SUP__bot,axiom,
! [A: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X6: dedekind_preal] : bot_bot_set_rat
@ A ) )
= bot_bot_set_rat ) ).
% SUP_bot
thf(fact_667_SUP__const,axiom,
! [A: set_Dedekind_preal,F: set_rat] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [I: dedekind_preal] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_668_SUP__const,axiom,
! [A: set_nat,F: set_rat] :
( ( A != bot_bot_set_nat )
=> ( ( comple3890839924845867745et_rat
@ ( image_nat_set_rat
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_669_SUP__const,axiom,
! [A: set_Dedekind_preal,F: $o] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [I: dedekind_preal] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_670_SUP__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_671_cSUP__const,axiom,
! [A: set_Dedekind_preal,C: set_rat] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X6: dedekind_preal] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_672_cSUP__const,axiom,
! [A: set_nat,C: set_rat] :
( ( A != bot_bot_set_nat )
=> ( ( comple3890839924845867745et_rat
@ ( image_nat_set_rat
@ ^ [X6: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_673_cSUP__const,axiom,
! [A: set_Dedekind_preal,C: $o] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [X6: dedekind_preal] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_674_cSUP__const,axiom,
! [A: set_nat,C: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X6: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_675_cSUP__const,axiom,
! [A: set_Dedekind_preal,C: nat] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( complete_Sup_Sup_nat
@ ( image_7207786835304483489al_nat
@ ^ [X6: dedekind_preal] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_676_cSUP__const,axiom,
! [A: set_nat,C: nat] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X6: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_677_UN__constant,axiom,
! [A: set_Dedekind_preal,C: set_Dedekind_preal] :
( ( ( A = bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= bot_bo4848840305443107682_preal ) )
& ( ( A != bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_678_UN__constant,axiom,
! [A: set_Dedekind_preal,C: set_nat] :
( ( ( A = bot_bo4848840305443107682_preal )
=> ( ( comple7399068483239264473et_nat
@ ( image_5739498456150602327et_nat
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bo4848840305443107682_preal )
=> ( ( comple7399068483239264473et_nat
@ ( image_5739498456150602327et_nat
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_679_UN__constant,axiom,
! [A: set_nat,C: set_Dedekind_preal] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple800427960086186211_preal
@ ( image_6010050456725241695_preal
@ ^ [Y5: nat] : C
@ A ) )
= bot_bo4848840305443107682_preal ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple800427960086186211_preal
@ ( image_6010050456725241695_preal
@ ^ [Y5: nat] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_680_UN__constant,axiom,
! [A: set_nat,C: set_nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : C
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_681_UN__constant,axiom,
! [A: set_Dedekind_preal,C: set_rat] :
( ( ( A = bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= bot_bot_set_rat ) )
& ( ( A != bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_682_UN__constant,axiom,
! [A: set_nat,C: set_rat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple3890839924845867745et_rat
@ ( image_nat_set_rat
@ ^ [Y5: nat] : C
@ A ) )
= bot_bot_set_rat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple3890839924845867745et_rat
@ ( image_nat_set_rat
@ ^ [Y5: nat] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_683_UNION__empty__conv_I2_J,axiom,
! [B: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) )
= bot_bot_set_rat )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
=> ( ( B @ X6 )
= bot_bot_set_rat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_684_UNION__empty__conv_I1_J,axiom,
! [B: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
=> ( ( B @ X6 )
= bot_bot_set_rat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_685_SUP__eqI,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > set_rat,X5: set_rat] :
( ! [I2: dedekind_preal] :
( ( member6871284927547481791_preal @ I2 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: set_rat] :
( ! [I3: dedekind_preal] :
( ( member6871284927547481791_preal @ I3 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_rat @ X5 @ Y3 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_686_SUP__eqI,axiom,
! [A: set_rat,F: rat > set_rat,X5: set_rat] :
( ! [I2: rat] :
( ( member_rat @ I2 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: set_rat] :
( ! [I3: rat] :
( ( member_rat @ I3 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_rat @ X5 @ Y3 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_687_SUP__eqI,axiom,
! [A: set_set_rat,F: set_rat > set_rat,X5: set_rat] :
( ! [I2: set_rat] :
( ( member_set_rat @ I2 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: set_rat] :
( ! [I3: set_rat] :
( ( member_set_rat @ I3 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_rat @ X5 @ Y3 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_688_SUP__eqI,axiom,
! [A: set_o,F: $o > set_rat,X5: set_rat] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: set_rat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_rat @ X5 @ Y3 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_689_SUP__eqI,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > $o,X5: $o] :
( ! [I2: dedekind_preal] :
( ( member6871284927547481791_preal @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: $o] :
( ! [I3: dedekind_preal] :
( ( member6871284927547481791_preal @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_o @ X5 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_690_SUP__eqI,axiom,
! [A: set_rat,F: rat > $o,X5: $o] :
( ! [I2: rat] :
( ( member_rat @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: $o] :
( ! [I3: rat] :
( ( member_rat @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_o @ X5 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_691_SUP__eqI,axiom,
! [A: set_set_rat,F: set_rat > $o,X5: $o] :
( ! [I2: set_rat] :
( ( member_set_rat @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: $o] :
( ! [I3: set_rat] :
( ( member_set_rat @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_o @ X5 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_692_SUP__eqI,axiom,
! [A: set_o,F: $o > $o,X5: $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X5 ) )
=> ( ! [Y3: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_o @ X5 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= X5 ) ) ) ).
% SUP_eqI
thf(fact_693_SUP__mono,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,F: dedekind_preal > set_rat,G: dedekind_preal > set_rat] :
( ! [N2: dedekind_preal] :
( ( member6871284927547481791_preal @ N2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_set_rat @ ( F @ N2 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_694_SUP__mono,axiom,
! [A: set_rat,B: set_Dedekind_preal,F: rat > set_rat,G: dedekind_preal > set_rat] :
( ! [N2: rat] :
( ( member_rat @ N2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_set_rat @ ( F @ N2 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_695_SUP__mono,axiom,
! [A: set_set_rat,B: set_Dedekind_preal,F: set_rat > set_rat,G: dedekind_preal > set_rat] :
( ! [N2: set_rat] :
( ( member_set_rat @ N2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_set_rat @ ( F @ N2 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_696_SUP__mono,axiom,
! [A: set_o,B: set_Dedekind_preal,F: $o > set_rat,G: dedekind_preal > set_rat] :
( ! [N2: $o] :
( ( member_o @ N2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_set_rat @ ( F @ N2 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_697_SUP__least,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > set_rat,U: set_rat] :
( ! [I2: dedekind_preal] :
( ( member6871284927547481791_preal @ I2 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_698_SUP__least,axiom,
! [A: set_rat,F: rat > set_rat,U: set_rat] :
( ! [I2: rat] :
( ( member_rat @ I2 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_699_SUP__least,axiom,
! [A: set_set_rat,F: set_rat > set_rat,U: set_rat] :
( ! [I2: set_rat] :
( ( member_set_rat @ I2 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_700_SUP__least,axiom,
! [A: set_o,F: $o > set_rat,U: set_rat] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_701_SUP__least,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > $o,U: $o] :
( ! [I2: dedekind_preal] :
( ( member6871284927547481791_preal @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_702_SUP__least,axiom,
! [A: set_rat,F: rat > $o,U: $o] :
( ! [I2: rat] :
( ( member_rat @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_703_SUP__least,axiom,
! [A: set_set_rat,F: set_rat > $o,U: $o] :
( ! [I2: set_rat] :
( ( member_set_rat @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_704_SUP__least,axiom,
! [A: set_o,F: $o > $o,U: $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_705_SUP__lessD,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal,Y2: set_rat,I4: dedekind_preal] :
( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) @ Y2 )
=> ( ( member6871284927547481791_preal @ I4 @ A )
=> ( ord_less_set_rat @ ( F @ I4 ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_706_SUP__lessD,axiom,
! [F: rat > set_rat,A: set_rat,Y2: set_rat,I4: rat] :
( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) @ Y2 )
=> ( ( member_rat @ I4 @ A )
=> ( ord_less_set_rat @ ( F @ I4 ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_707_SUP__lessD,axiom,
! [F: set_rat > set_rat,A: set_set_rat,Y2: set_rat,I4: set_rat] :
( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A ) ) @ Y2 )
=> ( ( member_set_rat @ I4 @ A )
=> ( ord_less_set_rat @ ( F @ I4 ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_708_SUP__lessD,axiom,
! [F: $o > set_rat,A: set_o,Y2: set_rat,I4: $o] :
( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) @ Y2 )
=> ( ( member_o @ I4 @ A )
=> ( ord_less_set_rat @ ( F @ I4 ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_709_SUP__lessD,axiom,
! [F: dedekind_preal > $o,A: set_Dedekind_preal,Y2: $o,I4: dedekind_preal] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) ) @ Y2 )
=> ( ( member6871284927547481791_preal @ I4 @ A )
=> ( ord_less_o @ ( F @ I4 ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_710_SUP__lessD,axiom,
! [F: rat > $o,A: set_rat,Y2: $o,I4: rat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) ) @ Y2 )
=> ( ( member_rat @ I4 @ A )
=> ( ord_less_o @ ( F @ I4 ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_711_SUP__lessD,axiom,
! [F: set_rat > $o,A: set_set_rat,Y2: $o,I4: set_rat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ A ) ) @ Y2 )
=> ( ( member_set_rat @ I4 @ A )
=> ( ord_less_o @ ( F @ I4 ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_712_SUP__lessD,axiom,
! [F: $o > $o,A: set_o,Y2: $o,I4: $o] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ Y2 )
=> ( ( member_o @ I4 @ A )
=> ( ord_less_o @ ( F @ I4 ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_713_SUP__mono_H,axiom,
! [F: dedekind_preal > set_rat,G: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ! [X: dedekind_preal] : ( ord_less_eq_set_rat @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ A ) ) ) ) ).
% SUP_mono'
thf(fact_714_SUP__upper,axiom,
! [I4: dedekind_preal,A: set_Dedekind_preal,F: dedekind_preal > set_rat] :
( ( member6871284927547481791_preal @ I4 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I4 ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_715_SUP__upper,axiom,
! [I4: rat,A: set_rat,F: rat > set_rat] :
( ( member_rat @ I4 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I4 ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_716_SUP__upper,axiom,
! [I4: set_rat,A: set_set_rat,F: set_rat > set_rat] :
( ( member_set_rat @ I4 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I4 ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_717_SUP__upper,axiom,
! [I4: $o,A: set_o,F: $o > set_rat] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_rat @ ( F @ I4 ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_718_SUP__upper,axiom,
! [I4: dedekind_preal,A: set_Dedekind_preal,F: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_719_SUP__upper,axiom,
! [I4: rat,A: set_rat,F: rat > $o] :
( ( member_rat @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_720_SUP__upper,axiom,
! [I4: set_rat,A: set_set_rat,F: set_rat > $o] :
( ( member_set_rat @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_721_SUP__upper,axiom,
! [I4: $o,A: set_o,F: $o > $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_722_SUP__le__iff,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal,U: set_rat] :
( ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) @ U )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
=> ( ord_less_eq_set_rat @ ( F @ X6 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_723_SUP__upper2,axiom,
! [I4: dedekind_preal,A: set_Dedekind_preal,U: set_rat,F: dedekind_preal > set_rat] :
( ( member6871284927547481791_preal @ I4 @ A )
=> ( ( ord_less_eq_set_rat @ U @ ( F @ I4 ) )
=> ( ord_less_eq_set_rat @ U @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_724_SUP__upper2,axiom,
! [I4: rat,A: set_rat,U: set_rat,F: rat > set_rat] :
( ( member_rat @ I4 @ A )
=> ( ( ord_less_eq_set_rat @ U @ ( F @ I4 ) )
=> ( ord_less_eq_set_rat @ U @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_725_SUP__upper2,axiom,
! [I4: set_rat,A: set_set_rat,U: set_rat,F: set_rat > set_rat] :
( ( member_set_rat @ I4 @ A )
=> ( ( ord_less_eq_set_rat @ U @ ( F @ I4 ) )
=> ( ord_less_eq_set_rat @ U @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_726_SUP__upper2,axiom,
! [I4: $o,A: set_o,U: set_rat,F: $o > set_rat] :
( ( member_o @ I4 @ A )
=> ( ( ord_less_eq_set_rat @ U @ ( F @ I4 ) )
=> ( ord_less_eq_set_rat @ U @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_727_SUP__upper2,axiom,
! [I4: dedekind_preal,A: set_Dedekind_preal,U: $o,F: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ I4 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I4 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_728_SUP__upper2,axiom,
! [I4: rat,A: set_rat,U: $o,F: rat > $o] :
( ( member_rat @ I4 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I4 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_729_SUP__upper2,axiom,
! [I4: set_rat,A: set_set_rat,U: $o,F: set_rat > $o] :
( ( member_set_rat @ I4 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I4 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_730_SUP__upper2,axiom,
! [I4: $o,A: set_o,U: $o,F: $o > $o] :
( ( member_o @ I4 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I4 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_731_SUP__subset__mono,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,F: dedekind_preal > set_rat,G: dedekind_preal > set_rat] :
( ( ord_le7349499860212017814_preal @ A @ B )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_732_SUP__subset__mono,axiom,
! [A: set_set_rat,B: set_set_rat,F: set_rat > set_rat,G: set_rat > set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_733_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_rat,G: $o > set_rat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_734_SUP__subset__mono,axiom,
! [A: set_rat,B: set_rat,F: rat > set_rat,G: rat > set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_735_SUP__subset__mono,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,F: dedekind_preal > $o,G: dedekind_preal > $o] :
( ( ord_le7349499860212017814_preal @ A @ B )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_736_SUP__subset__mono,axiom,
! [A: set_set_rat,B: set_set_rat,F: set_rat > $o,G: set_rat > $o] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_set_rat_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_737_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_738_SUP__subset__mono,axiom,
! [A: set_rat,B: set_rat,F: rat > $o,G: rat > $o] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_rat_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_739_in__mono,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,X5: dedekind_preal] :
( ( ord_le7349499860212017814_preal @ A @ B )
=> ( ( member6871284927547481791_preal @ X5 @ A )
=> ( member6871284927547481791_preal @ X5 @ B ) ) ) ).
% in_mono
thf(fact_740_in__mono,axiom,
! [A: set_set_rat,B: set_set_rat,X5: set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ( member_set_rat @ X5 @ A )
=> ( member_set_rat @ X5 @ B ) ) ) ).
% in_mono
thf(fact_741_in__mono,axiom,
! [A: set_o,B: set_o,X5: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X5 @ A )
=> ( member_o @ X5 @ B ) ) ) ).
% in_mono
thf(fact_742_in__mono,axiom,
! [A: set_rat,B: set_rat,X5: rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( member_rat @ X5 @ A )
=> ( member_rat @ X5 @ B ) ) ) ).
% in_mono
thf(fact_743_subsetD,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,C: dedekind_preal] :
( ( ord_le7349499860212017814_preal @ A @ B )
=> ( ( member6871284927547481791_preal @ C @ A )
=> ( member6871284927547481791_preal @ C @ B ) ) ) ).
% subsetD
thf(fact_744_subsetD,axiom,
! [A: set_set_rat,B: set_set_rat,C: set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ( member_set_rat @ C @ A )
=> ( member_set_rat @ C @ B ) ) ) ).
% subsetD
thf(fact_745_subsetD,axiom,
! [A: set_o,B: set_o,C: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% subsetD
thf(fact_746_subsetD,axiom,
! [A: set_rat,B: set_rat,C: rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( member_rat @ C @ A )
=> ( member_rat @ C @ B ) ) ) ).
% subsetD
thf(fact_747_empty__def,axiom,
( bot_bot_set_set_rat
= ( collect_set_rat
@ ^ [X6: set_rat] : $false ) ) ).
% empty_def
thf(fact_748_empty__def,axiom,
( bot_bot_set_rat
= ( collect_rat
@ ^ [X6: rat] : $false ) ) ).
% empty_def
thf(fact_749_empty__def,axiom,
( bot_bo4848840305443107682_preal
= ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] : $false ) ) ).
% empty_def
thf(fact_750_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X6: nat] : $false ) ) ).
% empty_def
thf(fact_751_equalityE,axiom,
! [A: set_rat,B: set_rat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_rat @ A @ B )
=> ~ ( ord_less_eq_set_rat @ B @ A ) ) ) ).
% equalityE
thf(fact_752_subset__eq,axiom,
( ord_le7349499860212017814_preal
= ( ^ [A3: set_Dedekind_preal,B2: set_Dedekind_preal] :
! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A3 )
=> ( member6871284927547481791_preal @ X6 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_753_subset__eq,axiom,
( ord_le513522071413781156et_rat
= ( ^ [A3: set_set_rat,B2: set_set_rat] :
! [X6: set_rat] :
( ( member_set_rat @ X6 @ A3 )
=> ( member_set_rat @ X6 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_754_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
! [X6: $o] :
( ( member_o @ X6 @ A3 )
=> ( member_o @ X6 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_755_subset__eq,axiom,
( ord_less_eq_set_rat
= ( ^ [A3: set_rat,B2: set_rat] :
! [X6: rat] :
( ( member_rat @ X6 @ A3 )
=> ( member_rat @ X6 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_756_equalityD1,axiom,
! [A: set_rat,B: set_rat] :
( ( A = B )
=> ( ord_less_eq_set_rat @ A @ B ) ) ).
% equalityD1
thf(fact_757_equalityD2,axiom,
! [A: set_rat,B: set_rat] :
( ( A = B )
=> ( ord_less_eq_set_rat @ B @ A ) ) ).
% equalityD2
thf(fact_758_subset__iff,axiom,
( ord_le7349499860212017814_preal
= ( ^ [A3: set_Dedekind_preal,B2: set_Dedekind_preal] :
! [T2: dedekind_preal] :
( ( member6871284927547481791_preal @ T2 @ A3 )
=> ( member6871284927547481791_preal @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_759_subset__iff,axiom,
( ord_le513522071413781156et_rat
= ( ^ [A3: set_set_rat,B2: set_set_rat] :
! [T2: set_rat] :
( ( member_set_rat @ T2 @ A3 )
=> ( member_set_rat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_760_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
! [T2: $o] :
( ( member_o @ T2 @ A3 )
=> ( member_o @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_761_subset__iff,axiom,
( ord_less_eq_set_rat
= ( ^ [A3: set_rat,B2: set_rat] :
! [T2: rat] :
( ( member_rat @ T2 @ A3 )
=> ( member_rat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_762_subset__refl,axiom,
! [A: set_rat] : ( ord_less_eq_set_rat @ A @ A ) ).
% subset_refl
thf(fact_763_Collect__mono,axiom,
! [P: set_rat > $o,Q: set_rat > $o] :
( ! [X: set_rat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le513522071413781156et_rat @ ( collect_set_rat @ P ) @ ( collect_set_rat @ Q ) ) ) ).
% Collect_mono
thf(fact_764_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_765_Collect__mono,axiom,
! [P: rat > $o,Q: rat > $o] :
( ! [X: rat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_rat @ ( collect_rat @ P ) @ ( collect_rat @ Q ) ) ) ).
