TPTP Problem File: SLH0658^1.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_00911_029940__5699008_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1468 ( 684 unt; 192 typ; 0 def)
% Number of atoms : 3598 (1961 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 11969 ( 447 ~; 35 |; 326 &;9829 @)
% ( 0 <=>;1332 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 1267 (1267 >; 0 *; 0 +; 0 <<)
% Number of symbols : 180 ( 177 usr; 16 con; 0-2 aty)
% Number of variables : 3989 ( 596 ^;3279 !; 114 ?;3989 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:30:07.520
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J_J,type,
set_set_set_set_rat: $tType ).
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__Set__Oset_I_Eo_J_J_J,type,
set_set_set_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__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_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__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 ).
% Explicit typings (177)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
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__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__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__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_I_Eo_J_J,type,
comple4436988014476444997_set_o: set_set_set_o > set_set_o ).
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_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
comple5990833217840624973et_rat: set_set_set_set_rat > set_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__Rat__Orat,type,
condit1579696412822616692ve_rat: set_rat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_I_Eo_J,type,
condit9194965722646069980_set_o: set_set_o > $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_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
condit6513867565548536544et_rat: set_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_Oinverse__set,type,
dedekind_inverse_set: 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_Fields_Oinverse__class_Oinverse_001t__Dedekind____Real__Opreal,type,
invers3090987106763476162_preal: dedekind_preal > dedekind_preal ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
inverse_inverse_rat: rat > rat ).
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_Oone__class_Oone_001t__Dedekind____Real__Opreal,type,
one_on9143529541772854033_preal: dedekind_preal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
one_one_rat: rat ).
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__Rat__Orat,type,
zero_zero_rat: rat ).
thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
sup_sup_o: $o > $o > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
sup_sup_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
sup_su8384800326570918602_preal: set_Dedekind_preal > set_Dedekind_preal > set_Dedekind_preal ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Rat__Orat_J,type,
sup_sup_set_rat: set_rat > set_rat > set_rat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
sup_sup_set_set_o: set_set_o > set_set_o > set_set_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
sup_sup_set_set_rat: set_set_rat > set_set_rat > set_set_rat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
sup_su3634870876412137766et_rat: set_set_set_rat > set_set_set_rat > set_set_set_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__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__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__Rat__Orat_J,type,
bot_bot_set_rat: set_rat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
bot_bot_set_set_o: set_set_o ).
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__Rat__Orat_J_J,type,
bot_bot_set_set_rat: set_set_rat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
bot_bo6619408370577057422et_rat: set_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__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__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_001t__Set__Oset_It__Rat__Orat_J,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_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__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_Obind_001_Eo_001_Eo,type,
bind_o_o: set_o > ( $o > set_o ) > set_o ).
thf(sy_c_Set_Obind_001_Eo_001t__Dedekind____Real__Opreal,type,
bind_o1395191148525001937_preal: set_o > ( $o > set_Dedekind_preal ) > set_Dedekind_preal ).
thf(sy_c_Set_Obind_001_Eo_001t__Rat__Orat,type,
bind_o_rat: set_o > ( $o > set_rat ) > set_rat ).
thf(sy_c_Set_Obind_001t__Dedekind____Real__Opreal_001_Eo,type,
bind_D7840052183623059541real_o: set_Dedekind_preal > ( dedekind_preal > set_o ) > set_o ).
thf(sy_c_Set_Obind_001t__Dedekind____Real__Opreal_001t__Dedekind____Real__Opreal,type,
bind_D8752870763990849491_preal: set_Dedekind_preal > ( dedekind_preal > set_Dedekind_preal ) > set_Dedekind_preal ).
thf(sy_c_Set_Obind_001t__Dedekind____Real__Opreal_001t__Rat__Orat,type,
bind_D1371630106014942811al_rat: set_Dedekind_preal > ( dedekind_preal > set_rat ) > set_rat ).
thf(sy_c_Set_Obind_001t__Dedekind____Real__Opreal_001t__Set__Oset_It__Rat__Orat_J,type,
bind_D6606601606562128401et_rat: set_Dedekind_preal > ( dedekind_preal > set_set_rat ) > set_set_rat ).
thf(sy_c_Set_Obind_001t__Rat__Orat_001_Eo,type,
bind_rat_o: set_rat > ( rat > set_o ) > set_o ).
thf(sy_c_Set_Obind_001t__Rat__Orat_001t__Dedekind____Real__Opreal,type,
bind_r1911519903473321769_preal: set_rat > ( rat > set_Dedekind_preal ) > set_Dedekind_preal ).
thf(sy_c_Set_Obind_001t__Rat__Orat_001t__Rat__Orat,type,
bind_rat_rat: set_rat > ( rat > set_rat ) > set_rat ).
thf(sy_c_Set_Obind_001t__Rat__Orat_001t__Set__Oset_I_Eo_J,type,
bind_rat_set_o: set_rat > ( rat > set_set_o ) > set_set_o ).
thf(sy_c_Set_Obind_001t__Rat__Orat_001t__Set__Oset_It__Rat__Orat_J,type,
bind_rat_set_rat: set_rat > ( rat > set_set_rat ) > set_set_rat ).
thf(sy_c_Set_Obind_001t__Rat__Orat_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
bind_rat_set_set_rat: set_rat > ( rat > set_set_set_rat ) > set_set_set_rat ).
thf(sy_c_Set_Obind_001t__Set__Oset_It__Rat__Orat_J_001t__Rat__Orat,type,
bind_set_rat_rat: set_set_rat > ( set_rat > set_rat ) > set_rat ).
thf(sy_c_Set_Obind_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Rat__Orat_J,type,
bind_set_rat_set_rat: set_set_rat > ( set_rat > set_set_rat ) > set_set_rat ).
thf(sy_c_Set_Obind_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
bind_s6617489967869682855et_rat: set_set_rat > ( set_rat > set_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__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__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__Rat__Orat_J,type,
image_o_set_rat: ( $o > set_rat ) > set_o > set_set_rat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
image_o_set_set_rat: ( $o > set_set_rat ) > set_o > set_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__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__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_I_Eo_J_J,type,
image_9033230029842370631_set_o: ( dedekind_preal > set_set_o ) > set_Dedekind_preal > set_set_set_o ).
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__Rat__Orat_001_062_I_Eo_M_Eo_J,type,
image_rat_o_o: ( rat > $o > $o ) > set_rat > set_o_o ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001_062_It__Rat__Orat_M_Eo_J,type,
image_rat_rat_o: ( rat > rat > $o ) > set_rat > set_rat_o ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001_062_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
image_rat_set_rat_o: ( rat > set_rat > $o ) > set_rat > set_set_rat_o ).