% Collect_mono
thf(fact_766_subset__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ( ord_less_eq_set_rat @ B @ C2 )
=> ( ord_less_eq_set_rat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_767_set__eq__subset,axiom,
( ( ^ [Y4: set_rat,Z3: set_rat] : ( Y4 = Z3 ) )
= ( ^ [A3: set_rat,B2: set_rat] :
( ( ord_less_eq_set_rat @ A3 @ B2 )
& ( ord_less_eq_set_rat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_768_Collect__subset,axiom,
! [A: set_Dedekind_preal,P: dedekind_preal > $o] :
( ord_le7349499860212017814_preal
@ ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
& ( P @ X6 ) ) )
@ A ) ).
% Collect_subset
thf(fact_769_Collect__subset,axiom,
! [A: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X6: $o] :
( ( member_o @ X6 @ A )
& ( P @ X6 ) ) )
@ A ) ).
% Collect_subset
thf(fact_770_Collect__subset,axiom,
! [A: set_set_rat,P: set_rat > $o] :
( ord_le513522071413781156et_rat
@ ( collect_set_rat
@ ^ [X6: set_rat] :
( ( member_set_rat @ X6 @ A )
& ( P @ X6 ) ) )
@ A ) ).
% Collect_subset
thf(fact_771_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X6: nat] :
( ( member_nat @ X6 @ A )
& ( P @ X6 ) ) )
@ A ) ).
% Collect_subset
thf(fact_772_Collect__subset,axiom,
! [A: set_rat,P: rat > $o] :
( ord_less_eq_set_rat
@ ( collect_rat
@ ^ [X6: rat] :
( ( member_rat @ X6 @ A )
& ( P @ X6 ) ) )
@ A ) ).
% Collect_subset
thf(fact_773_less__eq__set__def,axiom,
( ord_le7349499860212017814_preal
= ( ^ [A3: set_Dedekind_preal,B2: set_Dedekind_preal] :
( ord_le5990576202697070023real_o
@ ^ [X6: dedekind_preal] : ( member6871284927547481791_preal @ X6 @ A3 )
@ ^ [X6: dedekind_preal] : ( member6871284927547481791_preal @ X6 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_774_less__eq__set__def,axiom,
( ord_le513522071413781156et_rat
= ( ^ [A3: set_set_rat,B2: set_set_rat] :
( ord_le3214917520859460321_rat_o
@ ^ [X6: set_rat] : ( member_set_rat @ X6 @ A3 )
@ ^ [X6: set_rat] : ( member_set_rat @ X6 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_775_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
( ord_less_eq_o_o
@ ^ [X6: $o] : ( member_o @ X6 @ A3 )
@ ^ [X6: $o] : ( member_o @ X6 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_776_less__eq__set__def,axiom,
( ord_less_eq_set_rat
= ( ^ [A3: set_rat,B2: set_rat] :
( ord_less_eq_rat_o
@ ^ [X6: rat] : ( member_rat @ X6 @ A3 )
@ ^ [X6: rat] : ( member_rat @ X6 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_777_Collect__mono__iff,axiom,
! [P: set_rat > $o,Q: set_rat > $o] :
( ( ord_le513522071413781156et_rat @ ( collect_set_rat @ P ) @ ( collect_set_rat @ Q ) )
= ( ! [X6: set_rat] :
( ( P @ X6 )
=> ( Q @ X6 ) ) ) ) ).
% Collect_mono_iff
thf(fact_778_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X6: nat] :
( ( P @ X6 )
=> ( Q @ X6 ) ) ) ) ).
% Collect_mono_iff
thf(fact_779_Collect__mono__iff,axiom,
! [P: rat > $o,Q: rat > $o] :
( ( ord_less_eq_set_rat @ ( collect_rat @ P ) @ ( collect_rat @ Q ) )
= ( ! [X6: rat] :
( ( P @ X6 )
=> ( Q @ X6 ) ) ) ) ).
% Collect_mono_iff
thf(fact_780_UN__mono,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,F: dedekind_preal > set_rat,G: dedekind_preal > set_rat] :
( ( ord_le7349499860212017814_preal @ A @ B )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_781_UN__mono,axiom,
! [A: set_set_rat,B: set_set_rat,F: set_rat > set_rat,G: set_rat > set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_782_UN__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_rat,G: $o > set_rat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_783_UN__mono,axiom,
! [A: set_rat,B: set_rat,F: rat > set_rat,G: rat > set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_784_UN__empty,axiom,
! [B: dedekind_preal > set_Dedekind_preal] :
( ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B @ bot_bo4848840305443107682_preal ) )
= bot_bo4848840305443107682_preal ) ).
% UN_empty
thf(fact_785_UN__empty,axiom,
! [B: dedekind_preal > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_5739498456150602327et_nat @ B @ bot_bo4848840305443107682_preal ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_786_UN__empty,axiom,
! [B: nat > set_Dedekind_preal] :
( ( comple800427960086186211_preal @ ( image_6010050456725241695_preal @ B @ bot_bot_set_nat ) )
= bot_bo4848840305443107682_preal ) ).
% UN_empty
thf(fact_787_UN__empty,axiom,
! [B: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_788_UN__empty,axiom,
! [B: dedekind_preal > set_rat] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ bot_bo4848840305443107682_preal ) )
= bot_bot_set_rat ) ).
% UN_empty
thf(fact_789_UN__empty,axiom,
! [B: nat > set_rat] :
( ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ B @ bot_bot_set_nat ) )
= bot_bot_set_rat ) ).
% UN_empty
thf(fact_790_UN__least,axiom,
! [A: set_Dedekind_preal,B: dedekind_preal > set_rat,C2: set_rat] :
( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_less_eq_set_rat @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_791_UN__least,axiom,
! [A: set_rat,B: rat > set_rat,C2: set_rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_set_rat @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_792_UN__least,axiom,
! [A: set_set_rat,B: set_rat > set_rat,C2: set_rat] :
( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ( ord_less_eq_set_rat @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_793_UN__least,axiom,
! [A: set_o,B: $o > set_rat,C2: set_rat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_rat @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_794_UN__upper,axiom,
! [A2: dedekind_preal,A: set_Dedekind_preal,B: dedekind_preal > set_rat] :
( ( member6871284927547481791_preal @ A2 @ A )
=> ( ord_less_eq_set_rat @ ( B @ A2 ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_795_UN__upper,axiom,
! [A2: rat,A: set_rat,B: rat > set_rat] :
( ( member_rat @ A2 @ A )
=> ( ord_less_eq_set_rat @ ( B @ A2 ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_796_UN__upper,axiom,
! [A2: set_rat,A: set_set_rat,B: set_rat > set_rat] :
( ( member_set_rat @ A2 @ A )
=> ( ord_less_eq_set_rat @ ( B @ A2 ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_797_UN__upper,axiom,
! [A2: $o,A: set_o,B: $o > set_rat] :
( ( member_o @ A2 @ A )
=> ( ord_less_eq_set_rat @ ( B @ A2 ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_798_UN__empty2,axiom,
! [A: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X6: dedekind_preal] : bot_bot_set_rat
@ A ) )
= bot_bot_set_rat ) ).
% UN_empty2
thf(fact_799_Union__mono,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ ( comple3890839924845867745et_rat @ B ) ) ) ).
% Union_mono
thf(fact_800_Union__least,axiom,
! [A: set_set_rat,C2: set_rat] :
( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A )
=> ( ord_less_eq_set_rat @ X4 @ C2 ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ C2 ) ) ).
% Union_least
thf(fact_801_Union__upper,axiom,
! [B: set_rat,A: set_set_rat] :
( ( member_set_rat @ B @ A )
=> ( ord_less_eq_set_rat @ B @ ( comple3890839924845867745et_rat @ A ) ) ) ).
% Union_upper
thf(fact_802_UN__subset__iff,axiom,
! [A: dedekind_preal > set_rat,I5: set_Dedekind_preal,B: set_rat] :
( ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ A @ I5 ) ) @ B )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ I5 )
=> ( ord_less_eq_set_rat @ ( A @ X6 ) @ B ) ) ) ) ).
% UN_subset_iff
thf(fact_803_Union__subsetI,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ? [Y6: set_rat] :
( ( member_set_rat @ Y6 @ B )
& ( ord_less_eq_set_rat @ X @ Y6 ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ ( comple3890839924845867745et_rat @ B ) ) ) ).
% Union_subsetI
thf(fact_804_Compr__image__eq,axiom,
! [F: dedekind_preal > dedekind_preal,A: set_Dedekind_preal,P: dedekind_preal > $o] :
( ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ ( image_4420143364984939013_preal @ F @ A ) )
& ( P @ X6 ) ) )
= ( image_4420143364984939013_preal @ F
@ ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
& ( P @ ( F @ X6 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_805_Compr__image__eq,axiom,
! [F: $o > dedekind_preal,A: set_o,P: dedekind_preal > $o] :
( ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ ( image_3026614506929738499_preal @ F @ A ) )
& ( P @ X6 ) ) )
= ( image_3026614506929738499_preal @ F
@ ( collect_o
@ ^ [X6: $o] :
( ( member_o @ X6 @ A )
& ( P @ ( F @ X6 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_806_Compr__image__eq,axiom,
! [F: dedekind_preal > $o,A: set_Dedekind_preal,P: $o > $o] :
( ( collect_o
@ ^ [X6: $o] :
( ( member_o @ X6 @ ( image_248103505173020295real_o @ F @ A ) )
& ( P @ X6 ) ) )
= ( image_248103505173020295real_o @ F
@ ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
& ( P @ ( F @ X6 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_807_Compr__image__eq,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X6: $o] :
( ( member_o @ X6 @ ( image_o_o @ F @ A ) )
& ( P @ X6 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X6: $o] :
( ( member_o @ X6 @ A )
& ( P @ ( F @ X6 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_808_Compr__image__eq,axiom,
! [F: nat > dedekind_preal,A: set_nat,P: dedekind_preal > $o] :
( ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ ( image_8829711020507855231_preal @ F @ A ) )
& ( P @ X6 ) ) )
= ( image_8829711020507855231_preal @ F
@ ( collect_nat
@ ^ [X6: nat] :
( ( member_nat @ X6 @ A )
& ( P @ ( F @ X6 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_809_Compr__image__eq,axiom,
! [F: nat > $o,A: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X6: $o] :
( ( member_o @ X6 @ ( image_nat_o @ F @ A ) )
& ( P @ X6 ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X6: nat] :
( ( member_nat @ X6 @ A )
& ( P @ ( F @ X6 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_810_Compr__image__eq,axiom,
! [F: rat > dedekind_preal,A: set_rat,P: dedekind_preal > $o] :
( ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ ( image_7112546572676366711_preal @ F @ A ) )
& ( P @ X6 ) ) )
= ( image_7112546572676366711_preal @ F
@ ( collect_rat
@ ^ [X6: rat] :
( ( member_rat @ X6 @ A )
& ( P @ ( F @ X6 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_811_Compr__image__eq,axiom,
! [F: rat > $o,A: set_rat,P: $o > $o] :
( ( collect_o
@ ^ [X6: $o] :
( ( member_o @ X6 @ ( image_rat_o @ F @ A ) )
& ( P @ X6 ) ) )
= ( image_rat_o @ F
@ ( collect_rat
@ ^ [X6: rat] :
( ( member_rat @ X6 @ A )
& ( P @ ( F @ X6 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_812_Compr__image__eq,axiom,
! [F: dedekind_preal > nat,A: set_Dedekind_preal,P: nat > $o] :
( ( collect_nat
@ ^ [X6: nat] :
( ( member_nat @ X6 @ ( image_7207786835304483489al_nat @ F @ A ) )
& ( P @ X6 ) ) )
= ( image_7207786835304483489al_nat @ F
@ ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
& ( P @ ( F @ X6 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_813_Compr__image__eq,axiom,
! [F: $o > nat,A: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X6: nat] :
( ( member_nat @ X6 @ ( image_o_nat @ F @ A ) )
& ( P @ X6 ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X6: $o] :
( ( member_o @ X6 @ A )
& ( P @ ( F @ X6 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_814_image__image,axiom,
! [F: set_rat > set_rat,G: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( image_3939399684171694371et_rat @ F @ ( image_2231269897757205599et_rat @ G @ A ) )
= ( image_2231269897757205599et_rat
@ ^ [X6: dedekind_preal] : ( F @ ( G @ X6 ) )
@ A ) ) ).
% image_image
thf(fact_815_image__image,axiom,
! [F: dedekind_preal > set_rat,G: dedekind_preal > dedekind_preal,A: set_Dedekind_preal] :
( ( image_2231269897757205599et_rat @ F @ ( image_4420143364984939013_preal @ G @ A ) )
= ( image_2231269897757205599et_rat
@ ^ [X6: dedekind_preal] : ( F @ ( G @ X6 ) )
@ A ) ) ).
% image_image
thf(fact_816_imageE,axiom,
! [B3: dedekind_preal,F: dedekind_preal > dedekind_preal,A: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ B3 @ ( image_4420143364984939013_preal @ F @ A ) )
=> ~ ! [X: dedekind_preal] :
( ( B3
= ( F @ X ) )
=> ~ ( member6871284927547481791_preal @ X @ A ) ) ) ).
% imageE
thf(fact_817_imageE,axiom,
! [B3: dedekind_preal,F: rat > dedekind_preal,A: set_rat] :
( ( member6871284927547481791_preal @ B3 @ ( image_7112546572676366711_preal @ F @ A ) )
=> ~ ! [X: rat] :
( ( B3
= ( F @ X ) )
=> ~ ( member_rat @ X @ A ) ) ) ).
% imageE
thf(fact_818_imageE,axiom,
! [B3: dedekind_preal,F: $o > dedekind_preal,A: set_o] :
( ( member6871284927547481791_preal @ B3 @ ( image_3026614506929738499_preal @ F @ A ) )
=> ~ ! [X: $o] :
( ( B3
= ( F @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_819_imageE,axiom,
! [B3: rat,F: dedekind_preal > rat,A: set_Dedekind_preal] :
( ( member_rat @ B3 @ ( image_6572656775217987753al_rat @ F @ A ) )
=> ~ ! [X: dedekind_preal] :
( ( B3
= ( F @ X ) )
=> ~ ( member6871284927547481791_preal @ X @ A ) ) ) ).
% imageE
thf(fact_820_imageE,axiom,
! [B3: rat,F: rat > rat,A: set_rat] :
( ( member_rat @ B3 @ ( image_rat_rat @ F @ A ) )
=> ~ ! [X: rat] :
( ( B3
= ( F @ X ) )
=> ~ ( member_rat @ X @ A ) ) ) ).
% imageE
thf(fact_821_imageE,axiom,
! [B3: rat,F: $o > rat,A: set_o] :
( ( member_rat @ B3 @ ( image_o_rat @ F @ A ) )
=> ~ ! [X: $o] :
( ( B3
= ( F @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_822_imageE,axiom,
! [B3: $o,F: dedekind_preal > $o,A: set_Dedekind_preal] :
( ( member_o @ B3 @ ( image_248103505173020295real_o @ F @ A ) )
=> ~ ! [X: dedekind_preal] :
( ( B3
= ( F @ X ) )
=> ~ ( member6871284927547481791_preal @ X @ A ) ) ) ).
% imageE
thf(fact_823_imageE,axiom,
! [B3: $o,F: rat > $o,A: set_rat] :
( ( member_o @ B3 @ ( image_rat_o @ F @ A ) )
=> ~ ! [X: rat] :
( ( B3
= ( F @ X ) )
=> ~ ( member_rat @ X @ A ) ) ) ).
% imageE
thf(fact_824_imageE,axiom,
! [B3: $o,F: $o > $o,A: set_o] :
( ( member_o @ B3 @ ( image_o_o @ F @ A ) )
=> ~ ! [X: $o] :
( ( B3
= ( F @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_825_imageE,axiom,
! [B3: dedekind_preal,F: set_rat > dedekind_preal,A: set_set_rat] :
( ( member6871284927547481791_preal @ B3 @ ( image_6579680541524808769_preal @ F @ A ) )
=> ~ ! [X: set_rat] :
( ( B3
= ( F @ X ) )
=> ~ ( member_set_rat @ X @ A ) ) ) ).
% imageE
thf(fact_826_UN__extend__simps_I10_J,axiom,
! [B: set_rat > set_rat,F: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [A4: dedekind_preal] : ( B @ ( F @ A4 ) )
@ A ) )
= ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B @ ( image_2231269897757205599et_rat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_827_UN__extend__simps_I10_J,axiom,
! [B: dedekind_preal > set_rat,F: dedekind_preal > dedekind_preal,A: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [A4: dedekind_preal] : ( B @ ( F @ A4 ) )
@ A ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ ( image_4420143364984939013_preal @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_828_UN__extend__simps_I9_J,axiom,
! [C2: dedekind_preal > set_rat,B: dedekind_preal > set_Dedekind_preal,A: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X6: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C2 @ ( B @ X6 ) ) )
@ A ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C2 @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_829_UN__extend__simps_I9_J,axiom,
! [C2: rat > set_rat,B: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X6: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C2 @ ( B @ X6 ) ) )
@ A ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C2 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_830_UN__extend__simps_I8_J,axiom,
! [B: dedekind_preal > set_rat,A: set_se4284277844656040150_preal] :
( ( comple3890839924845867745et_rat
@ ( image_4042146516684777023et_rat
@ ^ [Y5: set_Dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ Y5 ) )
@ A ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ ( comple800427960086186211_preal @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_831_UN__extend__simps_I8_J,axiom,
! [B: rat > set_rat,A: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [Y5: set_rat] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B @ Y5 ) )
@ A ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B @ ( comple3890839924845867745et_rat @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_832_Inf_OINF__cong,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,C2: dedekind_preal > set_rat,D2: dedekind_preal > set_rat,Inf: set_set_rat > set_rat] :
( ( A = B )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Inf @ ( image_2231269897757205599et_rat @ C2 @ A ) )
= ( Inf @ ( image_2231269897757205599et_rat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_833_Sup_OSUP__cong,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,C2: dedekind_preal > set_rat,D2: dedekind_preal > set_rat,Sup: set_set_rat > set_rat] :
( ( A = B )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( Sup @ ( image_2231269897757205599et_rat @ C2 @ A ) )
= ( Sup @ ( image_2231269897757205599et_rat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_834_SUP__cong,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,C2: dedekind_preal > $o,D2: dedekind_preal > $o] :
( ( A = B )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_835_SUP__cong,axiom,
! [A: set_rat,B: set_rat,C2: rat > $o,D2: rat > $o] :
( ( A = B )
=> ( ! [X: rat] :
( ( member_rat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_rat_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_836_SUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > $o,D2: $o > $o] :
( ( A = B )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_837_SUP__cong,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,C2: dedekind_preal > nat,D2: dedekind_preal > nat] :
( ( A = B )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_7207786835304483489al_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_7207786835304483489al_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_838_SUP__cong,axiom,
! [A: set_rat,B: set_rat,C2: rat > nat,D2: rat > nat] :
( ( A = B )
=> ( ! [X: rat] :
( ( member_rat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_rat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_rat_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_839_SUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > nat,D2: $o > nat] :
( ( A = B )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_840_SUP__cong,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,C2: dedekind_preal > set_rat,D2: dedekind_preal > set_rat] :
( ( A = B )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C2 @ A ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_841_SUP__cong,axiom,
! [A: set_rat,B: set_rat,C2: rat > set_rat,D2: rat > set_rat] :
( ( A = B )
=> ( ! [X: rat] :
( ( member_rat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C2 @ A ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_842_SUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > set_rat,D2: $o > set_rat] :
( ( A = B )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ C2 @ A ) )
= ( comple3890839924845867745et_rat @ ( image_o_set_rat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_843_SUP__cong,axiom,
! [A: set_set_rat,B: set_set_rat,C2: set_rat > $o,D2: set_rat > $o] :
( ( A = B )
=> ( ! [X: set_rat] :
( ( member_set_rat @ X @ B )
=> ( ( C2 @ X )
= ( D2 @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_rat_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_rat_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_844_SUP__commute,axiom,
! [F: dedekind_preal > dedekind_preal > set_rat,B: set_Dedekind_preal,A: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [I: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ ( F @ I ) @ B ) )
@ A ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [J: dedekind_preal] :
( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [I: dedekind_preal] : ( F @ I @ J )
@ A ) )
@ B ) ) ) ).