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__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__Rat__Orat_J,type,
image_rat_set_rat: ( rat > set_rat ) > set_rat > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
image_rat_set_set_o: ( rat > set_set_o ) > set_rat > set_set_set_o ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
image_5169559123115255331et_rat: ( rat > set_set_rat ) > set_rat > set_set_set_rat ).
thf(sy_c_Set_Oimage_001t__Rat__Orat_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
image_8006369670470206425et_rat: ( rat > set_set_set_rat ) > set_rat > set_set_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_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_set_o_set_rat: ( set_o > set_rat ) > set_set_o > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
image_1318631078669633143et_rat: ( set_o > set_set_rat ) > set_set_o > set_set_set_rat ).
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__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_062_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
image_8508488739866243052_rat_o: ( set_rat > set_rat > $o ) > set_set_rat > set_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__Rat__Orat,type,
image_set_rat_rat: ( set_rat > rat ) > set_set_rat > set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_I_Eo_J,type,
image_set_rat_set_o: ( set_rat > set_o ) > set_set_rat > set_set_o ).
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__Rat__Orat_J_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
image_5893626272549079683_set_o: ( set_rat > set_set_o ) > set_set_rat > set_set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
image_9010471772881432025et_rat: ( set_rat > set_set_rat ) > set_set_rat > set_set_set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Rat__Orat_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_J,type,
image_6398182499340170127et_rat: ( set_rat > set_set_set_rat ) > set_set_rat > set_set_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_Oimage_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_001t__Set__Oset_I_Eo_J,type,
image_9166368412354806125_set_o: ( set_set_rat > set_o ) > set_set_set_rat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_001t__Set__Oset_It__Rat__Orat_J,type,
image_5585803834317650393et_rat: ( set_set_rat > set_rat ) > set_set_set_rat > set_set_rat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
image_1224367690689810063et_rat: ( set_set_rat > set_set_rat ) > set_set_set_rat > set_set_set_rat ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__Dedekind____Real__Opreal,type,
insert4264952892349296358_preal: dedekind_preal > set_Dedekind_preal > set_Dedekind_preal ).
thf(sy_c_Set_Oinsert_001t__Rat__Orat,type,
insert_rat: rat > set_rat > set_rat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
insert_set_o: set_o > set_set_o > set_set_o ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Rat__Orat_J,type,
insert_set_rat: set_rat > set_set_rat > set_set_rat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
insert_set_set_rat: set_set_rat > set_set_set_rat > set_set_set_rat ).
thf(sy_c_Set_Ois__empty_001_Eo,type,
is_empty_o: set_o > $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__Rat__Orat,type,
is_empty_rat: set_rat > $o ).
thf(sy_c_Set_Ois__singleton_001_Eo,type,
is_singleton_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001t__Dedekind____Real__Opreal,type,
is_sin4170067188624449034_preal: set_Dedekind_preal > $o ).
thf(sy_c_Set_Ois__singleton_001t__Rat__Orat,type,
is_singleton_rat: set_rat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Rat__Orat_J,type,
is_singleton_set_rat: set_set_rat > $o ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Othe__elem_001t__Dedekind____Real__Opreal,type,
the_el245288408674363275_preal: set_Dedekind_preal > dedekind_preal ).
thf(sy_c_Set_Othe__elem_001t__Rat__Orat,type,
the_elem_rat: set_rat > rat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_I_Eo_J,type,
the_elem_set_o: set_set_o > set_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_Set_Othe__elem_001t__Set__Oset_It__Set__Oset_It__Rat__Orat_J_J,type,
the_elem_set_set_rat: set_set_set_rat > set_set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001_Eo,type,
set_or6416164934427428222Than_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Dedekind____Real__Opreal,type,
set_or4526639430071246588_preal: dedekind_preal > set_Dedekind_preal ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Rat__Orat,type,
set_or575021546402375026an_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Set__Oset_It__Rat__Orat_J,type,
set_or6174011595382531368et_rat: set_rat > set_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__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__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_q____,type,
q: rat ).
% Relevant facts (1275)
thf(fact_0__092_060open_0620_A_060_Aq_092_060close_062,axiom,
ord_less_rat @ zero_zero_rat @ q ).
% \<open>0 < q\<close>
thf(fact_1__092_060open_062q_A_092_060notin_062_A_092_060Union_062_A_IRep__preal_A_096_AP_J_092_060close_062,axiom,
~ ( member_rat @ q @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ dedekind_Rep_preal @ p ) ) ) ).
% \<open>q \<notin> \<Union> (Rep_preal ` P)\<close>
thf(fact_2_assms_I2_J,axiom,
p != bot_bo4848840305443107682_preal ).
% assms(2)
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062q_O_A_092_060lbrakk_0620_A_060_Aq_059_Aq_A_092_060notin_062_A_092_060Union_062_A_IRep__preal_A_096_AP_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Q: rat] :
( ( ord_less_rat @ zero_zero_rat @ Q )
=> ( member_rat @ Q @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ dedekind_Rep_preal @ p ) ) ) ) ).
% \<open>\<And>thesis. (\<And>q. \<lbrakk>0 < q; q \<notin> \<Union> (Rep_preal ` P)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_Rep__preal__inject,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ( dedekind_Rep_preal @ X )
= ( dedekind_Rep_preal @ Y ) )
= ( X = Y ) ) ).
% Rep_preal_inject
thf(fact_5_mem__Rep__preal__Ex,axiom,
! [X2: dedekind_preal] :
? [X3: rat] : ( member_rat @ X3 @ ( dedekind_Rep_preal @ X2 ) ) ).
% mem_Rep_preal_Ex
thf(fact_6_calculation,axiom,
ord_less_set_rat @ bot_bot_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ dedekind_Rep_preal @ p ) ) ).
% calculation
thf(fact_7_UN__I,axiom,
! [A: rat,A2: set_rat,B: $o,B2: rat > set_o] :
( ( member_rat @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_8_UN__I,axiom,
! [A: $o,A2: set_o,B: $o,B2: $o > set_o] :
( ( member_o @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_9_UN__I,axiom,
! [A: dedekind_preal,A2: set_Dedekind_preal,B: rat,B2: dedekind_preal > set_rat] :
( ( member6871284927547481791_preal @ A @ A2 )
=> ( ( member_rat @ B @ ( B2 @ A ) )
=> ( member_rat @ B @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_10_UN__I,axiom,
! [A: rat,A2: set_rat,B: rat,B2: rat > set_rat] :
( ( member_rat @ A @ A2 )
=> ( ( member_rat @ B @ ( B2 @ A ) )
=> ( member_rat @ B @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_11_UN__I,axiom,
! [A: $o,A2: set_o,B: rat,B2: $o > set_rat] :
( ( member_o @ A @ A2 )
=> ( ( member_rat @ B @ ( B2 @ A ) )
=> ( member_rat @ B @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_12_UN__I,axiom,
! [A: rat,A2: set_rat,B: set_rat,B2: rat > set_set_rat] :
( ( member_rat @ A @ A2 )
=> ( ( member_set_rat @ B @ ( B2 @ A ) )
=> ( member_set_rat @ B @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_13_UN__I,axiom,
! [A: $o,A2: set_o,B: set_rat,B2: $o > set_set_rat] :
( ( member_o @ A @ A2 )
=> ( ( member_set_rat @ B @ ( B2 @ A ) )
=> ( member_set_rat @ B @ ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_14_UN__I,axiom,
! [A: set_rat,A2: set_set_rat,B: $o,B2: set_rat > set_o] :
( ( member_set_rat @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_set_rat_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_15_UN__I,axiom,
! [A: set_rat,A2: set_set_rat,B: rat,B2: set_rat > set_rat] :
( ( member_set_rat @ A @ A2 )
=> ( ( member_rat @ B @ ( B2 @ A ) )
=> ( member_rat @ B @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_16_UN__I,axiom,
! [A: set_rat,A2: set_set_rat,B: set_rat,B2: set_rat > set_set_rat] :
( ( member_set_rat @ A @ A2 )
=> ( ( member_set_rat @ B @ ( B2 @ A ) )
=> ( member_set_rat @ B @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_17_UN__iff,axiom,
! [B: rat,B2: set_rat > set_rat,A2: set_set_rat] :
( ( member_rat @ B @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ A2 ) ) )
= ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
& ( member_rat @ B @ ( B2 @ X4 ) ) ) ) ) ).
% UN_iff
thf(fact_18_UN__iff,axiom,
! [B: rat,B2: rat > set_rat,A2: set_rat] :
( ( member_rat @ B @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ( member_rat @ B @ ( B2 @ X4 ) ) ) ) ) ).
% UN_iff
thf(fact_19_UN__iff,axiom,
! [B: set_rat,B2: set_rat > set_set_rat,A2: set_set_rat] :
( ( member_set_rat @ B @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ A2 ) ) )
= ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
& ( member_set_rat @ B @ ( B2 @ X4 ) ) ) ) ) ).
% UN_iff
thf(fact_20_UN__iff,axiom,
! [B: set_rat,B2: rat > set_set_rat,A2: set_rat] :
( ( member_set_rat @ B @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ A2 ) ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ( member_set_rat @ B @ ( B2 @ X4 ) ) ) ) ) ).
% UN_iff
thf(fact_21_UN__iff,axiom,
! [B: $o,B2: rat > set_o,A2: set_rat] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ( member_o @ B @ ( B2 @ X4 ) ) ) ) ) ).
% UN_iff
thf(fact_22_UN__iff,axiom,
! [B: rat,B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( member_rat @ B @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) )
= ( ? [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
& ( member_rat @ B @ ( B2 @ X4 ) ) ) ) ) ).
% UN_iff
thf(fact_23_SUP__identity__eq,axiom,
! [A2: set_set_set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_1224367690689810063et_rat
@ ^ [X4: set_set_rat] : X4
@ A2 ) )
= ( comple3392050375588816791et_rat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_24_SUP__identity__eq,axiom,
! [A2: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o
@ ^ [X4: set_o] : X4
@ A2 ) )
= ( comple90263536869209701_set_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_25_SUP__identity__eq,axiom,
! [A2: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : X4
@ A2 ) )
= ( comple3890839924845867745et_rat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_26_SUP__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X4: $o] : X4
@ A2 ) )
= ( complete_Sup_Sup_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_27_UN__ball__bex__simps_I4_J,axiom,
! [B2: dedekind_preal > set_rat,A2: set_Dedekind_preal,P: rat > $o] :
( ( ? [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) )
& ( P @ X4 ) ) )
= ( ? [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
& ? [Y2: rat] :
( ( member_rat @ Y2 @ ( B2 @ X4 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_28_UN__ball__bex__simps_I4_J,axiom,
! [B2: set_rat > set_rat,A2: set_set_rat,P: rat > $o] :
( ( ? [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ A2 ) ) )
& ( P @ X4 ) ) )
= ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
& ? [Y2: rat] :
( ( member_rat @ Y2 @ ( B2 @ X4 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_29_UN__ball__bex__simps_I4_J,axiom,
! [B2: rat > set_rat,A2: set_rat,P: rat > $o] :
( ( ? [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) )
& ( P @ X4 ) ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ? [Y2: rat] :
( ( member_rat @ Y2 @ ( B2 @ X4 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_30_UN__ball__bex__simps_I4_J,axiom,
! [B2: set_rat > set_set_rat,A2: set_set_rat,P: set_rat > $o] :
( ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ A2 ) ) )
& ( P @ X4 ) ) )
= ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
& ? [Y2: set_rat] :
( ( member_set_rat @ Y2 @ ( B2 @ X4 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_31_UN__ball__bex__simps_I4_J,axiom,
! [B2: rat > set_set_rat,A2: set_rat,P: set_rat > $o] :
( ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ A2 ) ) )
& ( P @ X4 ) ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ? [Y2: set_rat] :
( ( member_set_rat @ Y2 @ ( B2 @ X4 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_32_UN__ball__bex__simps_I4_J,axiom,
! [B2: rat > set_o,A2: set_rat,P: $o > $o] :
( ( ? [X4: $o] :
( ( member_o @ X4 @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) )
& ( P @ X4 ) ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ? [Y2: $o] :
( ( member_o @ Y2 @ ( B2 @ X4 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_33_UN__ball__bex__simps_I2_J,axiom,
! [B2: dedekind_preal > set_rat,A2: set_Dedekind_preal,P: rat > $o] :
( ( ! [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) )
=> ( P @ X4 ) ) )
= ( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
=> ! [Y2: rat] :
( ( member_rat @ Y2 @ ( B2 @ X4 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_34_UN__ball__bex__simps_I2_J,axiom,
! [B2: set_rat > set_rat,A2: set_set_rat,P: rat > $o] :
( ( ! [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ A2 ) ) )
=> ( P @ X4 ) ) )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ! [Y2: rat] :
( ( member_rat @ Y2 @ ( B2 @ X4 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_35_UN__ball__bex__simps_I2_J,axiom,
! [B2: rat > set_rat,A2: set_rat,P: rat > $o] :
( ( ! [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) )
=> ( P @ X4 ) ) )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ! [Y2: rat] :
( ( member_rat @ Y2 @ ( B2 @ X4 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_36_UN__ball__bex__simps_I2_J,axiom,
! [B2: set_rat > set_set_rat,A2: set_set_rat,P: set_rat > $o] :
( ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ A2 ) ) )
=> ( P @ X4 ) ) )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ! [Y2: set_rat] :
( ( member_set_rat @ Y2 @ ( B2 @ X4 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_37_UN__ball__bex__simps_I2_J,axiom,
! [B2: rat > set_set_rat,A2: set_rat,P: set_rat > $o] :
( ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ A2 ) ) )
=> ( P @ X4 ) ) )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ! [Y2: set_rat] :
( ( member_set_rat @ Y2 @ ( B2 @ X4 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_38_UN__ball__bex__simps_I2_J,axiom,
! [B2: rat > set_o,A2: set_rat,P: $o > $o] :
( ( ! [X4: $o] :
( ( member_o @ X4 @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) )
=> ( P @ X4 ) ) )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ ( B2 @ X4 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_39_bex__UN,axiom,
! [B2: dedekind_preal > set_rat,A2: set_Dedekind_preal,P: rat > $o] :
( ( ? [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) )
& ( P @ X4 ) ) )
= ( ? [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
& ? [Y2: rat] :
( ( member_rat @ Y2 @ ( B2 @ X4 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_40_bex__UN,axiom,
! [B2: set_rat > set_rat,A2: set_set_rat,P: rat > $o] :
( ( ? [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ A2 ) ) )
& ( P @ X4 ) ) )
= ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
& ? [Y2: rat] :
( ( member_rat @ Y2 @ ( B2 @ X4 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_41_bex__UN,axiom,
! [B2: rat > set_rat,A2: set_rat,P: rat > $o] :