% SUP_commute
thf(fact_845_UN__E,axiom,
! [B3: dedekind_preal,B: dedekind_preal > set_Dedekind_preal,A: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ B3 @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B @ A ) ) )
=> ~ ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ~ ( member6871284927547481791_preal @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_846_UN__E,axiom,
! [B3: dedekind_preal,B: rat > set_Dedekind_preal,A: set_rat] :
( ( member6871284927547481791_preal @ B3 @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B @ A ) ) )
=> ~ ! [X: rat] :
( ( member_rat @ X @ A )
=> ~ ( member6871284927547481791_preal @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_847_UN__E,axiom,
! [B3: dedekind_preal,B: $o > set_Dedekind_preal,A: set_o] :
( ( member6871284927547481791_preal @ B3 @ ( comple800427960086186211_preal @ ( image_2856262260070713571_preal @ B @ A ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member6871284927547481791_preal @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_848_UN__E,axiom,
! [B3: $o,B: dedekind_preal > set_o,A: set_Dedekind_preal] :
( ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B @ A ) ) )
=> ~ ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ~ ( member_o @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_849_UN__E,axiom,
! [B3: $o,B: rat > set_o,A: set_rat] :
( ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B @ A ) ) )
=> ~ ! [X: rat] :
( ( member_rat @ X @ A )
=> ~ ( member_o @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_850_UN__E,axiom,
! [B3: $o,B: $o > set_o,A: set_o] :
( ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member_o @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_851_UN__E,axiom,
! [B3: rat,B: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( member_rat @ B3 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) )
=> ~ ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ~ ( member_rat @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_852_UN__E,axiom,
! [B3: rat,B: rat > set_rat,A: set_rat] :
( ( member_rat @ B3 @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B @ A ) ) )
=> ~ ! [X: rat] :
( ( member_rat @ X @ A )
=> ~ ( member_rat @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_853_UN__E,axiom,
! [B3: rat,B: $o > set_rat,A: set_o] :
( ( member_rat @ B3 @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B @ A ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member_rat @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_854_UN__E,axiom,
! [B3: dedekind_preal,B: set_rat > set_Dedekind_preal,A: set_set_rat] :
( ( member6871284927547481791_preal @ B3 @ ( comple800427960086186211_preal @ ( image_7647003310076334625_preal @ B @ A ) ) )
=> ~ ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ~ ( member6871284927547481791_preal @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_855_UnionE,axiom,
! [A: dedekind_preal,C2: set_se4284277844656040150_preal] :
( ( member6871284927547481791_preal @ A @ ( comple800427960086186211_preal @ C2 ) )
=> ~ ! [X4: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ A @ X4 )
=> ~ ( member2000626104285586591_preal @ X4 @ C2 ) ) ) ).
% UnionE
thf(fact_856_UnionE,axiom,
! [A: set_rat,C2: set_set_set_rat] :
( ( member_set_rat @ A @ ( comple3392050375588816791et_rat @ C2 ) )
=> ~ ! [X4: set_set_rat] :
( ( member_set_rat @ A @ X4 )
=> ~ ( member_set_set_rat @ X4 @ C2 ) ) ) ).
% UnionE
thf(fact_857_UnionE,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) )
=> ~ ! [X4: set_o] :
( ( member_o @ A @ X4 )
=> ~ ( member_set_o @ X4 @ C2 ) ) ) ).
% UnionE
thf(fact_858_UnionE,axiom,
! [A: rat,C2: set_set_rat] :
( ( member_rat @ A @ ( comple3890839924845867745et_rat @ C2 ) )
=> ~ ! [X4: set_rat] :
( ( member_rat @ A @ X4 )
=> ~ ( member_set_rat @ X4 @ C2 ) ) ) ).
% UnionE
thf(fact_859_image__UN,axiom,
! [F: rat > rat,B: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( image_rat_rat @ F @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X6: dedekind_preal] : ( image_rat_rat @ F @ ( B @ X6 ) )
@ A ) ) ) ).
% image_UN
thf(fact_860_SUP__UNION,axiom,
! [F: dedekind_preal > set_rat,G: dedekind_preal > set_Dedekind_preal,A: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ G @ A ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y5: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ ( G @ Y5 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_861_SUP__UNION,axiom,
! [F: rat > set_rat,G: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ A ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y5: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ ( G @ Y5 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_862_SUP__UNION,axiom,
! [F: rat > $o,G: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [Y5: dedekind_preal] : ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( G @ Y5 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_863_Sup__set__def,axiom,
( comple800427960086186211_preal
= ( ^ [A3: set_se4284277844656040150_preal] :
( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] : ( complete_Sup_Sup_o @ ( image_3678684046732248231real_o @ ( member6871284927547481791_preal @ X6 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_864_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X6: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X6 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_865_Sup__set__def,axiom,
( comple3392050375588816791et_rat
= ( ^ [A3: set_set_set_rat] :
( collect_set_rat
@ ^ [X6: set_rat] : ( complete_Sup_Sup_o @ ( image_set_set_rat_o @ ( member_set_rat @ X6 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_866_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X6: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X6 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_867_Sup__set__def,axiom,
( comple3890839924845867745et_rat
= ( ^ [A3: set_set_rat] :
( collect_rat
@ ^ [X6: rat] : ( complete_Sup_Sup_o @ ( image_set_rat_o @ ( member_rat @ X6 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_868_image__Union,axiom,
! [F: dedekind_preal > set_rat,S3: set_se4284277844656040150_preal] :
( ( image_2231269897757205599et_rat @ F @ ( comple800427960086186211_preal @ S3 ) )
= ( comple3392050375588816791et_rat @ ( image_1970760655392924405et_rat @ ( image_2231269897757205599et_rat @ F ) @ S3 ) ) ) ).
% image_Union
thf(fact_869_image__Union,axiom,
! [F: rat > rat,S3: set_set_rat] :
( ( image_rat_rat @ F @ ( comple3890839924845867745et_rat @ S3 ) )
= ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ ( image_rat_rat @ F ) @ S3 ) ) ) ).
% image_Union
thf(fact_870_UN__UN__flatten,axiom,
! [C2: dedekind_preal > set_rat,B: dedekind_preal > set_Dedekind_preal,A: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C2 @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B @ A ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y5: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C2 @ ( B @ Y5 ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_871_UN__UN__flatten,axiom,
! [C2: rat > set_rat,B: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C2 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y5: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C2 @ ( B @ Y5 ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_872_SUP__empty,axiom,
! [F: dedekind_preal > set_Dedekind_preal] :
( ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ F @ bot_bo4848840305443107682_preal ) )
= bot_bo4848840305443107682_preal ) ).
% SUP_empty
thf(fact_873_SUP__empty,axiom,
! [F: dedekind_preal > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_5739498456150602327et_nat @ F @ bot_bo4848840305443107682_preal ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_874_SUP__empty,axiom,
! [F: nat > set_Dedekind_preal] :
( ( comple800427960086186211_preal @ ( image_6010050456725241695_preal @ F @ bot_bot_set_nat ) )
= bot_bo4848840305443107682_preal ) ).
% SUP_empty
thf(fact_875_SUP__empty,axiom,
! [F: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_876_SUP__empty,axiom,
! [F: dedekind_preal > set_rat] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ bot_bo4848840305443107682_preal ) )
= bot_bot_set_rat ) ).
% SUP_empty
thf(fact_877_SUP__empty,axiom,
! [F: nat > set_rat] :
( ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ F @ bot_bot_set_nat ) )
= bot_bot_set_rat ) ).
% SUP_empty
thf(fact_878_SUP__empty,axiom,
! [F: dedekind_preal > $o] :
( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ bot_bo4848840305443107682_preal ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_879_SUP__empty,axiom,
! [F: nat > $o] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_880_SUP__constant,axiom,
! [A: set_Dedekind_preal,C: set_Dedekind_preal] :
( ( ( A = bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= bot_bo4848840305443107682_preal ) )
& ( ( A != bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_881_SUP__constant,axiom,
! [A: set_Dedekind_preal,C: set_nat] :
( ( ( A = bot_bo4848840305443107682_preal )
=> ( ( comple7399068483239264473et_nat
@ ( image_5739498456150602327et_nat
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bo4848840305443107682_preal )
=> ( ( comple7399068483239264473et_nat
@ ( image_5739498456150602327et_nat
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_882_SUP__constant,axiom,
! [A: set_nat,C: set_Dedekind_preal] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple800427960086186211_preal
@ ( image_6010050456725241695_preal
@ ^ [Y5: nat] : C
@ A ) )
= bot_bo4848840305443107682_preal ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple800427960086186211_preal
@ ( image_6010050456725241695_preal
@ ^ [Y5: nat] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_883_SUP__constant,axiom,
! [A: set_nat,C: set_nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : C
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_884_SUP__constant,axiom,
! [A: set_Dedekind_preal,C: set_rat] :
( ( ( A = bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= bot_bot_set_rat ) )
& ( ( A != bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_885_SUP__constant,axiom,
! [A: set_nat,C: set_rat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple3890839924845867745et_rat
@ ( image_nat_set_rat
@ ^ [Y5: nat] : C
@ A ) )
= bot_bot_set_rat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple3890839924845867745et_rat
@ ( image_nat_set_rat
@ ^ [Y5: nat] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_886_SUP__constant,axiom,
! [C: $o,A: set_Dedekind_preal] :
( ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [Y5: dedekind_preal] : C
@ A ) )
= ( ( ( A = bot_bo4848840305443107682_preal )
=> bot_bot_o )
& ( ( A != bot_bo4848840305443107682_preal )
=> C ) ) ) ).
% SUP_constant
thf(fact_887_SUP__constant,axiom,
! [C: $o,A: set_nat] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y5: nat] : C
@ A ) )
= ( ( ( A = bot_bot_set_nat )
=> bot_bot_o )
& ( ( A != bot_bot_set_nat )
=> C ) ) ) ).
% SUP_constant
thf(fact_888_less__set__def,axiom,
( ord_le1802228187270208418_preal
= ( ^ [A3: set_Dedekind_preal,B2: set_Dedekind_preal] :
( ord_le1914443497901267899real_o
@ ^ [X6: dedekind_preal] : ( member6871284927547481791_preal @ X6 @ A3 )
@ ^ [X6: dedekind_preal] : ( member6871284927547481791_preal @ X6 @ B2 ) ) ) ) ).
% less_set_def
thf(fact_889_less__set__def,axiom,
( ord_less_set_set_rat
= ( ^ [A3: set_set_rat,B2: set_set_rat] :
( ord_less_set_rat_o
@ ^ [X6: set_rat] : ( member_set_rat @ X6 @ A3 )
@ ^ [X6: set_rat] : ( member_set_rat @ X6 @ B2 ) ) ) ) ).
% less_set_def
thf(fact_890_less__set__def,axiom,
( ord_less_set_o
= ( ^ [A3: set_o,B2: set_o] :
( ord_less_o_o
@ ^ [X6: $o] : ( member_o @ X6 @ A3 )
@ ^ [X6: $o] : ( member_o @ X6 @ B2 ) ) ) ) ).
% less_set_def
thf(fact_891_less__set__def,axiom,
( ord_less_set_rat
= ( ^ [A3: set_rat,B2: set_rat] :
( ord_less_rat_o
@ ^ [X6: rat] : ( member_rat @ X6 @ A3 )
@ ^ [X6: rat] : ( member_rat @ X6 @ B2 ) ) ) ) ).
% less_set_def
thf(fact_892_SUP__eq,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,F: dedekind_preal > $o,G: dedekind_preal > $o] :
( ! [I2: dedekind_preal] :
( ( member6871284927547481791_preal @ I2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X3 ) ) ) )
=> ( ! [J2: dedekind_preal] :
( ( member6871284927547481791_preal @ J2 @ B )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X3 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_893_SUP__eq,axiom,
! [A: set_Dedekind_preal,B: set_rat,F: dedekind_preal > $o,G: rat > $o] :
( ! [I2: dedekind_preal] :
( ( member6871284927547481791_preal @ I2 @ A )
=> ? [X3: rat] :
( ( member_rat @ X3 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X3 ) ) ) )
=> ( ! [J2: rat] :
( ( member_rat @ J2 @ B )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X3 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_rat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_894_SUP__eq,axiom,
! [A: set_Dedekind_preal,B: set_o,F: dedekind_preal > $o,G: $o > $o] :
( ! [I2: dedekind_preal] :
( ( member6871284927547481791_preal @ I2 @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X3 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X3 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_895_SUP__eq,axiom,
! [A: set_rat,B: set_Dedekind_preal,F: rat > $o,G: dedekind_preal > $o] :
( ! [I2: rat] :
( ( member_rat @ I2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X3 ) ) ) )
=> ( ! [J2: dedekind_preal] :
( ( member6871284927547481791_preal @ J2 @ B )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X3 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_896_SUP__eq,axiom,
! [A: set_rat,B: set_rat,F: rat > $o,G: rat > $o] :
( ! [I2: rat] :
( ( member_rat @ I2 @ A )
=> ? [X3: rat] :
( ( member_rat @ X3 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X3 ) ) ) )
=> ( ! [J2: rat] :
( ( member_rat @ J2 @ B )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X3 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_rat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_897_SUP__eq,axiom,
! [A: set_rat,B: set_o,F: rat > $o,G: $o > $o] :
( ! [I2: rat] :
( ( member_rat @ I2 @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X3 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X3 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_898_SUP__eq,axiom,
! [A: set_o,B: set_Dedekind_preal,F: $o > $o,G: dedekind_preal > $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X3 ) ) ) )
=> ( ! [J2: dedekind_preal] :
( ( member6871284927547481791_preal @ J2 @ B )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X3 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_899_SUP__eq,axiom,
! [A: set_o,B: set_rat,F: $o > $o,G: rat > $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ? [X3: rat] :
( ( member_rat @ X3 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X3 ) ) ) )
=> ( ! [J2: rat] :
( ( member_rat @ J2 @ B )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X3 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_rat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_900_SUP__eq,axiom,
! [A: set_o,B: set_o,F: $o > $o,G: $o > $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X3 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X3 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_901_SUP__eq,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,F: dedekind_preal > set_rat,G: dedekind_preal > set_rat] :
( ! [I2: dedekind_preal] :
( ( member6871284927547481791_preal @ I2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_set_rat @ ( F @ I2 ) @ ( G @ X3 ) ) ) )
=> ( ! [J2: dedekind_preal] :
( ( member6871284927547481791_preal @ J2 @ B )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A )
& ( ord_less_eq_set_rat @ ( G @ J2 ) @ ( F @ X3 ) ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_902_SUP__eq__const,axiom,
! [I5: set_rat,F: rat > set_rat,X5: set_rat] :
( ( I5 != bot_bot_set_rat )
=> ( ! [I2: rat] :
( ( member_rat @ I2 @ I5 )
=> ( ( F @ I2 )
= X5 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ I5 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_903_SUP__eq__const,axiom,
! [I5: set_set_rat,F: set_rat > set_rat,X5: set_rat] :
( ( I5 != bot_bot_set_set_rat )
=> ( ! [I2: set_rat] :
( ( member_set_rat @ I2 @ I5 )
=> ( ( F @ I2 )
= X5 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ I5 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_904_SUP__eq__const,axiom,
! [I5: set_o,F: $o > set_rat,X5: set_rat] :
( ( I5 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I5 )
=> ( ( F @ I2 )
= X5 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ I5 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_905_SUP__eq__const,axiom,
! [I5: set_Dedekind_preal,F: dedekind_preal > set_rat,X5: set_rat] :
( ( I5 != bot_bo4848840305443107682_preal )
=> ( ! [I2: dedekind_preal] :
( ( member6871284927547481791_preal @ I2 @ I5 )
=> ( ( F @ I2 )
= X5 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ I5 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_906_SUP__eq__const,axiom,
! [I5: set_nat,F: nat > set_rat,X5: set_rat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ( F @ I2 )
= X5 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ F @ I5 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_907_SUP__eq__const,axiom,
! [I5: set_rat,F: rat > $o,X5: $o] :
( ( I5 != bot_bot_set_rat )
=> ( ! [I2: rat] :
( ( member_rat @ I2 @ I5 )
=> ( ( F @ I2 )
= X5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ I5 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_908_SUP__eq__const,axiom,
! [I5: set_set_rat,F: set_rat > $o,X5: $o] :
( ( I5 != bot_bot_set_set_rat )
=> ( ! [I2: set_rat] :
( ( member_set_rat @ I2 @ I5 )
=> ( ( F @ I2 )
= X5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ I5 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_909_SUP__eq__const,axiom,
! [I5: set_o,F: $o > $o,X5: $o] :
( ( I5 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I5 )
=> ( ( F @ I2 )
= X5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I5 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_910_SUP__eq__const,axiom,
! [I5: set_Dedekind_preal,F: dedekind_preal > $o,X5: $o] :
( ( I5 != bot_bo4848840305443107682_preal )
=> ( ! [I2: dedekind_preal] :
( ( member6871284927547481791_preal @ I2 @ I5 )
=> ( ( F @ I2 )
= X5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ I5 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_911_SUP__eq__const,axiom,
! [I5: set_nat,F: nat > $o,X5: $o] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ( F @ I2 )
= X5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I5 ) )
= X5 ) ) ) ).
% SUP_eq_const
thf(fact_912_imageI,axiom,
! [X5: dedekind_preal,A: set_Dedekind_preal,F: dedekind_preal > dedekind_preal] :
( ( member6871284927547481791_preal @ X5 @ A )
=> ( member6871284927547481791_preal @ ( F @ X5 ) @ ( image_4420143364984939013_preal @ F @ A ) ) ) ).
% imageI
thf(fact_913_imageI,axiom,
! [X5: dedekind_preal,A: set_Dedekind_preal,F: dedekind_preal > rat] :
( ( member6871284927547481791_preal @ X5 @ A )
=> ( member_rat @ ( F @ X5 ) @ ( image_6572656775217987753al_rat @ F @ A ) ) ) ).
% imageI
thf(fact_914_imageI,axiom,
! [X5: dedekind_preal,A: set_Dedekind_preal,F: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ X5 @ A )
=> ( member_o @ ( F @ X5 ) @ ( image_248103505173020295real_o @ F @ A ) ) ) ).
% imageI
thf(fact_915_imageI,axiom,
! [X5: rat,A: set_rat,F: rat > dedekind_preal] :
( ( member_rat @ X5 @ A )
=> ( member6871284927547481791_preal @ ( F @ X5 ) @ ( image_7112546572676366711_preal @ F @ A ) ) ) ).
% imageI
thf(fact_916_imageI,axiom,
! [X5: rat,A: set_rat,F: rat > rat] :
( ( member_rat @ X5 @ A )
=> ( member_rat @ ( F @ X5 ) @ ( image_rat_rat @ F @ A ) ) ) ).