( ( ? [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) )
& ( P @ X4 ) ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ? [Y2: rat] :
( ( member_rat @ Y2 @ ( B2 @ X4 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_42_bex__UN,axiom,
! [B2: set_rat > set_set_rat,A2: set_set_rat,P: set_rat > $o] :
( ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ A2 ) ) )
& ( P @ X4 ) ) )
= ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
& ? [Y2: set_rat] :
( ( member_set_rat @ Y2 @ ( B2 @ X4 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_43_bex__UN,axiom,
! [B2: rat > set_set_rat,A2: set_rat,P: set_rat > $o] :
( ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ A2 ) ) )
& ( P @ X4 ) ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ? [Y2: set_rat] :
( ( member_set_rat @ Y2 @ ( B2 @ X4 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_44_bex__UN,axiom,
! [B2: rat > set_o,A2: set_rat,P: $o > $o] :
( ( ? [X4: $o] :
( ( member_o @ X4 @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) )
& ( P @ X4 ) ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ? [Y2: $o] :
( ( member_o @ Y2 @ ( B2 @ X4 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_45_ball__UN,axiom,
! [B2: dedekind_preal > set_rat,A2: set_Dedekind_preal,P: rat > $o] :
( ( ! [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) )
=> ( P @ X4 ) ) )
= ( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
=> ! [Y2: rat] :
( ( member_rat @ Y2 @ ( B2 @ X4 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_46_ball__UN,axiom,
! [B2: set_rat > set_rat,A2: set_set_rat,P: rat > $o] :
( ( ! [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ A2 ) ) )
=> ( P @ X4 ) ) )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ! [Y2: rat] :
( ( member_rat @ Y2 @ ( B2 @ X4 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_47_ball__UN,axiom,
! [B2: rat > set_rat,A2: set_rat,P: rat > $o] :
( ( ! [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) )
=> ( P @ X4 ) ) )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ! [Y2: rat] :
( ( member_rat @ Y2 @ ( B2 @ X4 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_48_ball__UN,axiom,
! [B2: set_rat > set_set_rat,A2: set_set_rat,P: set_rat > $o] :
( ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ A2 ) ) )
=> ( P @ X4 ) ) )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ! [Y2: set_rat] :
( ( member_set_rat @ Y2 @ ( B2 @ X4 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_49_ball__UN,axiom,
! [B2: rat > set_set_rat,A2: set_rat,P: set_rat > $o] :
( ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ A2 ) ) )
=> ( P @ X4 ) ) )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ! [Y2: set_rat] :
( ( member_set_rat @ Y2 @ ( B2 @ X4 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_50_ball__UN,axiom,
! [B2: rat > set_o,A2: set_rat,P: $o > $o] :
( ( ! [X4: $o] :
( ( member_o @ X4 @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) )
=> ( P @ X4 ) ) )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ ( B2 @ X4 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_51_greaterThan__iff,axiom,
! [I: $o,K: $o] :
( ( member_o @ I @ ( set_or6416164934427428222Than_o @ K ) )
= ( ord_less_o @ K @ I ) ) ).
% greaterThan_iff
thf(fact_52_greaterThan__iff,axiom,
! [I: set_rat,K: set_rat] :
( ( member_set_rat @ I @ ( set_or6174011595382531368et_rat @ K ) )
= ( ord_less_set_rat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_53_greaterThan__iff,axiom,
! [I: dedekind_preal,K: dedekind_preal] :
( ( member6871284927547481791_preal @ I @ ( set_or4526639430071246588_preal @ K ) )
= ( ord_le5708704896291381698_preal @ K @ I ) ) ).
% greaterThan_iff
thf(fact_54_greaterThan__iff,axiom,
! [I: rat,K: rat] :
( ( member_rat @ I @ ( set_or575021546402375026an_rat @ K ) )
= ( ord_less_rat @ K @ I ) ) ).
% greaterThan_iff
thf(fact_55_SUP__UNION,axiom,
! [F: rat > $o,G: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [Y2: dedekind_preal] : ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_56_SUP__UNION,axiom,
! [F: rat > $o,G: rat > set_rat,A2: set_rat] :
( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_rat_o
@ ^ [Y2: rat] : ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_57_SUP__UNION,axiom,
! [F: $o > $o,G: rat > set_o,A2: set_rat] :
( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_rat_o
@ ^ [Y2: rat] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_58_SUP__UNION,axiom,
! [F: dedekind_preal > set_rat,G: dedekind_preal > set_Dedekind_preal,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ G @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y2: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_59_SUP__UNION,axiom,
! [F: dedekind_preal > set_rat,G: rat > set_Dedekind_preal,A2: set_rat] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ G @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [Y2: rat] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_60_SUP__UNION,axiom,
! [F: rat > set_rat,G: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y2: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_61_SUP__UNION,axiom,
! [F: rat > set_rat,G: rat > set_rat,A2: set_rat] :
( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ G @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [Y2: rat] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_62_SUP__UNION,axiom,
! [F: $o > set_rat,G: dedekind_preal > set_o,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ G @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y2: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_63_SUP__UNION,axiom,
! [F: $o > set_rat,G: rat > set_o,A2: set_rat] :
( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ G @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [Y2: rat] : ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_64_SUP__UNION,axiom,
! [F: rat > $o,G: set_rat > set_rat,A2: set_set_rat] :
( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_set_rat_o
@ ^ [Y2: set_rat] : ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_65_Union__iff,axiom,
! [A2: dedekind_preal,C: set_se4284277844656040150_preal] :
( ( member6871284927547481791_preal @ A2 @ ( comple800427960086186211_preal @ C ) )
= ( ? [X4: set_Dedekind_preal] :
( ( member2000626104285586591_preal @ X4 @ C )
& ( member6871284927547481791_preal @ A2 @ X4 ) ) ) ) ).
% Union_iff
thf(fact_66_Union__iff,axiom,
! [A2: rat,C: set_set_rat] :
( ( member_rat @ A2 @ ( comple3890839924845867745et_rat @ C ) )
= ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ C )
& ( member_rat @ A2 @ X4 ) ) ) ) ).
% Union_iff
thf(fact_67_Union__iff,axiom,
! [A2: set_rat,C: set_set_set_rat] :
( ( member_set_rat @ A2 @ ( comple3392050375588816791et_rat @ C ) )
= ( ? [X4: set_set_rat] :
( ( member_set_set_rat @ X4 @ C )
& ( member_set_rat @ A2 @ X4 ) ) ) ) ).