% imageI
thf(fact_917_imageI,axiom,
! [X5: rat,A: set_rat,F: rat > $o] :
( ( member_rat @ X5 @ A )
=> ( member_o @ ( F @ X5 ) @ ( image_rat_o @ F @ A ) ) ) ).
% imageI
thf(fact_918_imageI,axiom,
! [X5: $o,A: set_o,F: $o > dedekind_preal] :
( ( member_o @ X5 @ A )
=> ( member6871284927547481791_preal @ ( F @ X5 ) @ ( image_3026614506929738499_preal @ F @ A ) ) ) ).
% imageI
thf(fact_919_imageI,axiom,
! [X5: $o,A: set_o,F: $o > rat] :
( ( member_o @ X5 @ A )
=> ( member_rat @ ( F @ X5 ) @ ( image_o_rat @ F @ A ) ) ) ).
% imageI
thf(fact_920_imageI,axiom,
! [X5: $o,A: set_o,F: $o > $o] :
( ( member_o @ X5 @ A )
=> ( member_o @ ( F @ X5 ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_921_imageI,axiom,
! [X5: dedekind_preal,A: set_Dedekind_preal,F: dedekind_preal > set_rat] :
( ( member6871284927547481791_preal @ X5 @ A )
=> ( member_set_rat @ ( F @ X5 ) @ ( image_2231269897757205599et_rat @ F @ A ) ) ) ).
% imageI
thf(fact_922_image__iff,axiom,
! [Z4: set_rat,F: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( member_set_rat @ Z4 @ ( image_2231269897757205599et_rat @ F @ A ) )
= ( ? [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
& ( Z4
= ( F @ X6 ) ) ) ) ) ).
% image_iff
thf(fact_923_bex__imageD,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal,P: set_rat > $o] :
( ? [X3: set_rat] :
( ( member_set_rat @ X3 @ ( image_2231269897757205599et_rat @ F @ A ) )
& ( P @ X3 ) )
=> ? [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_924_image__cong,axiom,
! [M2: set_Dedekind_preal,N3: set_Dedekind_preal,F: dedekind_preal > set_rat,G: dedekind_preal > set_rat] :
( ( M2 = N3 )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ N3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_2231269897757205599et_rat @ F @ M2 )
= ( image_2231269897757205599et_rat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_925_ball__imageD,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal,P: set_rat > $o] :
( ! [X: set_rat] :
( ( member_set_rat @ X @ ( image_2231269897757205599et_rat @ F @ A ) )
=> ( P @ X ) )
=> ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_926_rev__image__eqI,axiom,
! [X5: dedekind_preal,A: set_Dedekind_preal,B3: dedekind_preal,F: dedekind_preal > dedekind_preal] :
( ( member6871284927547481791_preal @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member6871284927547481791_preal @ B3 @ ( image_4420143364984939013_preal @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_927_rev__image__eqI,axiom,
! [X5: dedekind_preal,A: set_Dedekind_preal,B3: rat,F: dedekind_preal > rat] :
( ( member6871284927547481791_preal @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_rat @ B3 @ ( image_6572656775217987753al_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_928_rev__image__eqI,axiom,
! [X5: dedekind_preal,A: set_Dedekind_preal,B3: $o,F: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_o @ B3 @ ( image_248103505173020295real_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_929_rev__image__eqI,axiom,
! [X5: rat,A: set_rat,B3: dedekind_preal,F: rat > dedekind_preal] :
( ( member_rat @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member6871284927547481791_preal @ B3 @ ( image_7112546572676366711_preal @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_930_rev__image__eqI,axiom,
! [X5: rat,A: set_rat,B3: rat,F: rat > rat] :
( ( member_rat @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_rat @ B3 @ ( image_rat_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_931_rev__image__eqI,axiom,
! [X5: rat,A: set_rat,B3: $o,F: rat > $o] :
( ( member_rat @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_o @ B3 @ ( image_rat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_932_rev__image__eqI,axiom,
! [X5: $o,A: set_o,B3: dedekind_preal,F: $o > dedekind_preal] :
( ( member_o @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member6871284927547481791_preal @ B3 @ ( image_3026614506929738499_preal @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_933_rev__image__eqI,axiom,
! [X5: $o,A: set_o,B3: rat,F: $o > rat] :
( ( member_o @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_rat @ B3 @ ( image_o_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_934_rev__image__eqI,axiom,
! [X5: $o,A: set_o,B3: $o,F: $o > $o] :
( ( member_o @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_o @ B3 @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_935_rev__image__eqI,axiom,
! [X5: dedekind_preal,A: set_Dedekind_preal,B3: set_rat,F: dedekind_preal > set_rat] :
( ( member6871284927547481791_preal @ X5 @ A )
=> ( ( B3
= ( F @ X5 ) )
=> ( member_set_rat @ B3 @ ( image_2231269897757205599et_rat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_936_subset__image__iff,axiom,
! [B: set_set_rat,F: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( ord_le513522071413781156et_rat @ B @ ( image_2231269897757205599et_rat @ F @ A ) )
= ( ? [AA: set_Dedekind_preal] :
( ( ord_le7349499860212017814_preal @ AA @ A )
& ( B
= ( image_2231269897757205599et_rat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_937_subset__image__iff,axiom,
! [B: set_rat,F: rat > rat,A: set_rat] :
( ( ord_less_eq_set_rat @ B @ ( image_rat_rat @ F @ A ) )
= ( ? [AA: set_rat] :
( ( ord_less_eq_set_rat @ AA @ A )
& ( B
= ( image_rat_rat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_938_image__subset__iff,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ ( image_2231269897757205599et_rat @ F @ A ) @ B )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
=> ( member_set_rat @ ( F @ X6 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_939_subset__imageE,axiom,
! [B: set_set_rat,F: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ( ord_le513522071413781156et_rat @ B @ ( image_2231269897757205599et_rat @ F @ A ) )
=> ~ ! [C4: set_Dedekind_preal] :
( ( ord_le7349499860212017814_preal @ C4 @ A )
=> ( B
!= ( image_2231269897757205599et_rat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_940_subset__imageE,axiom,
! [B: set_rat,F: rat > rat,A: set_rat] :
( ( ord_less_eq_set_rat @ B @ ( image_rat_rat @ F @ A ) )
=> ~ ! [C4: set_rat] :
( ( ord_less_eq_set_rat @ C4 @ A )
=> ( B
!= ( image_rat_rat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_941_image__subsetI,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > dedekind_preal,B: set_Dedekind_preal] :
( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( member6871284927547481791_preal @ ( F @ X ) @ B ) )
=> ( ord_le7349499860212017814_preal @ ( image_4420143364984939013_preal @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_942_image__subsetI,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > $o,B: set_o] :
( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_248103505173020295real_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_943_image__subsetI,axiom,
! [A: set_rat,F: rat > dedekind_preal,B: set_Dedekind_preal] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( member6871284927547481791_preal @ ( F @ X ) @ B ) )
=> ( ord_le7349499860212017814_preal @ ( image_7112546572676366711_preal @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_944_image__subsetI,axiom,
! [A: set_rat,F: rat > $o,B: set_o] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_rat_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_945_image__subsetI,axiom,
! [A: set_o,F: $o > dedekind_preal,B: set_Dedekind_preal] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member6871284927547481791_preal @ ( F @ X ) @ B ) )
=> ( ord_le7349499860212017814_preal @ ( image_3026614506929738499_preal @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_946_image__subsetI,axiom,
! [A: set_o,F: $o > $o,B: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_947_image__subsetI,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > rat,B: set_rat] :
( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( member_rat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_rat @ ( image_6572656775217987753al_rat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_948_image__subsetI,axiom,
! [A: set_rat,F: rat > rat,B: set_rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( member_rat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_rat @ ( image_rat_rat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_949_image__subsetI,axiom,
! [A: set_o,F: $o > rat,B: set_rat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_rat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_rat @ ( image_o_rat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_950_image__subsetI,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > set_rat,B: set_set_rat] :
( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( member_set_rat @ ( F @ X ) @ B ) )
=> ( ord_le513522071413781156et_rat @ ( image_2231269897757205599et_rat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_951_image__mono,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal,F: dedekind_preal > set_rat] :
( ( ord_le7349499860212017814_preal @ A @ B )
=> ( ord_le513522071413781156et_rat @ ( image_2231269897757205599et_rat @ F @ A ) @ ( image_2231269897757205599et_rat @ F @ B ) ) ) ).
% image_mono
thf(fact_952_image__mono,axiom,
! [A: set_rat,B: set_rat,F: rat > rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ord_less_eq_set_rat @ ( image_rat_rat @ F @ A ) @ ( image_rat_rat @ F @ B ) ) ) ).
% image_mono
thf(fact_953_Sup__eqI,axiom,
! [A: set_set_rat,X5: set_rat] :
( ! [Y3: set_rat] :
( ( member_set_rat @ Y3 @ A )
=> ( ord_less_eq_set_rat @ Y3 @ X5 ) )
=> ( ! [Y3: set_rat] :
( ! [Z2: set_rat] :
( ( member_set_rat @ Z2 @ A )
=> ( ord_less_eq_set_rat @ Z2 @ Y3 ) )
=> ( ord_less_eq_set_rat @ X5 @ Y3 ) )
=> ( ( comple3890839924845867745et_rat @ A )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_954_Sup__eqI,axiom,
! [A: set_o,X5: $o] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ A )
=> ( ord_less_eq_o @ Y3 @ X5 ) )
=> ( ! [Y3: $o] :
( ! [Z2: $o] :
( ( member_o @ Z2 @ A )
=> ( ord_less_eq_o @ Z2 @ Y3 ) )
=> ( ord_less_eq_o @ X5 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ A )
= X5 ) ) ) ).
% Sup_eqI
thf(fact_955_Sup__mono,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ! [A5: set_rat] :
( ( member_set_rat @ A5 @ A )
=> ? [X3: set_rat] :
( ( member_set_rat @ X3 @ B )
& ( ord_less_eq_set_rat @ A5 @ X3 ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ ( comple3890839924845867745et_rat @ B ) ) ) ).
% Sup_mono
thf(fact_956_Sup__mono,axiom,
! [A: set_o,B: set_o] :
( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ B )
& ( ord_less_eq_o @ A5 @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_mono
thf(fact_957_Sup__least,axiom,
! [A: set_set_rat,Z4: set_rat] :
( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ( ord_less_eq_set_rat @ X @ Z4 ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ Z4 ) ) ).
% Sup_least
thf(fact_958_Sup__least,axiom,
! [A: set_o,Z4: $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ X @ Z4 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z4 ) ) ).
% Sup_least
thf(fact_959_Sup__upper,axiom,
! [X5: set_rat,A: set_set_rat] :
( ( member_set_rat @ X5 @ A )
=> ( ord_less_eq_set_rat @ X5 @ ( comple3890839924845867745et_rat @ A ) ) ) ).
% Sup_upper
thf(fact_960_Sup__upper,axiom,
! [X5: $o,A: set_o] :
( ( member_o @ X5 @ A )
=> ( ord_less_eq_o @ X5 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_upper
thf(fact_961_Sup__le__iff,axiom,
! [A: set_set_rat,B3: set_rat] :
( ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ B3 )
= ( ! [X6: set_rat] :
( ( member_set_rat @ X6 @ A )
=> ( ord_less_eq_set_rat @ X6 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_962_Sup__le__iff,axiom,
! [A: set_o,B3: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ B3 )
= ( ! [X6: $o] :
( ( member_o @ X6 @ A )
=> ( ord_less_eq_o @ X6 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_963_Sup__upper2,axiom,
! [U: set_rat,A: set_set_rat,V: set_rat] :
( ( member_set_rat @ U @ A )
=> ( ( ord_less_eq_set_rat @ V @ U )
=> ( ord_less_eq_set_rat @ V @ ( comple3890839924845867745et_rat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_964_Sup__upper2,axiom,
! [U: $o,A: set_o,V: $o] :
( ( member_o @ U @ A )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_965_Sup__subset__mono,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ ( comple3890839924845867745et_rat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_966_Sup__subset__mono,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_subset_mono
thf(fact_967_cSup__eq__maximum,axiom,
! [Z4: set_rat,X2: set_set_rat] :
( ( member_set_rat @ Z4 @ X2 )
=> ( ! [X: set_rat] :
( ( member_set_rat @ X @ X2 )
=> ( ord_less_eq_set_rat @ X @ Z4 ) )
=> ( ( comple3890839924845867745et_rat @ X2 )
= Z4 ) ) ) ).
% cSup_eq_maximum
thf(fact_968_cSup__eq__maximum,axiom,
! [Z4: $o,X2: set_o] :
( ( member_o @ Z4 @ X2 )
=> ( ! [X: $o] :
( ( member_o @ X @ X2 )
=> ( ord_less_eq_o @ X @ Z4 ) )
=> ( ( complete_Sup_Sup_o @ X2 )
= Z4 ) ) ) ).
% cSup_eq_maximum
thf(fact_969_cSup__eq__maximum,axiom,
! [Z4: nat,X2: set_nat] :
( ( member_nat @ Z4 @ X2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ X2 )
=> ( ord_less_eq_nat @ X @ Z4 ) )
=> ( ( complete_Sup_Sup_nat @ X2 )
= Z4 ) ) ) ).
% cSup_eq_maximum
thf(fact_970_Union__empty,axiom,
( ( comple800427960086186211_preal @ bot_bo5050857909902108994_preal )
= bot_bo4848840305443107682_preal ) ).
% Union_empty
thf(fact_971_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_972_Union__empty,axiom,
( ( comple3890839924845867745et_rat @ bot_bot_set_set_rat )
= bot_bot_set_rat ) ).
% Union_empty
thf(fact_973_Union__empty__conv,axiom,
! [A: set_se4284277844656040150_preal] :
( ( ( comple800427960086186211_preal @ A )
= bot_bo4848840305443107682_preal )
= ( ! [X6: set_Dedekind_preal] :
( ( member2000626104285586591_preal @ X6 @ A )
=> ( X6 = bot_bo4848840305443107682_preal ) ) ) ) ).
% Union_empty_conv
thf(fact_974_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A )
=> ( X6 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_975_Union__empty__conv,axiom,
! [A: set_set_rat] :
( ( ( comple3890839924845867745et_rat @ A )
= bot_bot_set_rat )
= ( ! [X6: set_rat] :
( ( member_set_rat @ X6 @ A )
=> ( X6 = bot_bot_set_rat ) ) ) ) ).
% Union_empty_conv
thf(fact_976_empty__Union__conv,axiom,
! [A: set_se4284277844656040150_preal] :
( ( bot_bo4848840305443107682_preal
= ( comple800427960086186211_preal @ A ) )
= ( ! [X6: set_Dedekind_preal] :
( ( member2000626104285586591_preal @ X6 @ A )
=> ( X6 = bot_bo4848840305443107682_preal ) ) ) ) ).
% empty_Union_conv
thf(fact_977_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A )
=> ( X6 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_978_empty__Union__conv,axiom,
! [A: set_set_rat] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ A ) )
= ( ! [X6: set_rat] :
( ( member_set_rat @ X6 @ A )
=> ( X6 = bot_bot_set_rat ) ) ) ) ).
% empty_Union_conv
thf(fact_979_preal__sup,axiom,
! [P: set_Dedekind_preal,Y: dedekind_preal] :
( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ P )
=> ( ord_le5604041210740703414_preal @ X4 @ Y ) )
=> ( ( P != bot_bo4848840305443107682_preal )
=> ( dedekind_cut @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ dedekind_Rep_preal @ P ) ) ) ) ) ).
% preal_sup
thf(fact_980_SUP__eq__iff,axiom,
! [I5: set_rat,C: set_rat,F: rat > set_rat] :
( ( I5 != bot_bot_set_rat )
=> ( ! [I2: rat] :
( ( member_rat @ I2 @ I5 )
=> ( ord_less_eq_set_rat @ C @ ( F @ I2 ) ) )
=> ( ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ I5 ) )
= C )
= ( ! [X6: rat] :
( ( member_rat @ X6 @ I5 )
=> ( ( F @ X6 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_981_SUP__eq__iff,axiom,
! [I5: set_set_rat,C: set_rat,F: set_rat > set_rat] :
( ( I5 != bot_bot_set_set_rat )
=> ( ! [I2: set_rat] :
( ( member_set_rat @ I2 @ I5 )
=> ( ord_less_eq_set_rat @ C @ ( F @ I2 ) ) )
=> ( ( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ I5 ) )
= C )
= ( ! [X6: set_rat] :
( ( member_set_rat @ X6 @ I5 )
=> ( ( F @ X6 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_982_SUP__eq__iff,axiom,
! [I5: set_o,C: set_rat,F: $o > set_rat] :
( ( I5 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I5 )
=> ( ord_less_eq_set_rat @ C @ ( F @ I2 ) ) )
=> ( ( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ I5 ) )
= C )
= ( ! [X6: $o] :
( ( member_o @ X6 @ I5 )
=> ( ( F @ X6 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_983_SUP__eq__iff,axiom,
! [I5: set_Dedekind_preal,C: set_rat,F: dedekind_preal > set_rat] :
( ( I5 != bot_bo4848840305443107682_preal )
=> ( ! [I2: dedekind_preal] :
( ( member6871284927547481791_preal @ I2 @ I5 )
=> ( ord_less_eq_set_rat @ C @ ( F @ I2 ) ) )
=> ( ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ I5 ) )
= C )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ I5 )
=> ( ( F @ X6 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_984_SUP__eq__iff,axiom,
! [I5: set_nat,C: set_rat,F: nat > set_rat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_set_rat @ C @ ( F @ I2 ) ) )
=> ( ( ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ F @ I5 ) )
= C )
= ( ! [X6: nat] :
( ( member_nat @ X6 @ I5 )
=> ( ( F @ X6 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_985_SUP__eq__iff,axiom,
! [I5: set_rat,C: $o,F: rat > $o] :
( ( I5 != bot_bot_set_rat )
=> ( ! [I2: rat] :
( ( member_rat @ I2 @ I5 )
=> ( ord_less_eq_o @ C @ ( F @ I2 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ I5 ) )
= C )
= ( ! [X6: rat] :
( ( member_rat @ X6 @ I5 )
=> ( ( F @ X6 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_986_SUP__eq__iff,axiom,
! [I5: set_set_rat,C: $o,F: set_rat > $o] :
( ( I5 != bot_bot_set_set_rat )
=> ( ! [I2: set_rat] :
( ( member_set_rat @ I2 @ I5 )
=> ( ord_less_eq_o @ C @ ( F @ I2 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ I5 ) )
= C )
= ( ! [X6: set_rat] :
( ( member_set_rat @ X6 @ I5 )
=> ( ( F @ X6 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_987_SUP__eq__iff,axiom,
! [I5: set_o,C: $o,F: $o > $o] :
( ( I5 != bot_bot_set_o )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I5 )
=> ( ord_less_eq_o @ C @ ( F @ I2 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I5 ) )
= C )
= ( ! [X6: $o] :
( ( member_o @ X6 @ I5 )
=> ( ( F @ X6 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_988_SUP__eq__iff,axiom,
! [I5: set_Dedekind_preal,C: $o,F: dedekind_preal > $o] :
( ( I5 != bot_bo4848840305443107682_preal )
=> ( ! [I2: dedekind_preal] :
( ( member6871284927547481791_preal @ I2 @ I5 )
=> ( ord_less_eq_o @ C @ ( F @ I2 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ I5 ) )
= C )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ I5 )
=> ( ( F @ X6 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_989_SUP__eq__iff,axiom,
! [I5: set_nat,C: $o,F: nat > $o] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_o @ C @ ( F @ I2 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I5 ) )
= C )
= ( ! [X6: nat] :
( ( member_nat @ X6 @ I5 )
=> ( ( F @ X6 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_990_cSUP__least,axiom,
! [A: set_rat,F: rat > $o,M2: $o] :
( ( A != bot_bot_set_rat )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ M2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_991_cSUP__least,axiom,
! [A: set_o,F: $o > $o,M2: $o] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ M2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_992_cSUP__least,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > $o,M2: $o] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ M2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_993_cSUP__least,axiom,
! [A: set_nat,F: nat > $o,M2: $o] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ M2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_994_cSUP__least,axiom,
! [A: set_rat,F: rat > nat,M2: nat] :
( ( A != bot_bot_set_rat )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ M2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_rat_nat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_995_cSUP__least,axiom,
! [A: set_o,F: $o > nat,M2: nat] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ M2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_996_cSUP__least,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > nat,M2: nat] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ M2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_7207786835304483489al_nat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_997_cSUP__least,axiom,
! [A: set_nat,F: nat > nat,M2: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ M2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_998_cSUP__least,axiom,
! [A: set_rat,F: rat > set_rat,M2: set_rat] :
( ( A != bot_bot_set_rat )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ M2 ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_999_cSUP__least,axiom,
! [A: set_o,F: $o > set_rat,M2: set_rat] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ M2 ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_1000_verit__comp__simplify1_I2_J,axiom,
! [A2: dedekind_preal] : ( ord_le5604041210740703414_preal @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1001_verit__comp__simplify1_I2_J,axiom,
! [A2: set_rat] : ( ord_less_eq_set_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1002_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1003_verit__comp__simplify1_I2_J,axiom,
! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1004_verit__la__disequality,axiom,
! [A2: dedekind_preal,B3: dedekind_preal] :
( ( A2 = B3 )
| ~ ( ord_le5604041210740703414_preal @ A2 @ B3 )
| ~ ( ord_le5604041210740703414_preal @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_1005_verit__la__disequality,axiom,
! [A2: nat,B3: nat] :
( ( A2 = B3 )
| ~ ( ord_less_eq_nat @ A2 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_1006_verit__la__disequality,axiom,
! [A2: rat,B3: rat] :
( ( A2 = B3 )
| ~ ( ord_less_eq_rat @ A2 @ B3 )
| ~ ( ord_less_eq_rat @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_1007_verit__comp__simplify1_I1_J,axiom,
! [A2: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_1008_verit__comp__simplify1_I1_J,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_1009_verit__comp__simplify1_I1_J,axiom,
! [A2: rat] :
~ ( ord_less_rat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_1010_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_1011_less__eq__Sup,axiom,
! [A: set_set_rat,U: set_rat] :
( ! [V2: set_rat] :
( ( member_set_rat @ V2 @ A )
=> ( ord_less_eq_set_rat @ U @ V2 ) )
=> ( ( A != bot_bot_set_set_rat )
=> ( ord_less_eq_set_rat @ U @ ( comple3890839924845867745et_rat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1012_less__eq__Sup,axiom,
! [A: set_o,U: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A )
=> ( ord_less_eq_o @ U @ V2 ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1013_cSup__least,axiom,
! [X2: set_set_rat,Z4: set_rat] :
( ( X2 != bot_bot_set_set_rat )
=> ( ! [X: set_rat] :
( ( member_set_rat @ X @ X2 )
=> ( ord_less_eq_set_rat @ X @ Z4 ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ X2 ) @ Z4 ) ) ) ).