% Union_iff
thf(fact_68_Union__iff,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) )
= ( ? [X4: set_o] :
( ( member_set_o @ X4 @ C )
& ( member_o @ A2 @ X4 ) ) ) ) ).
% Union_iff
thf(fact_69_UnionI,axiom,
! [X2: set_Dedekind_preal,C: set_se4284277844656040150_preal,A2: dedekind_preal] :
( ( member2000626104285586591_preal @ X2 @ C )
=> ( ( member6871284927547481791_preal @ A2 @ X2 )
=> ( member6871284927547481791_preal @ A2 @ ( comple800427960086186211_preal @ C ) ) ) ) ).
% UnionI
thf(fact_70_UnionI,axiom,
! [X2: set_rat,C: set_set_rat,A2: rat] :
( ( member_set_rat @ X2 @ C )
=> ( ( member_rat @ A2 @ X2 )
=> ( member_rat @ A2 @ ( comple3890839924845867745et_rat @ C ) ) ) ) ).
% UnionI
thf(fact_71_UnionI,axiom,
! [X2: set_set_rat,C: set_set_set_rat,A2: set_rat] :
( ( member_set_set_rat @ X2 @ C )
=> ( ( member_set_rat @ A2 @ X2 )
=> ( member_set_rat @ A2 @ ( comple3392050375588816791et_rat @ C ) ) ) ) ).
% UnionI
thf(fact_72_UnionI,axiom,
! [X2: set_o,C: set_set_o,A2: $o] :
( ( member_set_o @ X2 @ C )
=> ( ( member_o @ A2 @ X2 )
=> ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) ) ) ) ).
% UnionI
thf(fact_73_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_rat,P: rat > $o] :
( ( ! [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ A2 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ! [Y2: rat] :
( ( member_rat @ Y2 @ X4 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_74_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_set_rat,P: set_rat > $o] :
( ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ A2 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: set_set_rat] :
( ( member_set_set_rat @ X4 @ A2 )
=> ! [Y2: set_rat] :
( ( member_set_rat @ Y2 @ X4 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_75_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_o,P: $o > $o] :
( ( ! [X4: $o] :
( ( member_o @ X4 @ ( comple90263536869209701_set_o @ A2 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A2 )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ X4 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_76_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_rat,P: rat > $o] :
( ( ? [X4: rat] :
( ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ A2 ) )
& ( P @ X4 ) ) )
= ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
& ? [Y2: rat] :
( ( member_rat @ Y2 @ X4 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_77_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_set_rat,P: set_rat > $o] :
( ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ A2 ) )
& ( P @ X4 ) ) )
= ( ? [X4: set_set_rat] :
( ( member_set_set_rat @ X4 @ A2 )
& ? [Y2: set_rat] :
( ( member_set_rat @ Y2 @ X4 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_78_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_o,P: $o > $o] :
( ( ? [X4: $o] :
( ( member_o @ X4 @ ( comple90263536869209701_set_o @ A2 ) )
& ( P @ X4 ) ) )
= ( ? [X4: set_o] :
( ( member_set_o @ X4 @ A2 )
& ? [Y2: $o] :
( ( member_o @ Y2 @ X4 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_79_greaterThan__eq__iff,axiom,
! [X: rat,Y: rat] :
( ( ( set_or575021546402375026an_rat @ X )
= ( set_or575021546402375026an_rat @ Y ) )
= ( X = Y ) ) ).
% greaterThan_eq_iff
thf(fact_80_Sup__bot__conv_I1_J,axiom,
! [A2: set_se4284277844656040150_preal] :
( ( ( comple800427960086186211_preal @ A2 )
= bot_bo4848840305443107682_preal )
= ( ! [X4: set_Dedekind_preal] :
( ( member2000626104285586591_preal @ X4 @ A2 )
=> ( X4 = bot_bo4848840305443107682_preal ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_81_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_rat] :
( ( ( comple3890839924845867745et_rat @ A2 )
= bot_bot_set_rat )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ( X4 = bot_bot_set_rat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_82_Sup__bot__conv_I1_J,axiom,
! [A2: set_o] :
( ( ( complete_Sup_Sup_o @ A2 )
= bot_bot_o )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( X4 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_83_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_set_rat] :
( ( ( comple3392050375588816791et_rat @ A2 )
= bot_bot_set_set_rat )
= ( ! [X4: set_set_rat] :
( ( member_set_set_rat @ X4 @ A2 )
=> ( X4 = bot_bot_set_set_rat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_84_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_o] :
( ( ( comple90263536869209701_set_o @ A2 )
= bot_bot_set_o )
= ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A2 )
=> ( X4 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_85_Sup__bot__conv_I2_J,axiom,
! [A2: set_se4284277844656040150_preal] :
( ( bot_bo4848840305443107682_preal
= ( comple800427960086186211_preal @ A2 ) )
= ( ! [X4: set_Dedekind_preal] :
( ( member2000626104285586591_preal @ X4 @ A2 )
=> ( X4 = bot_bo4848840305443107682_preal ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_86_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_rat] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ A2 ) )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ( X4 = bot_bot_set_rat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_87_Sup__bot__conv_I2_J,axiom,
! [A2: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A2 ) )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( X4 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_88_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_set_rat] :
( ( bot_bot_set_set_rat
= ( comple3392050375588816791et_rat @ A2 ) )
= ( ! [X4: set_set_rat] :
( ( member_set_set_rat @ X4 @ A2 )
=> ( X4 = bot_bot_set_set_rat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_89_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A2 ) )
= ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A2 )
=> ( X4 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_90_Sup__empty,axiom,
( ( comple800427960086186211_preal @ bot_bo5050857909902108994_preal )
= bot_bo4848840305443107682_preal ) ).
% Sup_empty
thf(fact_91_Sup__empty,axiom,
( ( comple3890839924845867745et_rat @ bot_bot_set_set_rat )
= bot_bot_set_rat ) ).
% Sup_empty
thf(fact_92_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_93_Sup__empty,axiom,
( ( comple3392050375588816791et_rat @ bot_bo6619408370577057422et_rat )
= bot_bot_set_set_rat ) ).
% Sup_empty
thf(fact_94_Sup__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Sup_empty
thf(fact_95_SUP__bot,axiom,
! [A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : bot_bot_set_rat
@ A2 ) )
= bot_bot_set_rat ) ).
% SUP_bot
thf(fact_96_SUP__bot,axiom,
! [A2: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : bot_bot_set_rat
@ A2 ) )
= bot_bot_set_rat ) ).
% SUP_bot
thf(fact_97_SUP__bot,axiom,
! [A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : bot_bot_set_rat
@ A2 ) )
= bot_bot_set_rat ) ).
% SUP_bot
thf(fact_98_SUP__bot,axiom,
! [A2: set_set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_9010471772881432025et_rat
@ ^ [X4: set_rat] : bot_bot_set_set_rat
@ A2 ) )
= bot_bot_set_set_rat ) ).