% cSup_least
thf(fact_1014_cSup__least,axiom,
! [X2: set_o,Z4: $o] :
( ( X2 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ X2 )
=> ( ord_less_eq_o @ X @ Z4 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X2 ) @ Z4 ) ) ) ).
% cSup_least
thf(fact_1015_cSup__least,axiom,
! [X2: set_nat,Z4: nat] :
( ( X2 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X2 )
=> ( ord_less_eq_nat @ X @ Z4 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X2 ) @ Z4 ) ) ) ).
% cSup_least
thf(fact_1016_cSup__eq__non__empty,axiom,
! [X2: set_set_rat,A2: set_rat] :
( ( X2 != bot_bot_set_set_rat )
=> ( ! [X: set_rat] :
( ( member_set_rat @ X @ X2 )
=> ( ord_less_eq_set_rat @ X @ A2 ) )
=> ( ! [Y3: set_rat] :
( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ X2 )
=> ( ord_less_eq_set_rat @ X3 @ Y3 ) )
=> ( ord_less_eq_set_rat @ A2 @ Y3 ) )
=> ( ( comple3890839924845867745et_rat @ X2 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1017_cSup__eq__non__empty,axiom,
! [X2: set_o,A2: $o] :
( ( X2 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ X2 )
=> ( ord_less_eq_o @ X @ A2 ) )
=> ( ! [Y3: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ X2 )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ord_less_eq_o @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ X2 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1018_cSup__eq__non__empty,axiom,
! [X2: set_nat,A2: nat] :
( ( X2 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X2 )
=> ( ord_less_eq_nat @ X @ A2 ) )
=> ( ! [Y3: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ X2 )
=> ( ord_less_eq_nat @ X3 @ Y3 ) )
=> ( ord_less_eq_nat @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup_nat @ X2 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1019_less__cSupE,axiom,
! [Y2: nat,X2: set_nat] :
( ( ord_less_nat @ Y2 @ ( complete_Sup_Sup_nat @ X2 ) )
=> ( ( X2 != bot_bot_set_nat )
=> ~ ! [X: nat] :
( ( member_nat @ X @ X2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ) ) ).
% less_cSupE
thf(fact_1020_less__cSupD,axiom,
! [X2: set_nat,Z4: nat] :
( ( X2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z4 @ ( complete_Sup_Sup_nat @ X2 ) )
=> ? [X: nat] :
( ( member_nat @ X @ X2 )
& ( ord_less_nat @ Z4 @ X ) ) ) ) ).
% less_cSupD
thf(fact_1021_SUP__UN__eq,axiom,
! [R2: dedekind_preal > set_rat,S3: set_Dedekind_preal] :
( ( comple4580332206425622756_rat_o
@ ( image_8369558857164962022_rat_o
@ ^ [I: dedekind_preal,X6: rat] : ( member_rat @ X6 @ ( R2 @ I ) )
@ S3 ) )
= ( ^ [X6: rat] : ( member_rat @ X6 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ R2 @ S3 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_1022_UN__constant__eq,axiom,
! [A2: dedekind_preal,A: set_Dedekind_preal,F: dedekind_preal > set_rat,C: set_rat] :
( ( member6871284927547481791_preal @ A2 @ A )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ( F @ X )
= C ) )
=> ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1023_UN__constant__eq,axiom,
! [A2: rat,A: set_rat,F: rat > set_rat,C: set_rat] :
( ( member_rat @ A2 @ A )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ( F @ X )
= C ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1024_UN__constant__eq,axiom,
! [A2: set_rat,A: set_set_rat,F: set_rat > set_rat,C: set_rat] :
( ( member_set_rat @ A2 @ A )
=> ( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ( ( F @ X )
= C ) )
=> ( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1025_UN__constant__eq,axiom,
! [A2: $o,A: set_o,F: $o > set_rat,C: set_rat] :
( ( member_o @ A2 @ A )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( F @ X )
= C ) )
=> ( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1026_bot__empty__eq,axiom,
( bot_bot_rat_o
= ( ^ [X6: rat] : ( member_rat @ X6 @ bot_bot_set_rat ) ) ) ).
% bot_empty_eq
thf(fact_1027_bot__empty__eq,axiom,
( bot_bot_set_rat_o
= ( ^ [X6: set_rat] : ( member_set_rat @ X6 @ bot_bot_set_set_rat ) ) ) ).
% bot_empty_eq
thf(fact_1028_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X6: $o] : ( member_o @ X6 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_1029_bot__empty__eq,axiom,
( bot_bo4815710026557797371real_o
= ( ^ [X6: dedekind_preal] : ( member6871284927547481791_preal @ X6 @ bot_bo4848840305443107682_preal ) ) ) ).
% bot_empty_eq
thf(fact_1030_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X6: nat] : ( member_nat @ X6 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1031_Collect__empty__eq__bot,axiom,
! [P: set_rat > $o] :
( ( ( collect_set_rat @ P )
= bot_bot_set_set_rat )
= ( P = bot_bot_set_rat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1032_Collect__empty__eq__bot,axiom,
! [P: rat > $o] :
( ( ( collect_rat @ P )
= bot_bot_set_rat )
= ( P = bot_bot_rat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1033_Collect__empty__eq__bot,axiom,
! [P: dedekind_preal > $o] :
( ( ( collec1132657498972982273_preal @ P )
= bot_bo4848840305443107682_preal )
= ( P = bot_bo4815710026557797371real_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1034_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1035_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > dedekind_preal,B: set_Dedekind_preal] :
( ! [X: nat] :
( ( P @ X )
=> ( member6871284927547481791_preal @ ( F @ X ) @ B ) )
=> ( ord_le7349499860212017814_preal @ ( image_8829711020507855231_preal @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1036_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > $o,B: set_o] :
( ! [X: nat] :
( ( P @ X )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1037_image__Collect__subsetI,axiom,
! [P: rat > $o,F: rat > dedekind_preal,B: set_Dedekind_preal] :
( ! [X: rat] :
( ( P @ X )
=> ( member6871284927547481791_preal @ ( F @ X ) @ B ) )
=> ( ord_le7349499860212017814_preal @ ( image_7112546572676366711_preal @ F @ ( collect_rat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1038_image__Collect__subsetI,axiom,
! [P: rat > $o,F: rat > $o,B: set_o] :
( ! [X: rat] :
( ( P @ X )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_rat_o @ F @ ( collect_rat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1039_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > rat,B: set_rat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_rat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1040_image__Collect__subsetI,axiom,
! [P: rat > $o,F: rat > rat,B: set_rat] :
( ! [X: rat] :
( ( P @ X )
=> ( member_rat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_rat @ ( image_rat_rat @ F @ ( collect_rat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1041_image__Collect__subsetI,axiom,
! [P: dedekind_preal > $o,F: dedekind_preal > set_rat,B: set_set_rat] :
( ! [X: dedekind_preal] :
( ( P @ X )
=> ( member_set_rat @ ( F @ X ) @ B ) )
=> ( ord_le513522071413781156et_rat @ ( image_2231269897757205599et_rat @ F @ ( collec1132657498972982273_preal @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1042_image__Collect__subsetI,axiom,
! [P: set_rat > $o,F: set_rat > dedekind_preal,B: set_Dedekind_preal] :
( ! [X: set_rat] :
( ( P @ X )
=> ( member6871284927547481791_preal @ ( F @ X ) @ B ) )
=> ( ord_le7349499860212017814_preal @ ( image_6579680541524808769_preal @ F @ ( collect_set_rat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1043_image__Collect__subsetI,axiom,
! [P: set_rat > $o,F: set_rat > $o,B: set_o] :
( ! [X: set_rat] :
( ( P @ X )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_set_rat_o @ F @ ( collect_set_rat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1044_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > set_rat,B: set_set_rat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_set_rat @ ( F @ X ) @ B ) )
=> ( ord_le513522071413781156et_rat @ ( image_nat_set_rat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1045_subset__emptyI,axiom,
! [A: set_set_rat] :
( ! [X: set_rat] :
~ ( member_set_rat @ X @ A )
=> ( ord_le513522071413781156et_rat @ A @ bot_bot_set_set_rat ) ) ).
% subset_emptyI
thf(fact_1046_subset__emptyI,axiom,
! [A: set_o] :
( ! [X: $o] :
~ ( member_o @ X @ A )
=> ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_1047_subset__emptyI,axiom,
! [A: set_Dedekind_preal] :
( ! [X: dedekind_preal] :
~ ( member6871284927547481791_preal @ X @ A )
=> ( ord_le7349499860212017814_preal @ A @ bot_bo4848840305443107682_preal ) ) ).
% subset_emptyI
thf(fact_1048_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1049_subset__emptyI,axiom,
! [A: set_rat] :
( ! [X: rat] :
~ ( member_rat @ X @ A )
=> ( ord_less_eq_set_rat @ A @ bot_bot_set_rat ) ) ).
% subset_emptyI
thf(fact_1050_all__subset__image,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal,P: set_set_rat > $o] :
( ( ! [B2: set_set_rat] :
( ( ord_le513522071413781156et_rat @ B2 @ ( image_2231269897757205599et_rat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_Dedekind_preal] :
( ( ord_le7349499860212017814_preal @ B2 @ A )
=> ( P @ ( image_2231269897757205599et_rat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_1051_all__subset__image,axiom,
! [F: rat > rat,A: set_rat,P: set_rat > $o] :
( ( ! [B2: set_rat] :
( ( ord_less_eq_set_rat @ B2 @ ( image_rat_rat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_rat] :
( ( ord_less_eq_set_rat @ B2 @ A )
=> ( P @ ( image_rat_rat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_1052_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1053_Sup__SUP__eq,axiom,
( comple7848071142344908218real_o
= ( ^ [S4: set_Dedekind_preal_o,X6: dedekind_preal] : ( member6871284927547481791_preal @ X6 @ ( comple800427960086186211_preal @ ( image_8702985839003149832_preal @ collec1132657498972982273_preal @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1054_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S4: set_o_o,X6: $o] : ( member_o @ X6 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1055_Sup__SUP__eq,axiom,
( comple3057484590954014638_rat_o
= ( ^ [S4: set_set_rat_o,X6: set_rat] : ( member_set_rat @ X6 @ ( comple3392050375588816791et_rat @ ( image_3676206661936088128et_rat @ collect_set_rat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1056_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S4: set_nat_o,X6: nat] : ( member_nat @ X6 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1057_Sup__SUP__eq,axiom,
( comple4580332206425622756_rat_o
= ( ^ [S4: set_rat_o,X6: rat] : ( member_rat @ X6 @ ( comple3890839924845867745et_rat @ ( image_rat_o_set_rat @ collect_rat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1058_Collect__restrict,axiom,
! [X2: set_Dedekind_preal,P: dedekind_preal > $o] :
( ord_le7349499860212017814_preal
@ ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ X2 )
& ( P @ X6 ) ) )
@ X2 ) ).
% Collect_restrict
thf(fact_1059_Collect__restrict,axiom,
! [X2: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X6: $o] :
( ( member_o @ X6 @ X2 )
& ( P @ X6 ) ) )
@ X2 ) ).
% Collect_restrict
thf(fact_1060_Collect__restrict,axiom,
! [X2: set_set_rat,P: set_rat > $o] :
( ord_le513522071413781156et_rat
@ ( collect_set_rat
@ ^ [X6: set_rat] :
( ( member_set_rat @ X6 @ X2 )
& ( P @ X6 ) ) )
@ X2 ) ).
% Collect_restrict
thf(fact_1061_Collect__restrict,axiom,
! [X2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X6: nat] :
( ( member_nat @ X6 @ X2 )
& ( P @ X6 ) ) )
@ X2 ) ).
% Collect_restrict
thf(fact_1062_Collect__restrict,axiom,
! [X2: set_rat,P: rat > $o] :
( ord_less_eq_set_rat
@ ( collect_rat
@ ^ [X6: rat] :
( ( member_rat @ X6 @ X2 )
& ( P @ X6 ) ) )
@ X2 ) ).
% Collect_restrict
thf(fact_1063_prop__restrict,axiom,
! [X5: dedekind_preal,Z5: set_Dedekind_preal,X2: set_Dedekind_preal,P: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ X5 @ Z5 )
=> ( ( ord_le7349499860212017814_preal @ Z5
@ ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ X2 )
& ( P @ X6 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_1064_prop__restrict,axiom,
! [X5: $o,Z5: set_o,X2: set_o,P: $o > $o] :
( ( member_o @ X5 @ Z5 )
=> ( ( ord_less_eq_set_o @ Z5
@ ( collect_o
@ ^ [X6: $o] :
( ( member_o @ X6 @ X2 )
& ( P @ X6 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_1065_prop__restrict,axiom,
! [X5: set_rat,Z5: set_set_rat,X2: set_set_rat,P: set_rat > $o] :
( ( member_set_rat @ X5 @ Z5 )
=> ( ( ord_le513522071413781156et_rat @ Z5
@ ( collect_set_rat
@ ^ [X6: set_rat] :
( ( member_set_rat @ X6 @ X2 )
& ( P @ X6 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_1066_prop__restrict,axiom,
! [X5: nat,Z5: set_nat,X2: set_nat,P: nat > $o] :
( ( member_nat @ X5 @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X6: nat] :
( ( member_nat @ X6 @ X2 )
& ( P @ X6 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_1067_prop__restrict,axiom,
! [X5: rat,Z5: set_rat,X2: set_rat,P: rat > $o] :
( ( member_rat @ X5 @ Z5 )
=> ( ( ord_less_eq_set_rat @ Z5
@ ( collect_rat
@ ^ [X6: rat] :
( ( member_rat @ X6 @ X2 )
& ( P @ X6 ) ) ) )
=> ( P @ X5 ) ) ) ).