% SUP_bot
thf(fact_99_SUP__bot,axiom,
! [A2: set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [X4: rat] : bot_bot_set_set_rat
@ A2 ) )
= bot_bot_set_set_rat ) ).
% SUP_bot
thf(fact_100_SUP__bot,axiom,
! [A2: set_rat] :
( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : bot_bot_set_o
@ A2 ) )
= bot_bot_set_o ) ).
% SUP_bot
thf(fact_101_SUP__bot__conv_I1_J,axiom,
! [B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) )
= bot_bot_set_rat )
= ( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_rat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_102_SUP__bot__conv_I1_J,axiom,
! [B2: set_rat > set_rat,A2: set_set_rat] :
( ( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ A2 ) )
= bot_bot_set_rat )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_rat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_103_SUP__bot__conv_I1_J,axiom,
! [B2: rat > set_rat,A2: set_rat] :
( ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) )
= bot_bot_set_rat )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_rat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_104_SUP__bot__conv_I1_J,axiom,
! [B2: set_rat > set_set_rat,A2: set_set_rat] :
( ( ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ A2 ) )
= bot_bot_set_set_rat )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_set_rat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_105_SUP__bot__conv_I1_J,axiom,
! [B2: rat > set_set_rat,A2: set_rat] :
( ( ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ A2 ) )
= bot_bot_set_set_rat )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_set_rat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_106_SUP__bot__conv_I1_J,axiom,
! [B2: rat > set_o,A2: set_rat] :
( ( ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) )
= bot_bot_set_o )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_o ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_107_SUP__bot__conv_I2_J,axiom,
! [B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) )
= ( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_rat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_108_SUP__bot__conv_I2_J,axiom,
! [B2: set_rat > set_rat,A2: set_set_rat] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ A2 ) ) )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_rat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_109_SUP__bot__conv_I2_J,axiom,
! [B2: rat > set_rat,A2: set_rat] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_rat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_110_SUP__bot__conv_I2_J,axiom,
! [B2: set_rat > set_set_rat,A2: set_set_rat] :
( ( bot_bot_set_set_rat
= ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ A2 ) ) )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_set_rat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_111_SUP__bot__conv_I2_J,axiom,
! [B2: rat > set_set_rat,A2: set_rat] :
( ( bot_bot_set_set_rat
= ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ A2 ) ) )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_set_rat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_112_SUP__bot__conv_I2_J,axiom,
! [B2: rat > set_o,A2: set_rat] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_o ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_113_SUP__const,axiom,
! [A2: set_Dedekind_preal,F: $o] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [I2: dedekind_preal] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_114_SUP__const,axiom,
! [A2: set_rat,F: $o] :
( ( A2 != bot_bot_set_rat )
=> ( ( complete_Sup_Sup_o
@ ( image_rat_o
@ ^ [I2: rat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_115_SUP__const,axiom,
! [A2: set_o,F: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [I2: $o] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_116_SUP__const,axiom,
! [A2: set_Dedekind_preal,F: set_rat] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [I2: dedekind_preal] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_117_SUP__const,axiom,
! [A2: set_rat,F: set_rat] :
( ( A2 != bot_bot_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [I2: rat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_118_SUP__const,axiom,
! [A2: set_o,F: set_rat] :
( ( A2 != bot_bot_set_o )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [I2: $o] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_119_SUP__const,axiom,
! [A2: set_Dedekind_preal,F: set_o] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [I2: dedekind_preal] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_120_SUP__const,axiom,
! [A2: set_rat,F: set_o] :
( ( A2 != bot_bot_set_rat )
=> ( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [I2: rat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_121_SUP__const,axiom,
! [A2: set_o,F: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [I2: $o] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_122_SUP__const,axiom,
! [A2: set_set_rat,F: set_rat] :
( ( A2 != bot_bot_set_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [I2: set_rat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_123_UN__constant,axiom,
! [A2: set_Dedekind_preal,C2: set_Dedekind_preal] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= bot_bo4848840305443107682_preal ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_124_UN__constant,axiom,
! [A2: set_rat,C2: set_Dedekind_preal] :
( ( ( A2 = bot_bot_set_rat )
=> ( ( comple800427960086186211_preal
@ ( image_1093189101101737815_preal
@ ^ [Y2: rat] : C2
@ A2 ) )
= bot_bo4848840305443107682_preal ) )
& ( ( A2 != bot_bot_set_rat )
=> ( ( comple800427960086186211_preal
@ ( image_1093189101101737815_preal
@ ^ [Y2: rat] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_125_UN__constant,axiom,
! [A2: set_o,C2: set_Dedekind_preal] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple800427960086186211_preal
@ ( image_2856262260070713571_preal
@ ^ [Y2: $o] : C2
@ A2 ) )
= bot_bo4848840305443107682_preal ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple800427960086186211_preal
@ ( image_2856262260070713571_preal
@ ^ [Y2: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_126_UN__constant,axiom,
! [A2: set_Dedekind_preal,C2: set_rat] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_127_UN__constant,axiom,
! [A2: set_rat,C2: set_rat] :
( ( ( A2 = bot_bot_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [Y2: rat] : C2
@ A2 ) )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bot_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [Y2: rat] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_128_UN__constant,axiom,
! [A2: set_o,C2: set_rat] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [Y2: $o] : C2
@ A2 ) )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [Y2: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_129_UN__constant,axiom,
! [A2: set_Dedekind_preal,C2: set_o] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_130_UN__constant,axiom,
! [A2: set_rat,C2: set_o] :
( ( ( A2 = bot_bot_set_rat )
=> ( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [Y2: rat] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_rat )
=> ( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [Y2: rat] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_131_UN__constant,axiom,
! [A2: set_o,C2: set_o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_132_UN__constant,axiom,
! [A2: set_set_rat,C2: set_rat] :
( ( ( A2 = bot_bot_set_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [Y2: set_rat] : C2
@ A2 ) )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bot_set_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [Y2: set_rat] : C2
@ A2 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_133_Union__empty,axiom,
( ( comple800427960086186211_preal @ bot_bo5050857909902108994_preal )
= bot_bo4848840305443107682_preal ) ).
% Union_empty
thf(fact_134_Union__empty,axiom,
( ( comple3890839924845867745et_rat @ bot_bot_set_set_rat )
= bot_bot_set_rat ) ).
% Union_empty
thf(fact_135_Union__empty,axiom,
( ( comple3392050375588816791et_rat @ bot_bo6619408370577057422et_rat )
= bot_bot_set_set_rat ) ).
% Union_empty
thf(fact_136_Union__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Union_empty
thf(fact_137_Sup__set__def,axiom,
( comple800427960086186211_preal
= ( ^ [A3: set_se4284277844656040150_preal] :
( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] : ( complete_Sup_Sup_o @ ( image_3678684046732248231real_o @ ( member6871284927547481791_preal @ X4 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_138_Sup__set__def,axiom,
( comple3890839924845867745et_rat
= ( ^ [A3: set_set_rat] :
( collect_rat
@ ^ [X4: rat] : ( complete_Sup_Sup_o @ ( image_set_rat_o @ ( member_rat @ X4 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_139_Sup__set__def,axiom,
( comple3392050375588816791et_rat
= ( ^ [A3: set_set_set_rat] :
( collect_set_rat
@ ^ [X4: set_rat] : ( complete_Sup_Sup_o @ ( image_set_set_rat_o @ ( member_set_rat @ X4 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_140_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X4: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X4 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_141_empty__Union__conv,axiom,
! [A2: set_se4284277844656040150_preal] :
( ( bot_bo4848840305443107682_preal
= ( comple800427960086186211_preal @ A2 ) )
= ( ! [X4: set_Dedekind_preal] :
( ( member2000626104285586591_preal @ X4 @ A2 )
=> ( X4 = bot_bo4848840305443107682_preal ) ) ) ) ).
% empty_Union_conv
thf(fact_142_empty__Union__conv,axiom,
! [A2: set_set_rat] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ A2 ) )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ( X4 = bot_bot_set_rat ) ) ) ) ).
% empty_Union_conv
thf(fact_143_empty__Union__conv,axiom,
! [A2: set_set_set_rat] :
( ( bot_bot_set_set_rat
= ( comple3392050375588816791et_rat @ A2 ) )
= ( ! [X4: set_set_rat] :
( ( member_set_set_rat @ X4 @ A2 )
=> ( X4 = bot_bot_set_set_rat ) ) ) ) ).
% empty_Union_conv
thf(fact_144_empty__Union__conv,axiom,
! [A2: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A2 ) )
= ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A2 )
=> ( X4 = bot_bot_set_o ) ) ) ) ).
% empty_Union_conv
thf(fact_145_Union__empty__conv,axiom,
! [A2: set_se4284277844656040150_preal] :
( ( ( comple800427960086186211_preal @ A2 )
= bot_bo4848840305443107682_preal )
= ( ! [X4: set_Dedekind_preal] :
( ( member2000626104285586591_preal @ X4 @ A2 )
=> ( X4 = bot_bo4848840305443107682_preal ) ) ) ) ).
% Union_empty_conv
thf(fact_146_Union__empty__conv,axiom,
! [A2: set_set_rat] :
( ( ( comple3890839924845867745et_rat @ A2 )
= bot_bot_set_rat )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ( X4 = bot_bot_set_rat ) ) ) ) ).
% Union_empty_conv
thf(fact_147_Union__empty__conv,axiom,
! [A2: set_set_set_rat] :
( ( ( comple3392050375588816791et_rat @ A2 )
= bot_bot_set_set_rat )
= ( ! [X4: set_set_rat] :
( ( member_set_set_rat @ X4 @ A2 )
=> ( X4 = bot_bot_set_set_rat ) ) ) ) ).
% Union_empty_conv
thf(fact_148_Union__empty__conv,axiom,
! [A2: set_set_o] :
( ( ( comple90263536869209701_set_o @ A2 )
= bot_bot_set_o )
= ( ! [X4: set_o] :
( ( member_set_o @ X4 @ A2 )
=> ( X4 = bot_bot_set_o ) ) ) ) ).
% Union_empty_conv
thf(fact_149_greaterThan__non__empty,axiom,
! [X: rat] :
( ( set_or575021546402375026an_rat @ X )
!= bot_bot_set_rat ) ).
% greaterThan_non_empty
thf(fact_150_SUP__constant,axiom,
! [C2: $o,A2: set_Dedekind_preal] :
( ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= ( ( ( A2 = bot_bo4848840305443107682_preal )
=> bot_bot_o )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_151_SUP__constant,axiom,
! [C2: $o,A2: set_rat] :
( ( complete_Sup_Sup_o
@ ( image_rat_o
@ ^ [Y2: rat] : C2
@ A2 ) )
= ( ( ( A2 = bot_bot_set_rat )
=> bot_bot_o )
& ( ( A2 != bot_bot_set_rat )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_152_SUP__constant,axiom,
! [C2: $o,A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y2: $o] : C2
@ A2 ) )
= ( ( ( A2 = bot_bot_set_o )
=> bot_bot_o )
& ( ( A2 != bot_bot_set_o )
=> C2 ) ) ) ).
% SUP_constant
thf(fact_153_SUP__constant,axiom,
! [A2: set_Dedekind_preal,C2: set_Dedekind_preal] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= bot_bo4848840305443107682_preal ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_154_SUP__constant,axiom,
! [A2: set_rat,C2: set_Dedekind_preal] :
( ( ( A2 = bot_bot_set_rat )
=> ( ( comple800427960086186211_preal
@ ( image_1093189101101737815_preal
@ ^ [Y2: rat] : C2
@ A2 ) )
= bot_bo4848840305443107682_preal ) )
& ( ( A2 != bot_bot_set_rat )
=> ( ( comple800427960086186211_preal
@ ( image_1093189101101737815_preal
@ ^ [Y2: rat] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_155_SUP__constant,axiom,
! [A2: set_o,C2: set_Dedekind_preal] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple800427960086186211_preal
@ ( image_2856262260070713571_preal
@ ^ [Y2: $o] : C2
@ A2 ) )
= bot_bo4848840305443107682_preal ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple800427960086186211_preal
@ ( image_2856262260070713571_preal
@ ^ [Y2: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_156_SUP__constant,axiom,
! [A2: set_Dedekind_preal,C2: set_rat] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_157_SUP__constant,axiom,
! [A2: set_rat,C2: set_rat] :
( ( ( A2 = bot_bot_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [Y2: rat] : C2
@ A2 ) )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bot_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [Y2: rat] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_158_SUP__constant,axiom,
! [A2: set_o,C2: set_rat] :
( ( ( A2 = bot_bot_set_o )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [Y2: $o] : C2
@ A2 ) )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [Y2: $o] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_159_SUP__constant,axiom,
! [A2: set_Dedekind_preal,C2: set_o] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= bot_bot_set_o ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [Y2: dedekind_preal] : C2
@ A2 ) )
= C2 ) ) ) ).
% SUP_constant
thf(fact_160_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_161_SUP__empty,axiom,
! [F: rat > $o] :
( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ bot_bot_set_rat ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_162_SUP__empty,axiom,
! [F: $o > $o] :
( ( complete_Sup_Sup_o @ ( image_o_o @ F @ bot_bot_set_o ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_163_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_164_SUP__empty,axiom,
! [F: rat > set_Dedekind_preal] :
( ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ F @ bot_bot_set_rat ) )
= bot_bo4848840305443107682_preal ) ).
% SUP_empty
thf(fact_165_SUP__empty,axiom,
! [F: $o > set_Dedekind_preal] :
( ( comple800427960086186211_preal @ ( image_2856262260070713571_preal @ F @ bot_bot_set_o ) )
= bot_bo4848840305443107682_preal ) ).