% prop_restrict
thf(fact_1068_conj__subset__def,axiom,
! [A: set_set_rat,P: set_rat > $o,Q: set_rat > $o] :
( ( ord_le513522071413781156et_rat @ A
@ ( collect_set_rat
@ ^ [X6: set_rat] :
( ( P @ X6 )
& ( Q @ X6 ) ) ) )
= ( ( ord_le513522071413781156et_rat @ A @ ( collect_set_rat @ P ) )
& ( ord_le513522071413781156et_rat @ A @ ( collect_set_rat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1069_conj__subset__def,axiom,
! [A: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X6: nat] :
( ( P @ X6 )
& ( Q @ X6 ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1070_conj__subset__def,axiom,
! [A: set_rat,P: rat > $o,Q: rat > $o] :
( ( ord_less_eq_set_rat @ A
@ ( collect_rat
@ ^ [X6: rat] :
( ( P @ X6 )
& ( Q @ X6 ) ) ) )
= ( ( ord_less_eq_set_rat @ A @ ( collect_rat @ P ) )
& ( ord_less_eq_set_rat @ A @ ( collect_rat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1071_pred__subset__eq,axiom,
! [R3: set_Dedekind_preal,S3: set_Dedekind_preal] :
( ( ord_le5990576202697070023real_o
@ ^ [X6: dedekind_preal] : ( member6871284927547481791_preal @ X6 @ R3 )
@ ^ [X6: dedekind_preal] : ( member6871284927547481791_preal @ X6 @ S3 ) )
= ( ord_le7349499860212017814_preal @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_1072_pred__subset__eq,axiom,
! [R3: set_set_rat,S3: set_set_rat] :
( ( ord_le3214917520859460321_rat_o
@ ^ [X6: set_rat] : ( member_set_rat @ X6 @ R3 )
@ ^ [X6: set_rat] : ( member_set_rat @ X6 @ S3 ) )
= ( ord_le513522071413781156et_rat @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_1073_pred__subset__eq,axiom,
! [R3: set_o,S3: set_o] :
( ( ord_less_eq_o_o
@ ^ [X6: $o] : ( member_o @ X6 @ R3 )
@ ^ [X6: $o] : ( member_o @ X6 @ S3 ) )
= ( ord_less_eq_set_o @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_1074_pred__subset__eq,axiom,
! [R3: set_rat,S3: set_rat] :
( ( ord_less_eq_rat_o
@ ^ [X6: rat] : ( member_rat @ X6 @ R3 )
@ ^ [X6: rat] : ( member_rat @ X6 @ S3 ) )
= ( ord_less_eq_set_rat @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_1075_SUP__Sup__eq,axiom,
! [S3: set_se4284277844656040150_preal] :
( ( comple7848071142344908218real_o
@ ( image_7211531428116543256real_o
@ ^ [I: set_Dedekind_preal,X6: dedekind_preal] : ( member6871284927547481791_preal @ X6 @ I )
@ S3 ) )
= ( ^ [X6: dedekind_preal] : ( member6871284927547481791_preal @ X6 @ ( comple800427960086186211_preal @ S3 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1076_SUP__Sup__eq,axiom,
! [S3: set_set_set_rat] :
( ( comple3057484590954014638_rat_o
@ ( image_5922895781192582838_rat_o
@ ^ [I: set_set_rat,X6: set_rat] : ( member_set_rat @ X6 @ I )
@ S3 ) )
= ( ^ [X6: set_rat] : ( member_set_rat @ X6 @ ( comple3392050375588816791et_rat @ S3 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1077_SUP__Sup__eq,axiom,
! [S3: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I: set_o,X6: $o] : ( member_o @ X6 @ I )
@ S3 ) )
= ( ^ [X6: $o] : ( member_o @ X6 @ ( comple90263536869209701_set_o @ S3 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1078_SUP__Sup__eq,axiom,
! [S3: set_set_rat] :
( ( comple4580332206425622756_rat_o
@ ( image_set_rat_rat_o
@ ^ [I: set_rat,X6: rat] : ( member_rat @ X6 @ I )
@ S3 ) )
= ( ^ [X6: rat] : ( member_rat @ X6 @ ( comple3890839924845867745et_rat @ S3 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1079_image__Fpow__mono,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ ( image_2231269897757205599et_rat @ F @ A ) @ B )
=> ( ord_le8552383839478139994et_rat @ ( image_1970760655392924405et_rat @ ( image_2231269897757205599et_rat @ F ) @ ( finite6657831709359810172_preal @ A ) ) @ ( finite_Fpow_set_rat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_1080_bot_Oordering__top__axioms,axiom,
( orderi8630438233043249959_preal
@ ^ [X6: set_Dedekind_preal,Y5: set_Dedekind_preal] : ( ord_le7349499860212017814_preal @ Y5 @ X6 )
@ ^ [X6: set_Dedekind_preal,Y5: set_Dedekind_preal] : ( ord_le1802228187270208418_preal @ Y5 @ X6 )
@ bot_bo4848840305443107682_preal ) ).
% bot.ordering_top_axioms
thf(fact_1081_bot_Oordering__top__axioms,axiom,
( ordering_top_set_nat
@ ^ [X6: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ Y5 @ X6 )
@ ^ [X6: set_nat,Y5: set_nat] : ( ord_less_set_nat @ Y5 @ X6 )
@ bot_bot_set_nat ) ).
% bot.ordering_top_axioms
thf(fact_1082_bot_Oordering__top__axioms,axiom,
( ordering_top_set_rat
@ ^ [X6: set_rat,Y5: set_rat] : ( ord_less_eq_set_rat @ Y5 @ X6 )
@ ^ [X6: set_rat,Y5: set_rat] : ( ord_less_set_rat @ Y5 @ X6 )
@ bot_bot_set_rat ) ).
% bot.ordering_top_axioms
thf(fact_1083_bot_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X6: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X6 )
@ ^ [X6: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X6 )
@ bot_bot_nat ) ).
% bot.ordering_top_axioms
thf(fact_1084_cSUP__UNION,axiom,
! [A: set_rat,B: rat > set_Dedekind_preal,F: dedekind_preal > $o] :
( ( A != bot_bot_set_rat )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ( B @ X )
!= bot_bo4848840305443107682_preal ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X6: rat] : ( image_248103505173020295real_o @ F @ ( B @ X6 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_rat_o
@ ^ [X6: rat] : ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( B @ X6 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1085_cSUP__UNION,axiom,
! [A: set_o,B: $o > set_Dedekind_preal,F: dedekind_preal > $o] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( B @ X )
!= bot_bo4848840305443107682_preal ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X6: $o] : ( image_248103505173020295real_o @ F @ ( B @ X6 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( comple800427960086186211_preal @ ( image_2856262260070713571_preal @ B @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X6: $o] : ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( B @ X6 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1086_cSUP__UNION,axiom,
! [A: set_rat,B: rat > set_nat,F: nat > $o] :
( ( A != bot_bot_set_rat )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_nat ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X6: rat] : ( image_nat_o @ F @ ( B @ X6 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_rat_set_nat @ B @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_rat_o
@ ^ [X6: rat] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( B @ X6 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1087_cSUP__UNION,axiom,
! [A: set_o,B: $o > set_nat,F: nat > $o] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_nat ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X6: $o] : ( image_nat_o @ F @ ( B @ X6 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X6: $o] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( B @ X6 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1088_cSUP__UNION,axiom,
! [A: set_Dedekind_preal,B: dedekind_preal > set_Dedekind_preal,F: dedekind_preal > $o] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ( B @ X )
!= bot_bo4848840305443107682_preal ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X6: dedekind_preal] : ( image_248103505173020295real_o @ F @ ( B @ X6 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [X6: dedekind_preal] : ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( B @ X6 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1089_cSUP__UNION,axiom,
! [A: set_Dedekind_preal,B: dedekind_preal > set_nat,F: nat > $o] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_nat ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X6: dedekind_preal] : ( image_nat_o @ F @ ( B @ X6 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_5739498456150602327et_nat @ B @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [X6: dedekind_preal] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( B @ X6 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1090_cSUP__UNION,axiom,
! [A: set_nat,B: nat > set_Dedekind_preal,F: dedekind_preal > $o] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B @ X )
!= bot_bo4848840305443107682_preal ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X6: nat] : ( image_248103505173020295real_o @ F @ ( B @ X6 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( comple800427960086186211_preal @ ( image_6010050456725241695_preal @ B @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X6: nat] : ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( B @ X6 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1091_cSUP__UNION,axiom,
! [A: set_nat,B: nat > set_nat,F: nat > $o] :
( ( A != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_nat ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X6: nat] : ( image_nat_o @ F @ ( B @ X6 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X6: nat] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( B @ X6 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1092_cSUP__UNION,axiom,
! [A: set_rat,B: rat > set_rat,F: rat > $o] :
( ( A != bot_bot_set_rat )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_rat ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X6: rat] : ( image_rat_o @ F @ ( B @ X6 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_rat_o
@ ^ [X6: rat] : ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( B @ X6 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1093_cSUP__UNION,axiom,
! [A: set_o,B: $o > set_rat,F: rat > $o] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ( B @ X )
!= bot_bot_set_rat ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X6: $o] : ( image_rat_o @ F @ ( B @ X6 ) )
@ A ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X6: $o] : ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( B @ X6 ) ) )
@ A ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_1094_the__elem__image__unique,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > set_rat,X5: dedekind_preal] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ! [Y3: dedekind_preal] :
( ( member6871284927547481791_preal @ Y3 @ A )
=> ( ( F @ Y3 )
= ( F @ X5 ) ) )
=> ( ( the_elem_set_rat @ ( image_2231269897757205599et_rat @ F @ A ) )
= ( F @ X5 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1095_cSUP__subset__mono,axiom,
! [A: set_o,G: $o > $o,B: set_o,F: $o > $o] :
( ( A != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ G @ B ) )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1096_cSUP__subset__mono,axiom,
! [A: set_Dedekind_preal,G: dedekind_preal > $o,B: set_Dedekind_preal,F: dedekind_preal > $o] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( condit5488710616941104124bove_o @ ( image_248103505173020295real_o @ G @ B ) )
=> ( ( ord_le7349499860212017814_preal @ A @ B )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1097_cSUP__subset__mono,axiom,
! [A: set_nat,G: nat > $o,B: set_nat,F: nat > $o] :
( ( A != bot_bot_set_nat )
=> ( ( condit5488710616941104124bove_o @ ( image_nat_o @ G @ B ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1098_cSUP__subset__mono,axiom,
! [A: set_rat,G: rat > $o,B: set_rat,F: rat > $o] :
( ( A != bot_bot_set_rat )
=> ( ( condit5488710616941104124bove_o @ ( image_rat_o @ G @ B ) )
=> ( ( ord_less_eq_set_rat @ A @ B )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_rat_o @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1099_cSUP__subset__mono,axiom,
! [A: set_o,G: $o > nat,B: set_o,F: $o > nat] :
( ( A != bot_bot_set_o )
=> ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ G @ B ) )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_o_nat @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1100_cSUP__subset__mono,axiom,
! [A: set_Dedekind_preal,G: dedekind_preal > nat,B: set_Dedekind_preal,F: dedekind_preal > nat] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( condit2214826472909112428ve_nat @ ( image_7207786835304483489al_nat @ G @ B ) )
=> ( ( ord_le7349499860212017814_preal @ A @ B )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_7207786835304483489al_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_7207786835304483489al_nat @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1101_cSUP__subset__mono,axiom,
! [A: set_nat,G: nat > nat,B: set_nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1102_cSUP__subset__mono,axiom,
! [A: set_rat,G: rat > nat,B: set_rat,F: rat > nat] :
( ( A != bot_bot_set_rat )
=> ( ( condit2214826472909112428ve_nat @ ( image_rat_nat @ G @ B ) )
=> ( ( ord_less_eq_set_rat @ A @ B )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_rat_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_rat_nat @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1103_cSUP__subset__mono,axiom,
! [A: set_o,G: $o > set_rat,B: set_o,F: $o > set_rat] :
( ( A != bot_bot_set_o )
=> ( ( condit1969311730731577898et_rat @ ( image_o_set_rat @ G @ B ) )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1104_cSUP__subset__mono,axiom,
! [A: set_Dedekind_preal,G: dedekind_preal > set_rat,B: set_Dedekind_preal,F: dedekind_preal > set_rat] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ G @ B ) )
=> ( ( ord_le7349499860212017814_preal @ A @ B )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_less_eq_set_rat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1105_image__Pow__mono,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal,B: set_set_rat] :
( ( ord_le513522071413781156et_rat @ ( image_2231269897757205599et_rat @ F @ A ) @ B )
=> ( ord_le8552383839478139994et_rat @ ( image_1970760655392924405et_rat @ ( image_2231269897757205599et_rat @ F ) @ ( pow_Dedekind_preal @ A ) ) @ ( pow_set_rat @ B ) ) ) ).
% image_Pow_mono
thf(fact_1106_bdd__above_OI,axiom,
! [A: set_o,M2: $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ X @ M2 ) )
=> ( condit5488710616941104124bove_o @ A ) ) ).
% bdd_above.I
thf(fact_1107_bdd__above_OI,axiom,
! [A: set_Dedekind_preal,M2: dedekind_preal] :
( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_le5604041210740703414_preal @ X @ M2 ) )
=> ( condit1512905060068503930_preal @ A ) ) ).
% bdd_above.I
thf(fact_1108_bdd__above_OI,axiom,
! [A: set_set_rat,M2: set_rat] :
( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ( ord_less_eq_set_rat @ X @ M2 ) )
=> ( condit1969311730731577898et_rat @ A ) ) ).
% bdd_above.I
thf(fact_1109_bdd__above_OI,axiom,
! [A: set_rat,M2: rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_rat @ X @ M2 ) )
=> ( condit1579696412822616692ve_rat @ A ) ) ).
% bdd_above.I
thf(fact_1110_bdd__above_OI,axiom,
! [A: set_nat,M2: nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ M2 ) )
=> ( condit2214826472909112428ve_nat @ A ) ) ).
% bdd_above.I
thf(fact_1111_bdd__above__empty,axiom,
condit1512905060068503930_preal @ bot_bo4848840305443107682_preal ).
% bdd_above_empty
thf(fact_1112_bdd__above__empty,axiom,
condit2214826472909112428ve_nat @ bot_bot_set_nat ).
% bdd_above_empty
thf(fact_1113_PowI,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( member_set_rat @ A @ ( pow_rat @ B ) ) ) ).
% PowI
thf(fact_1114_Pow__iff,axiom,
! [A: set_rat,B: set_rat] :
( ( member_set_rat @ A @ ( pow_rat @ B ) )
= ( ord_less_eq_set_rat @ A @ B ) ) ).
% Pow_iff
thf(fact_1115_Union__Pow__eq,axiom,
! [A: set_rat] :
( ( comple3890839924845867745et_rat @ ( pow_rat @ A ) )
= A ) ).
% Union_Pow_eq
thf(fact_1116_Pow__top,axiom,
! [A: set_rat] : ( member_set_rat @ A @ ( pow_rat @ A ) ) ).
% Pow_top
thf(fact_1117_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A2 )
=> ( A2 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_1118_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A2 != Top )
= ( Less @ A2 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_1119_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A2 )
= ( A2 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_1120_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A2 ) ) ).
% ordering_top.extremum_strict
thf(fact_1121_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A2: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A2 @ Top ) ) ).
% ordering_top.extremum
thf(fact_1122_bdd__above_OE,axiom,
! [A: set_o] :
( ( condit5488710616941104124bove_o @ A )
=> ~ ! [M3: $o] :
~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ X3 @ M3 ) ) ) ).
% bdd_above.E
thf(fact_1123_bdd__above_OE,axiom,
! [A: set_Dedekind_preal] :
( ( condit1512905060068503930_preal @ A )
=> ~ ! [M3: dedekind_preal] :
~ ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A )
=> ( ord_le5604041210740703414_preal @ X3 @ M3 ) ) ) ).
% bdd_above.E
thf(fact_1124_bdd__above_OE,axiom,
! [A: set_set_rat] :
( ( condit1969311730731577898et_rat @ A )
=> ~ ! [M3: set_rat] :
~ ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
=> ( ord_less_eq_set_rat @ X3 @ M3 ) ) ) ).
% bdd_above.E
thf(fact_1125_bdd__above_OE,axiom,
! [A: set_rat] :
( ( condit1579696412822616692ve_rat @ A )
=> ~ ! [M3: rat] :
~ ! [X3: rat] :
( ( member_rat @ X3 @ A )
=> ( ord_less_eq_rat @ X3 @ M3 ) ) ) ).
% bdd_above.E
thf(fact_1126_bdd__above_OE,axiom,
! [A: set_nat] :
( ( condit2214826472909112428ve_nat @ A )
=> ~ ! [M3: nat] :
~ ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ).
% bdd_above.E
thf(fact_1127_bdd__above_Ounfold,axiom,
( condit1512905060068503930_preal
= ( ^ [A3: set_Dedekind_preal] :
? [M4: dedekind_preal] :
! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A3 )
=> ( ord_le5604041210740703414_preal @ X6 @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1128_bdd__above_Ounfold,axiom,
( condit1969311730731577898et_rat
= ( ^ [A3: set_set_rat] :
? [M4: set_rat] :
! [X6: set_rat] :
( ( member_set_rat @ X6 @ A3 )
=> ( ord_less_eq_set_rat @ X6 @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1129_bdd__above_Ounfold,axiom,
( condit1579696412822616692ve_rat
= ( ^ [A3: set_rat] :
? [M4: rat] :
! [X6: rat] :
( ( member_rat @ X6 @ A3 )
=> ( ord_less_eq_rat @ X6 @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1130_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A3: set_nat] :
? [M4: nat] :
! [X6: nat] :
( ( member_nat @ X6 @ A3 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1131_bdd__above__mono,axiom,
! [B: set_rat,A: set_rat] :
( ( condit1579696412822616692ve_rat @ B )
=> ( ( ord_less_eq_set_rat @ A @ B )
=> ( condit1579696412822616692ve_rat @ A ) ) ) ).
% bdd_above_mono
thf(fact_1132_bdd__above__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( condit2214826472909112428ve_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( condit2214826472909112428ve_nat @ A ) ) ) ).
% bdd_above_mono
thf(fact_1133_Pow__bottom,axiom,
! [B: set_rat] : ( member_set_rat @ bot_bot_set_rat @ ( pow_rat @ B ) ) ).
% Pow_bottom
thf(fact_1134_Pow__bottom,axiom,
! [B: set_Dedekind_preal] : ( member2000626104285586591_preal @ bot_bo4848840305443107682_preal @ ( pow_Dedekind_preal @ B ) ) ).
% Pow_bottom
thf(fact_1135_Pow__bottom,axiom,
! [B: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B ) ) ).
% Pow_bottom
thf(fact_1136_PowD,axiom,
! [A: set_rat,B: set_rat] :
( ( member_set_rat @ A @ ( pow_rat @ B ) )
=> ( ord_less_eq_set_rat @ A @ B ) ) ).
% PowD
thf(fact_1137_Pow__def,axiom,
( pow_rat
= ( ^ [A3: set_rat] :
( collect_set_rat
@ ^ [B2: set_rat] : ( ord_less_eq_set_rat @ B2 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_1138_bdd__above_OI2,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > dedekind_preal,M2: dedekind_preal] :
( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ M2 ) )
=> ( condit1512905060068503930_preal @ ( image_4420143364984939013_preal @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1139_bdd__above_OI2,axiom,
! [A: set_rat,F: rat > dedekind_preal,M2: dedekind_preal] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ M2 ) )
=> ( condit1512905060068503930_preal @ ( image_7112546572676366711_preal @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1140_bdd__above_OI2,axiom,
! [A: set_o,F: $o > dedekind_preal,M2: dedekind_preal] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ M2 ) )
=> ( condit1512905060068503930_preal @ ( image_3026614506929738499_preal @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1141_bdd__above_OI2,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > rat,M2: rat] :
( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_less_eq_rat @ ( F @ X ) @ M2 ) )
=> ( condit1579696412822616692ve_rat @ ( image_6572656775217987753al_rat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1142_bdd__above_OI2,axiom,
! [A: set_rat,F: rat > rat,M2: rat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_rat @ ( F @ X ) @ M2 ) )
=> ( condit1579696412822616692ve_rat @ ( image_rat_rat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1143_bdd__above_OI2,axiom,
! [A: set_o,F: $o > rat,M2: rat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_rat @ ( F @ X ) @ M2 ) )
=> ( condit1579696412822616692ve_rat @ ( image_o_rat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1144_bdd__above_OI2,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > nat,M2: nat] :
( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ M2 ) )
=> ( condit2214826472909112428ve_nat @ ( image_7207786835304483489al_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1145_bdd__above_OI2,axiom,
! [A: set_rat,F: rat > nat,M2: nat] :
( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ M2 ) )
=> ( condit2214826472909112428ve_nat @ ( image_rat_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1146_bdd__above_OI2,axiom,
! [A: set_o,F: $o > nat,M2: nat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ M2 ) )
=> ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1147_bdd__above_OI2,axiom,
! [A: set_set_rat,F: set_rat > dedekind_preal,M2: dedekind_preal] :
( ! [X: set_rat] :
( ( member_set_rat @ X @ A )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ M2 ) )
=> ( condit1512905060068503930_preal @ ( image_6579680541524808769_preal @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1148_cSup__upper2,axiom,
! [X5: set_rat,X2: set_set_rat,Y2: set_rat] :
( ( member_set_rat @ X5 @ X2 )
=> ( ( ord_less_eq_set_rat @ Y2 @ X5 )
=> ( ( condit1969311730731577898et_rat @ X2 )
=> ( ord_less_eq_set_rat @ Y2 @ ( comple3890839924845867745et_rat @ X2 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1149_cSup__upper2,axiom,
! [X5: $o,X2: set_o,Y2: $o] :
( ( member_o @ X5 @ X2 )
=> ( ( ord_less_eq_o @ Y2 @ X5 )
=> ( ( condit5488710616941104124bove_o @ X2 )
=> ( ord_less_eq_o @ Y2 @ ( complete_Sup_Sup_o @ X2 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1150_cSup__upper2,axiom,
! [X5: nat,X2: set_nat,Y2: nat] :
( ( member_nat @ X5 @ X2 )
=> ( ( ord_less_eq_nat @ Y2 @ X5 )
=> ( ( condit2214826472909112428ve_nat @ X2 )
=> ( ord_less_eq_nat @ Y2 @ ( complete_Sup_Sup_nat @ X2 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1151_cSup__upper,axiom,
! [X5: set_rat,X2: set_set_rat] :
( ( member_set_rat @ X5 @ X2 )
=> ( ( condit1969311730731577898et_rat @ X2 )
=> ( ord_less_eq_set_rat @ X5 @ ( comple3890839924845867745et_rat @ X2 ) ) ) ) ).
% cSup_upper
thf(fact_1152_cSup__upper,axiom,
! [X5: $o,X2: set_o] :
( ( member_o @ X5 @ X2 )
=> ( ( condit5488710616941104124bove_o @ X2 )
=> ( ord_less_eq_o @ X5 @ ( complete_Sup_Sup_o @ X2 ) ) ) ) ).
% cSup_upper
thf(fact_1153_cSup__upper,axiom,
! [X5: nat,X2: set_nat] :
( ( member_nat @ X5 @ X2 )
=> ( ( condit2214826472909112428ve_nat @ X2 )
=> ( ord_less_eq_nat @ X5 @ ( complete_Sup_Sup_nat @ X2 ) ) ) ) ).
% cSup_upper
thf(fact_1154_empty__in__Fpow,axiom,
! [A: set_rat] : ( member_set_rat @ bot_bot_set_rat @ ( finite_Fpow_rat @ A ) ) ).
% empty_in_Fpow
thf(fact_1155_empty__in__Fpow,axiom,
! [A: set_Dedekind_preal] : ( member2000626104285586591_preal @ bot_bo4848840305443107682_preal @ ( finite6657831709359810172_preal @ A ) ) ).
% empty_in_Fpow
thf(fact_1156_empty__in__Fpow,axiom,
! [A: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_1157_Pow__mono,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ord_le513522071413781156et_rat @ ( pow_rat @ A ) @ ( pow_rat @ B ) ) ) ).
% Pow_mono
thf(fact_1158_subset__Pow__Union,axiom,
! [A: set_set_rat] : ( ord_le513522071413781156et_rat @ A @ ( pow_rat @ ( comple3890839924845867745et_rat @ A ) ) ) ).
% subset_Pow_Union
thf(fact_1159_image__Pow__surj,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal,B: set_set_rat] :
( ( ( image_2231269897757205599et_rat @ F @ A )
= B )
=> ( ( image_1970760655392924405et_rat @ ( image_2231269897757205599et_rat @ F ) @ ( pow_Dedekind_preal @ A ) )
= ( pow_set_rat @ B ) ) ) ).
% image_Pow_surj
thf(fact_1160_cSUP__upper,axiom,
! [X5: dedekind_preal,A: set_Dedekind_preal,F: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ X5 @ A )
=> ( ( condit5488710616941104124bove_o @ ( image_248103505173020295real_o @ F @ A ) )
=> ( ord_less_eq_o @ ( F @ X5 ) @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1161_cSUP__upper,axiom,
! [X5: rat,A: set_rat,F: rat > $o] :
( ( member_rat @ X5 @ A )
=> ( ( condit5488710616941104124bove_o @ ( image_rat_o @ F @ A ) )
=> ( ord_less_eq_o @ ( F @ X5 ) @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1162_cSUP__upper,axiom,
! [X5: $o,A: set_o,F: $o > $o] :
( ( member_o @ X5 @ A )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A ) )
=> ( ord_less_eq_o @ ( F @ X5 ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1163_cSUP__upper,axiom,
! [X5: dedekind_preal,A: set_Dedekind_preal,F: dedekind_preal > nat] :
( ( member6871284927547481791_preal @ X5 @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_7207786835304483489al_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( complete_Sup_Sup_nat @ ( image_7207786835304483489al_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1164_cSUP__upper,axiom,
! [X5: rat,A: set_rat,F: rat > nat] :
( ( member_rat @ X5 @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_rat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( complete_Sup_Sup_nat @ ( image_rat_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1165_cSUP__upper,axiom,
! [X5: $o,A: set_o,F: $o > nat] :
( ( member_o @ X5 @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1166_cSUP__upper,axiom,
! [X5: dedekind_preal,A: set_Dedekind_preal,F: dedekind_preal > set_rat] :
( ( member6871284927547481791_preal @ X5 @ A )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ F @ A ) )
=> ( ord_less_eq_set_rat @ ( F @ X5 ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1167_cSUP__upper,axiom,
! [X5: rat,A: set_rat,F: rat > set_rat] :
( ( member_rat @ X5 @ A )
=> ( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ F @ A ) )
=> ( ord_less_eq_set_rat @ ( F @ X5 ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1168_cSUP__upper,axiom,
! [X5: $o,A: set_o,F: $o > set_rat] :
( ( member_o @ X5 @ A )
=> ( ( condit1969311730731577898et_rat @ ( image_o_set_rat @ F @ A ) )
=> ( ord_less_eq_set_rat @ ( F @ X5 ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1169_cSUP__upper,axiom,
! [X5: set_rat,A: set_set_rat,F: set_rat > $o] :
( ( member_set_rat @ X5 @ A )
=> ( ( condit5488710616941104124bove_o @ ( image_set_rat_o @ F @ A ) )
=> ( ord_less_eq_o @ ( F @ X5 ) @ ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1170_cSUP__upper2,axiom,
! [F: dedekind_preal > $o,A: set_Dedekind_preal,X5: dedekind_preal,U: $o] :
( ( condit5488710616941104124bove_o @ ( image_248103505173020295real_o @ F @ A ) )
=> ( ( member6871284927547481791_preal @ X5 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ X5 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1171_cSUP__upper2,axiom,
! [F: rat > $o,A: set_rat,X5: rat,U: $o] :
( ( condit5488710616941104124bove_o @ ( image_rat_o @ F @ A ) )
=> ( ( member_rat @ X5 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ X5 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1172_cSUP__upper2,axiom,
! [F: $o > $o,A: set_o,X5: $o,U: $o] :
( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A ) )
=> ( ( member_o @ X5 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ X5 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1173_cSUP__upper2,axiom,
! [F: dedekind_preal > nat,A: set_Dedekind_preal,X5: dedekind_preal,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_7207786835304483489al_nat @ F @ A ) )
=> ( ( member6871284927547481791_preal @ X5 @ A )
=> ( ( ord_less_eq_nat @ U @ ( F @ X5 ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_7207786835304483489al_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1174_cSUP__upper2,axiom,
! [F: rat > nat,A: set_rat,X5: rat,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_rat_nat @ F @ A ) )
=> ( ( member_rat @ X5 @ A )
=> ( ( ord_less_eq_nat @ U @ ( F @ X5 ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_rat_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1175_cSUP__upper2,axiom,
! [F: $o > nat,A: set_o,X5: $o,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A ) )
=> ( ( member_o @ X5 @ A )
=> ( ( ord_less_eq_nat @ U @ ( F @ X5 ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1176_cSUP__upper2,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal,X5: dedekind_preal,U: set_rat] :
( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ F @ A ) )
=> ( ( member6871284927547481791_preal @ X5 @ A )
=> ( ( ord_less_eq_set_rat @ U @ ( F @ X5 ) )
=> ( ord_less_eq_set_rat @ U @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1177_cSUP__upper2,axiom,
! [F: rat > set_rat,A: set_rat,X5: rat,U: set_rat] :
( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ F @ A ) )
=> ( ( member_rat @ X5 @ A )
=> ( ( ord_less_eq_set_rat @ U @ ( F @ X5 ) )
=> ( ord_less_eq_set_rat @ U @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1178_cSUP__upper2,axiom,
! [F: $o > set_rat,A: set_o,X5: $o,U: set_rat] :
( ( condit1969311730731577898et_rat @ ( image_o_set_rat @ F @ A ) )
=> ( ( member_o @ X5 @ A )
=> ( ( ord_less_eq_set_rat @ U @ ( F @ X5 ) )
=> ( ord_less_eq_set_rat @ U @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1179_cSUP__upper2,axiom,
! [F: set_rat > $o,A: set_set_rat,X5: set_rat,U: $o] :
( ( condit5488710616941104124bove_o @ ( image_set_rat_o @ F @ A ) )
=> ( ( member_set_rat @ X5 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ X5 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1180_cSup__mono,axiom,
! [B: set_set_rat,A: set_set_rat] :
( ( B != bot_bot_set_set_rat )
=> ( ( condit1969311730731577898et_rat @ A )
=> ( ! [B5: set_rat] :
( ( member_set_rat @ B5 @ B )
=> ? [X3: set_rat] :
( ( member_set_rat @ X3 @ A )
& ( ord_less_eq_set_rat @ B5 @ X3 ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ B ) @ ( comple3890839924845867745et_rat @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1181_cSup__mono,axiom,
! [B: set_o,A: set_o] :
( ( B != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A )
=> ( ! [B5: $o] :
( ( member_o @ B5 @ B )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ( ord_less_eq_o @ B5 @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ B ) @ ( complete_Sup_Sup_o @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1182_cSup__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( B != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ B )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ B5 @ X3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B ) @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1183_cSup__le__iff,axiom,
! [S3: set_set_rat,A2: set_rat] :
( ( S3 != bot_bot_set_set_rat )
=> ( ( condit1969311730731577898et_rat @ S3 )
=> ( ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ S3 ) @ A2 )
= ( ! [X6: set_rat] :
( ( member_set_rat @ X6 @ S3 )
=> ( ord_less_eq_set_rat @ X6 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1184_cSup__le__iff,axiom,
! [S3: set_o,A2: $o] :
( ( S3 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ S3 )
=> ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ S3 ) @ A2 )
= ( ! [X6: $o] :
( ( member_o @ X6 @ S3 )
=> ( ord_less_eq_o @ X6 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1185_cSup__le__iff,axiom,
! [S3: set_nat,A2: nat] :
( ( S3 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S3 )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S3 ) @ A2 )
= ( ! [X6: nat] :
( ( member_nat @ X6 @ S3 )
=> ( ord_less_eq_nat @ X6 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1186_less__cSup__iff,axiom,
! [X2: set_nat,Y2: nat] :
( ( X2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X2 )
=> ( ( ord_less_nat @ Y2 @ ( complete_Sup_Sup_nat @ X2 ) )
= ( ? [X6: nat] :
( ( member_nat @ X6 @ X2 )
& ( ord_less_nat @ Y2 @ X6 ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_1187_cSUP__lessD,axiom,
! [F: dedekind_preal > $o,A: set_Dedekind_preal,Y2: $o,I4: dedekind_preal] :
( ( condit5488710616941104124bove_o @ ( image_248103505173020295real_o @ F @ A ) )
=> ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) ) @ Y2 )
=> ( ( member6871284927547481791_preal @ I4 @ A )
=> ( ord_less_o @ ( F @ I4 ) @ Y2 ) ) ) ) ).
% cSUP_lessD
thf(fact_1188_cSUP__lessD,axiom,
! [F: rat > $o,A: set_rat,Y2: $o,I4: rat] :
( ( condit5488710616941104124bove_o @ ( image_rat_o @ F @ A ) )
=> ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A ) ) @ Y2 )
=> ( ( member_rat @ I4 @ A )
=> ( ord_less_o @ ( F @ I4 ) @ Y2 ) ) ) ) ).
% cSUP_lessD
thf(fact_1189_cSUP__lessD,axiom,
! [F: $o > $o,A: set_o,Y2: $o,I4: $o] :
( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A ) )
=> ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ Y2 )
=> ( ( member_o @ I4 @ A )
=> ( ord_less_o @ ( F @ I4 ) @ Y2 ) ) ) ) ).
% cSUP_lessD
thf(fact_1190_cSUP__lessD,axiom,
! [F: dedekind_preal > nat,A: set_Dedekind_preal,Y2: nat,I4: dedekind_preal] :
( ( condit2214826472909112428ve_nat @ ( image_7207786835304483489al_nat @ F @ A ) )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_7207786835304483489al_nat @ F @ A ) ) @ Y2 )
=> ( ( member6871284927547481791_preal @ I4 @ A )
=> ( ord_less_nat @ ( F @ I4 ) @ Y2 ) ) ) ) ).
% cSUP_lessD
thf(fact_1191_cSUP__lessD,axiom,
! [F: rat > nat,A: set_rat,Y2: nat,I4: rat] :
( ( condit2214826472909112428ve_nat @ ( image_rat_nat @ F @ A ) )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_rat_nat @ F @ A ) ) @ Y2 )
=> ( ( member_rat @ I4 @ A )
=> ( ord_less_nat @ ( F @ I4 ) @ Y2 ) ) ) ) ).
% cSUP_lessD
thf(fact_1192_cSUP__lessD,axiom,
! [F: $o > nat,A: set_o,Y2: nat,I4: $o] :
( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A ) )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A ) ) @ Y2 )
=> ( ( member_o @ I4 @ A )
=> ( ord_less_nat @ ( F @ I4 ) @ Y2 ) ) ) ) ).
% cSUP_lessD
thf(fact_1193_cSUP__lessD,axiom,
! [F: dedekind_preal > set_rat,A: set_Dedekind_preal,Y2: set_rat,I4: dedekind_preal] :
( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ F @ A ) )
=> ( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) @ Y2 )
=> ( ( member6871284927547481791_preal @ I4 @ A )
=> ( ord_less_set_rat @ ( F @ I4 ) @ Y2 ) ) ) ) ).
% cSUP_lessD
thf(fact_1194_cSUP__lessD,axiom,
! [F: rat > set_rat,A: set_rat,Y2: set_rat,I4: rat] :
( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ F @ A ) )
=> ( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) @ Y2 )
=> ( ( member_rat @ I4 @ A )
=> ( ord_less_set_rat @ ( F @ I4 ) @ Y2 ) ) ) ) ).
% cSUP_lessD
thf(fact_1195_cSUP__lessD,axiom,
! [F: $o > set_rat,A: set_o,Y2: set_rat,I4: $o] :
( ( condit1969311730731577898et_rat @ ( image_o_set_rat @ F @ A ) )
=> ( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) @ Y2 )
=> ( ( member_o @ I4 @ A )
=> ( ord_less_set_rat @ ( F @ I4 ) @ Y2 ) ) ) ) ).
% cSUP_lessD
thf(fact_1196_cSUP__lessD,axiom,
! [F: set_rat > $o,A: set_set_rat,Y2: $o,I4: set_rat] :
( ( condit5488710616941104124bove_o @ ( image_set_rat_o @ F @ A ) )
=> ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ A ) ) @ Y2 )
=> ( ( member_set_rat @ I4 @ A )
=> ( ord_less_o @ ( F @ I4 ) @ Y2 ) ) ) ) ).
% cSUP_lessD
thf(fact_1197_cSUP__mono,axiom,
! [A: set_rat,G: dedekind_preal > set_rat,B: set_Dedekind_preal,F: rat > set_rat] :
( ( A != bot_bot_set_rat )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ G @ B ) )
=> ( ! [N2: rat] :
( ( member_rat @ N2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_set_rat @ ( F @ N2 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1198_cSUP__mono,axiom,
! [A: set_set_rat,G: dedekind_preal > set_rat,B: set_Dedekind_preal,F: set_rat > set_rat] :
( ( A != bot_bot_set_set_rat )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ G @ B ) )
=> ( ! [N2: set_rat] :
( ( member_set_rat @ N2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_set_rat @ ( F @ N2 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1199_cSUP__mono,axiom,
! [A: set_o,G: dedekind_preal > set_rat,B: set_Dedekind_preal,F: $o > set_rat] :
( ( A != bot_bot_set_o )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ G @ B ) )
=> ( ! [N2: $o] :
( ( member_o @ N2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_set_rat @ ( F @ N2 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1200_cSUP__mono,axiom,
! [A: set_Dedekind_preal,G: dedekind_preal > set_rat,B: set_Dedekind_preal,F: dedekind_preal > set_rat] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ G @ B ) )
=> ( ! [N2: dedekind_preal] :
( ( member6871284927547481791_preal @ N2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_set_rat @ ( F @ N2 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1201_cSUP__mono,axiom,
! [A: set_nat,G: dedekind_preal > set_rat,B: set_Dedekind_preal,F: nat > set_rat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ G @ B ) )
=> ( ! [N2: nat] :
( ( member_nat @ N2 @ A )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B )
& ( ord_less_eq_set_rat @ ( F @ N2 ) @ ( G @ X3 ) ) ) )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ F @ A ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ B ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_1202_cSUP__le__iff,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > set_rat,U: set_rat] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ F @ A ) )
=> ( ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A ) ) @ U )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
=> ( ord_less_eq_set_rat @ ( F @ X6 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1203_cSUP__le__iff,axiom,
! [A: set_nat,F: nat > set_rat,U: set_rat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1969311730731577898et_rat @ ( image_nat_set_rat @ F @ A ) )
=> ( ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ F @ A ) ) @ U )
= ( ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( ord_less_eq_set_rat @ ( F @ X6 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1204_cSUP__le__iff,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > $o,U: $o] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( condit5488710616941104124bove_o @ ( image_248103505173020295real_o @ F @ A ) )
=> ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A ) ) @ U )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
=> ( ord_less_eq_o @ ( F @ X6 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1205_cSUP__le__iff,axiom,
! [A: set_nat,F: nat > $o,U: $o] :
( ( A != bot_bot_set_nat )
=> ( ( condit5488710616941104124bove_o @ ( image_nat_o @ F @ A ) )
=> ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ U )
= ( ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( ord_less_eq_o @ ( F @ X6 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1206_cSUP__le__iff,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > nat,U: nat] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( condit2214826472909112428ve_nat @ ( image_7207786835304483489al_nat @ F @ A ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_7207786835304483489al_nat @ F @ A ) ) @ U )
= ( ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1207_cSUP__le__iff,axiom,
! [A: set_nat,F: nat > nat,U: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ U )
= ( ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1208_cSup__subset__mono,axiom,
! [A: set_set_rat,B: set_set_rat] :
( ( A != bot_bot_set_set_rat )
=> ( ( condit1969311730731577898et_rat @ B )
=> ( ( ord_le513522071413781156et_rat @ A @ B )
=> ( ord_less_eq_set_rat @ ( comple3890839924845867745et_rat @ A ) @ ( comple3890839924845867745et_rat @ B ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1209_cSup__subset__mono,axiom,
! [A: set_o,B: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1210_cSup__subset__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1211_UN__Pow__subset,axiom,
! [B: dedekind_preal > set_rat,A: set_Dedekind_preal] :
( ord_le513522071413781156et_rat
@ ( comple3392050375588816791et_rat
@ ( image_8268678932589273365et_rat
@ ^ [X6: dedekind_preal] : ( pow_rat @ ( B @ X6 ) )
@ A ) )
@ ( pow_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B @ A ) ) ) ) ).
% UN_Pow_subset
thf(fact_1212_Fpow__mono,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_eq_set_rat @ A @ B )
=> ( ord_le513522071413781156et_rat @ ( finite_Fpow_rat @ A ) @ ( finite_Fpow_rat @ B ) ) ) ).
% Fpow_mono
thf(fact_1213_less__cSUP__iff,axiom,
! [A: set_Dedekind_preal,F: dedekind_preal > nat,A2: nat] :
( ( A != bot_bo4848840305443107682_preal )
=> ( ( condit2214826472909112428ve_nat @ ( image_7207786835304483489al_nat @ F @ A ) )
=> ( ( ord_less_nat @ A2 @ ( complete_Sup_Sup_nat @ ( image_7207786835304483489al_nat @ F @ A ) ) )
= ( ? [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
& ( ord_less_nat @ A2 @ ( F @ X6 ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_1214_less__cSUP__iff,axiom,
! [A: set_nat,F: nat > nat,A2: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_nat @ A2 @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) )
= ( ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_nat @ A2 @ ( F @ X6 ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_1215_subset__CollectI,axiom,
! [B: set_Dedekind_preal,A: set_Dedekind_preal,Q: dedekind_preal > $o,P: dedekind_preal > $o] :
( ( ord_le7349499860212017814_preal @ B @ A )
=> ( ! [X: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le7349499860212017814_preal
@ ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ B )
& ( Q @ X6 ) ) )
@ ( collec1132657498972982273_preal
@ ^ [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A )
& ( P @ X6 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1216_subset__CollectI,axiom,
! [B: set_o,A: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X6: $o] :
( ( member_o @ X6 @ B )
& ( Q @ X6 ) ) )
@ ( collect_o
@ ^ [X6: $o] :
( ( member_o @ X6 @ A )
& ( P @ X6 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1217_subset__CollectI,axiom,
! [B: set_set_rat,A: set_set_rat,Q: set_rat > $o,P: set_rat > $o] :
( ( ord_le513522071413781156et_rat @ B @ A )
=> ( ! [X: set_rat] :
( ( member_set_rat @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le513522071413781156et_rat
@ ( collect_set_rat
@ ^ [X6: set_rat] :
( ( member_set_rat @ X6 @ B )
& ( Q @ X6 ) ) )
@ ( collect_set_rat
@ ^ [X6: set_rat] :
( ( member_set_rat @ X6 @ A )
& ( P @ X6 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1218_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X6: nat] :
( ( member_nat @ X6 @ B )
& ( Q @ X6 ) ) )
@ ( collect_nat
@ ^ [X6: nat] :
( ( member_nat @ X6 @ A )
& ( P @ X6 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1219_subset__CollectI,axiom,
! [B: set_rat,A: set_rat,Q: rat > $o,P: rat > $o] :
( ( ord_less_eq_set_rat @ B @ A )
=> ( ! [X: rat] :
( ( member_rat @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_rat
@ ( collect_rat
@ ^ [X6: rat] :
( ( member_rat @ X6 @ B )
& ( Q @ X6 ) ) )
@ ( collect_rat
@ ^ [X6: rat] :
( ( member_rat @ X6 @ A )
& ( P @ X6 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1220_subset__Collect__iff,axiom,
! [B: set_set_rat,A: set_set_rat,P: set_rat > $o] :
( ( ord_le513522071413781156et_rat @ B @ A )
=> ( ( ord_le513522071413781156et_rat @ B
@ ( collect_set_rat
@ ^ [X6: set_rat] :
( ( member_set_rat @ X6 @ A )
& ( P @ X6 ) ) ) )
= ( ! [X6: set_rat] :
( ( member_set_rat @ X6 @ B )
=> ( P @ X6 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1221_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X6: nat] :
( ( member_nat @ X6 @ A )
& ( P @ X6 ) ) ) )
= ( ! [X6: nat] :
( ( member_nat @ X6 @ B )
=> ( P @ X6 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1222_subset__Collect__iff,axiom,
! [B: set_rat,A: set_rat,P: rat > $o] :
( ( ord_less_eq_set_rat @ B @ A )
=> ( ( ord_less_eq_set_rat @ B
@ ( collect_rat
@ ^ [X6: rat] :
( ( member_rat @ X6 @ A )
& ( P @ X6 ) ) ) )
= ( ! [X6: rat] :
( ( member_rat @ X6 @ B )
=> ( P @ X6 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1223_preal__mult__def,axiom,
( times_3000655703912201937_preal
= ( ^ [R: dedekind_preal,S: dedekind_preal] : ( dedekind_Abs_preal @ ( dedekind_mult_set @ ( dedekind_Rep_preal @ R ) @ ( dedekind_Rep_preal @ S ) ) ) ) ) ).
% preal_mult_def
thf(fact_1224_preal__add__def,axiom,
( plus_p3173629198307831117_preal
= ( ^ [R: dedekind_preal,S: dedekind_preal] : ( dedekind_Abs_preal @ ( dedekind_add_set @ ( dedekind_Rep_preal @ R ) @ ( dedekind_Rep_preal @ S ) ) ) ) ) ).
% preal_add_def
thf(fact_1225_preal__add__less__cancel__left,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S2 ) )
= ( ord_le5708704896291381698_preal @ R2 @ S2 ) ) ).
% preal_add_less_cancel_left
thf(fact_1226_preal__add__less__cancel__right,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S2 @ T ) )
= ( ord_le5708704896291381698_preal @ R2 @ S2 ) ) ).
% preal_add_less_cancel_right
thf(fact_1227_preal__add__le__cancel__right,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S2 @ T ) )
= ( ord_le5604041210740703414_preal @ R2 @ S2 ) ) ).
% preal_add_le_cancel_right
thf(fact_1228_preal__add__le__cancel__left,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S2 ) )
= ( ord_le5604041210740703414_preal @ R2 @ S2 ) ) ).
% preal_add_le_cancel_left
thf(fact_1229_mem__Rep__preal__add__iff,axiom,
! [Z4: rat,R2: dedekind_preal,S2: dedekind_preal] :
( ( member_rat @ Z4 @ ( dedekind_Rep_preal @ ( plus_p3173629198307831117_preal @ R2 @ S2 ) ) )
= ( ? [X6: rat] :
( ( member_rat @ X6 @ ( dedekind_Rep_preal @ R2 ) )
& ? [Y5: rat] :
( ( member_rat @ Y5 @ ( dedekind_Rep_preal @ S2 ) )
& ( Z4
= ( plus_plus_rat @ X6 @ Y5 ) ) ) ) ) ) ).
% mem_Rep_preal_add_iff
thf(fact_1230_preal__add__mult__distrib2,axiom,
! [W2: dedekind_preal,X5: dedekind_preal,Y2: dedekind_preal] :
( ( times_3000655703912201937_preal @ W2 @ ( plus_p3173629198307831117_preal @ X5 @ Y2 ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W2 @ X5 ) @ ( times_3000655703912201937_preal @ W2 @ Y2 ) ) ) ).
% preal_add_mult_distrib2
thf(fact_1231_preal__add__right__cancel,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S2: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ R2 @ T )
= ( plus_p3173629198307831117_preal @ S2 @ T ) )
=> ( R2 = S2 ) ) ).
% preal_add_right_cancel
thf(fact_1232_preal__add__mult__distrib,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal,W2: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ X5 @ Y2 ) @ W2 )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X5 @ W2 ) @ ( times_3000655703912201937_preal @ Y2 @ W2 ) ) ) ).
% preal_add_mult_distrib
thf(fact_1233_preal__add__left__cancel,axiom,
! [C: dedekind_preal,A2: dedekind_preal,B3: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ C @ A2 )
= ( plus_p3173629198307831117_preal @ C @ B3 ) )
=> ( A2 = B3 ) ) ).
% preal_add_left_cancel
thf(fact_1234_preal__mult__commute,axiom,
( times_3000655703912201937_preal
= ( ^ [X6: dedekind_preal,Y5: dedekind_preal] : ( times_3000655703912201937_preal @ Y5 @ X6 ) ) ) ).
% preal_mult_commute
thf(fact_1235_preal__add__commute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [X6: dedekind_preal,Y5: dedekind_preal] : ( plus_p3173629198307831117_preal @ Y5 @ X6 ) ) ) ).
% preal_add_commute
thf(fact_1236_preal__mult__assoc,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal,Z4: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ X5 @ Y2 ) @ Z4 )
= ( times_3000655703912201937_preal @ X5 @ ( times_3000655703912201937_preal @ Y2 @ Z4 ) ) ) ).
% preal_mult_assoc
thf(fact_1237_preal__add__assoc,axiom,
! [X5: dedekind_preal,Y2: dedekind_preal,Z4: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ X5 @ Y2 ) @ Z4 )
= ( plus_p3173629198307831117_preal @ X5 @ ( plus_p3173629198307831117_preal @ Y2 @ Z4 ) ) ) ).
% preal_add_assoc
thf(fact_1238_distrib__subset1,axiom,
! [W2: dedekind_preal,X5: dedekind_preal,Y2: dedekind_preal] : ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ ( times_3000655703912201937_preal @ W2 @ ( plus_p3173629198307831117_preal @ X5 @ Y2 ) ) ) @ ( dedekind_Rep_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W2 @ X5 ) @ ( times_3000655703912201937_preal @ W2 @ Y2 ) ) ) ) ).
% distrib_subset1
thf(fact_1239_distrib__subset2,axiom,
! [W2: dedekind_preal,X5: dedekind_preal,Y2: dedekind_preal] : ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W2 @ X5 ) @ ( times_3000655703912201937_preal @ W2 @ Y2 ) ) ) @ ( dedekind_Rep_preal @ ( times_3000655703912201937_preal @ W2 @ ( plus_p3173629198307831117_preal @ X5 @ Y2 ) ) ) ) ).
% distrib_subset2
thf(fact_1240_preal__add__less2__mono1,axiom,
! [R2: dedekind_preal,S2: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S2 )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S2 @ T ) ) ) ).
% preal_add_less2_mono1
thf(fact_1241_preal__add__less2__mono2,axiom,
! [R2: dedekind_preal,S2: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S2 )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S2 ) ) ) ).
% preal_add_less2_mono2
thf(fact_1242_preal__self__less__add__left,axiom,
! [R2: dedekind_preal,S2: dedekind_preal] : ( ord_le5708704896291381698_preal @ R2 @ ( plus_p3173629198307831117_preal @ R2 @ S2 ) ) ).
% preal_self_less_add_left
thf(fact_1243_preal__add__left__less__cancel,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S2 ) )
=> ( ord_le5708704896291381698_preal @ R2 @ S2 ) ) ).
% preal_add_left_less_cancel
thf(fact_1244_preal__add__right__less__cancel,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S2 @ T ) )
=> ( ord_le5708704896291381698_preal @ R2 @ S2 ) ) ).
% preal_add_right_less_cancel
thf(fact_1245_Rep__preal__sum__not__subset,axiom,
! [R2: dedekind_preal,S2: dedekind_preal] :
~ ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ ( plus_p3173629198307831117_preal @ R2 @ S2 ) ) @ ( dedekind_Rep_preal @ R2 ) ) ).
% Rep_preal_sum_not_subset
thf(fact_1246_Rep__preal__self__subset,axiom,
! [R2: dedekind_preal,S2: dedekind_preal] : ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ R2 ) @ ( dedekind_Rep_preal @ ( plus_p3173629198307831117_preal @ R2 @ S2 ) ) ) ).
% Rep_preal_self_subset
thf(fact_1247_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1248_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I4: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I4 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1249_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1250_bdd__above__nat,axiom,
condit2214826472909112428ve_nat = finite_finite_nat ).
% bdd_above_nat
thf(fact_1251_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1252_preal__add__mult__distrib__mean,axiom,
! [A2: rat,W2: dedekind_preal,B3: rat,D3: rat,X5: dedekind_preal,E: rat,Y2: dedekind_preal] :
( ( member_rat @ A2 @ ( dedekind_Rep_preal @ W2 ) )
=> ( ( member_rat @ B3 @ ( dedekind_Rep_preal @ W2 ) )
=> ( ( member_rat @ D3 @ ( dedekind_Rep_preal @ X5 ) )
=> ( ( member_rat @ E @ ( dedekind_Rep_preal @ Y2 ) )
=> ? [X: rat] :
( ( member_rat @ X @ ( dedekind_Rep_preal @ W2 ) )
& ( ( plus_plus_rat @ ( times_times_rat @ A2 @ D3 ) @ ( times_times_rat @ B3 @ E ) )
= ( times_times_rat @ X @ ( plus_plus_rat @ D3 @ E ) ) ) ) ) ) ) ) ).
% preal_add_mult_distrib_mean
thf(fact_1253_mem__Rep__preal__mult__iff,axiom,
! [Z4: rat,R2: dedekind_preal,S2: dedekind_preal] :
( ( member_rat @ Z4 @ ( dedekind_Rep_preal @ ( times_3000655703912201937_preal @ R2 @ S2 ) ) )
= ( ? [X6: rat] :
( ( member_rat @ X6 @ ( dedekind_Rep_preal @ R2 ) )
& ? [Y5: rat] :
( ( member_rat @ Y5 @ ( dedekind_Rep_preal @ S2 ) )
& ( Z4
= ( times_times_rat @ X6 @ Y5 ) ) ) ) ) ) ).
% mem_Rep_preal_mult_iff
thf(fact_1254_less__add__left,axiom,
! [R2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S2 )
=> ( ( plus_p3173629198307831117_preal @ R2 @ ( minus_7336623429200594941_preal @ S2 @ R2 ) )
= S2 ) ) ).
% less_add_left
thf(fact_1255_preal__diff__def,axiom,
( minus_7336623429200594941_preal
= ( ^ [R: dedekind_preal,S: dedekind_preal] : ( dedekind_Abs_preal @ ( dedekind_diff_set @ ( dedekind_Rep_preal @ R ) @ ( dedekind_Rep_preal @ S ) ) ) ) ) ).
% preal_diff_def
thf(fact_1256_mem__Rep__preal__diff__iff,axiom,
! [R2: dedekind_preal,S2: dedekind_preal,Z4: rat] :
( ( ord_le5708704896291381698_preal @ R2 @ S2 )
=> ( ( member_rat @ Z4 @ ( dedekind_Rep_preal @ ( minus_7336623429200594941_preal @ S2 @ R2 ) ) )
= ( ? [X6: rat] :
( ( ord_less_rat @ zero_zero_rat @ X6 )
& ( ord_less_rat @ zero_zero_rat @ Z4 )
& ~ ( member_rat @ X6 @ ( dedekind_Rep_preal @ R2 ) )
& ( member_rat @ ( plus_plus_rat @ X6 @ Z4 ) @ ( dedekind_Rep_preal @ S2 ) ) ) ) ) ) ).
% mem_Rep_preal_diff_iff
thf(fact_1257_not__in__Rep__preal__ub,axiom,
! [X5: rat,Xa: dedekind_preal,Y2: rat] :
( ~ ( member_rat @ X5 @ ( dedekind_Rep_preal @ Xa ) )
=> ( ( member_rat @ Y2 @ ( dedekind_Rep_preal @ Xa ) )
=> ( ( ord_less_rat @ zero_zero_rat @ X5 )
=> ( ord_less_rat @ Y2 @ X5 ) ) ) ) ).
% not_in_Rep_preal_ub
thf(fact_1258_Rep__preal__exists__bound,axiom,
! [X2: dedekind_preal] :
? [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
& ~ ( member_rat @ X @ ( dedekind_Rep_preal @ X2 ) ) ) ).
% Rep_preal_exists_bound
thf(fact_1259_preal__downwards__closed_H,axiom,
! [A: set_rat,Y2: rat,Z4: rat] :
( ( dedekind_cut @ A )
=> ( ( member_rat @ Y2 @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ Z4 )
=> ( ( ord_less_eq_rat @ Z4 @ Y2 )
=> ( member_rat @ Z4 @ A ) ) ) ) ) ).
% preal_downwards_closed'
thf(fact_1260_preal__downwards__closed,axiom,
! [A: set_rat,Y2: rat,Z4: rat] :
( ( dedekind_cut @ A )
=> ( ( member_rat @ Y2 @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ Z4 )
=> ( ( ord_less_rat @ Z4 @ Y2 )
=> ( member_rat @ Z4 @ A ) ) ) ) ) ).
% preal_downwards_closed
thf(fact_1261_preal__exists__bound,axiom,
! [A: set_rat] :
( ( dedekind_cut @ A )
=> ? [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
& ~ ( member_rat @ X @ A ) ) ) ).
% preal_exists_bound
thf(fact_1262_not__in__preal__ub,axiom,
! [A: set_rat,X5: rat,Y2: rat] :
( ( dedekind_cut @ A )
=> ( ~ ( member_rat @ X5 @ A )
=> ( ( member_rat @ Y2 @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ X5 )
=> ( ord_less_rat @ Y2 @ X5 ) ) ) ) ) ).
% not_in_preal_ub
thf(fact_1263_preal__nonempty,axiom,
! [A: set_rat] :
( ( dedekind_cut @ A )
=> ? [X: rat] :
( ( member_rat @ X @ A )
& ( ord_less_rat @ zero_zero_rat @ X ) ) ) ).
% preal_nonempty
thf(fact_1264_preal__imp__pos,axiom,
! [A: set_rat,R2: rat] :
( ( dedekind_cut @ A )
=> ( ( member_rat @ R2 @ A )
=> ( ord_less_rat @ zero_zero_rat @ R2 ) ) ) ).
% preal_imp_pos
thf(fact_1265_rat__mem__preal,axiom,
! [Q3: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q3 )
=> ( dedekind_cut
@ ( collect_rat
@ ^ [R: rat] :
( ( ord_less_rat @ zero_zero_rat @ R )
& ( ord_less_rat @ R @ Q3 ) ) ) ) ) ).
% rat_mem_preal
thf(fact_1266_Gleason9__34__contra,axiom,
! [A: set_rat,U: rat,Y2: rat] :
( ( dedekind_cut @ A )
=> ( ! [X: rat] :
( ( member_rat @ X @ A )
=> ( member_rat @ ( plus_plus_rat @ X @ U ) @ A ) )
=> ( ( ord_less_rat @ zero_zero_rat @ U )
=> ( ( ord_less_rat @ zero_zero_rat @ Y2 )
=> ( member_rat @ Y2 @ A ) ) ) ) ) ).
% Gleason9_34_contra
thf(fact_1267_Gleason9__34,axiom,
! [A: set_rat,U: rat] :
( ( dedekind_cut @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ U )
=> ? [X: rat] :
( ( member_rat @ X @ A )
& ~ ( member_rat @ ( plus_plus_rat @ X @ U ) @ A ) ) ) ) ).
% Gleason9_34
thf(fact_1268_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ~ ! [S5: rat] :
( ( ord_less_rat @ zero_zero_rat @ S5 )
=> ! [T3: rat] :
( ( ord_less_rat @ zero_zero_rat @ T3 )
=> ( R2
!= ( plus_plus_rat @ S5 @ T3 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_1269_less__eq__rat__def,axiom,
( ord_less_eq_rat
= ( ^ [X6: rat,Y5: rat] :
( ( ord_less_rat @ X6 @ Y5 )
| ( X6 = Y5 ) ) ) ) ).
% less_eq_rat_def
thf(fact_1270_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X6: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X6 )
@ ^ [X6: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X6 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1271_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
% Conjectures (1)
thf(conj_0,conjecture,
ord_le5708704896291381698_preal @ z @ ( dedekind_psup @ p ) ).
%------------------------------------------------------------------------------