% SUP_empty
thf(fact_166_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_167_SUP__empty,axiom,
! [F: rat > set_rat] :
( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ bot_bot_set_rat ) )
= bot_bot_set_rat ) ).
% SUP_empty
thf(fact_168_SUP__empty,axiom,
! [F: $o > set_rat] :
( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ bot_bot_set_o ) )
= bot_bot_set_rat ) ).
% SUP_empty
thf(fact_169_SUP__empty,axiom,
! [F: dedekind_preal > set_o] :
( ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ bot_bo4848840305443107682_preal ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_170_SUP__eq__const,axiom,
! [I3: set_Dedekind_preal,F: dedekind_preal > $o,X: $o] :
( ( I3 != bot_bo4848840305443107682_preal )
=> ( ! [I4: dedekind_preal] :
( ( member6871284927547481791_preal @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_171_SUP__eq__const,axiom,
! [I3: set_rat,F: rat > $o,X: $o] :
( ( I3 != bot_bot_set_rat )
=> ( ! [I4: rat] :
( ( member_rat @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_172_SUP__eq__const,axiom,
! [I3: set_o,F: $o > $o,X: $o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_173_SUP__eq__const,axiom,
! [I3: set_Dedekind_preal,F: dedekind_preal > set_rat,X: set_rat] :
( ( I3 != bot_bo4848840305443107682_preal )
=> ( ! [I4: dedekind_preal] :
( ( member6871284927547481791_preal @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_174_SUP__eq__const,axiom,
! [I3: set_rat,F: rat > set_rat,X: set_rat] :
( ( I3 != bot_bot_set_rat )
=> ( ! [I4: rat] :
( ( member_rat @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_175_SUP__eq__const,axiom,
! [I3: set_o,F: $o > set_rat,X: set_rat] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_176_SUP__eq__const,axiom,
! [I3: set_set_rat,F: set_rat > $o,X: $o] :
( ( I3 != bot_bot_set_set_rat )
=> ( ! [I4: set_rat] :
( ( member_set_rat @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_177_SUP__eq__const,axiom,
! [I3: set_Dedekind_preal,F: dedekind_preal > set_o,X: set_o] :
( ( I3 != bot_bo4848840305443107682_preal )
=> ( ! [I4: dedekind_preal] :
( ( member6871284927547481791_preal @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_178_SUP__eq__const,axiom,
! [I3: set_rat,F: rat > set_o,X: set_o] :
( ( I3 != bot_bot_set_rat )
=> ( ! [I4: rat] :
( ( member_rat @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_179_SUP__eq__const,axiom,
! [I3: set_o,F: $o > set_o,X: set_o] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_180_UN__empty2,axiom,
! [A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : bot_bot_set_rat
@ A2 ) )
= bot_bot_set_rat ) ).
% UN_empty2
thf(fact_181_UN__empty2,axiom,
! [A2: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : bot_bot_set_rat
@ A2 ) )
= bot_bot_set_rat ) ).
% UN_empty2
thf(fact_182_UN__empty2,axiom,
! [A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : bot_bot_set_rat
@ A2 ) )
= bot_bot_set_rat ) ).
% UN_empty2
thf(fact_183_UN__empty2,axiom,
! [A2: set_set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_9010471772881432025et_rat
@ ^ [X4: set_rat] : bot_bot_set_set_rat
@ A2 ) )
= bot_bot_set_set_rat ) ).
% UN_empty2
thf(fact_184_UN__empty2,axiom,
! [A2: set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [X4: rat] : bot_bot_set_set_rat
@ A2 ) )
= bot_bot_set_set_rat ) ).
% UN_empty2
thf(fact_185_UN__empty2,axiom,
! [A2: set_rat] :
( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : bot_bot_set_o
@ A2 ) )
= bot_bot_set_o ) ).
% UN_empty2
thf(fact_186_UN__empty,axiom,
! [B2: dedekind_preal > set_Dedekind_preal] :
( ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B2 @ bot_bo4848840305443107682_preal ) )
= bot_bo4848840305443107682_preal ) ).
% UN_empty
thf(fact_187_UN__empty,axiom,
! [B2: rat > set_Dedekind_preal] :
( ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B2 @ bot_bot_set_rat ) )
= bot_bo4848840305443107682_preal ) ).
% UN_empty
thf(fact_188_UN__empty,axiom,
! [B2: $o > set_Dedekind_preal] :
( ( comple800427960086186211_preal @ ( image_2856262260070713571_preal @ B2 @ bot_bot_set_o ) )
= bot_bo4848840305443107682_preal ) ).
% UN_empty
thf(fact_189_UN__empty,axiom,
! [B2: dedekind_preal > set_rat] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ bot_bo4848840305443107682_preal ) )
= bot_bot_set_rat ) ).
% UN_empty
thf(fact_190_UN__empty,axiom,
! [B2: rat > set_rat] :
( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ bot_bot_set_rat ) )
= bot_bot_set_rat ) ).
% UN_empty
thf(fact_191_UN__empty,axiom,
! [B2: $o > set_rat] :
( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ bot_bot_set_o ) )
= bot_bot_set_rat ) ).
% UN_empty
thf(fact_192_UN__empty,axiom,
! [B2: dedekind_preal > set_o] :
( ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ bot_bo4848840305443107682_preal ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_193_UN__empty,axiom,
! [B2: rat > set_o] :
( ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ bot_bot_set_rat ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_194_UN__empty,axiom,
! [B2: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_195_UN__empty,axiom,
! [B2: set_rat > set_rat] :
( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ bot_bot_set_set_rat ) )
= bot_bot_set_rat ) ).
% UN_empty
thf(fact_196_UNION__empty__conv_I1_J,axiom,
! [B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) )
= ( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_rat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_197_UNION__empty__conv_I1_J,axiom,
! [B2: set_rat > set_rat,A2: set_set_rat] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ A2 ) ) )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_rat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_198_UNION__empty__conv_I1_J,axiom,
! [B2: rat > set_rat,A2: set_rat] :
( ( bot_bot_set_rat
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_rat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_199_UNION__empty__conv_I1_J,axiom,
! [B2: set_rat > set_set_rat,A2: set_set_rat] :
( ( bot_bot_set_set_rat
= ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ A2 ) ) )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_set_rat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_200_UNION__empty__conv_I1_J,axiom,
! [B2: rat > set_set_rat,A2: set_rat] :
( ( bot_bot_set_set_rat
= ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ A2 ) ) )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_set_rat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_201_UNION__empty__conv_I1_J,axiom,
! [B2: rat > set_o,A2: set_rat] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_o ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_202_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_203_mem__Collect__eq,axiom,
! [A: dedekind_preal,P: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ A @ ( collec1132657498972982273_preal @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_204_mem__Collect__eq,axiom,
! [A: set_rat,P: set_rat > $o] :
( ( member_set_rat @ A @ ( collect_set_rat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_205_mem__Collect__eq,axiom,
! [A: rat,P: rat > $o] :
( ( member_rat @ A @ ( collect_rat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_206_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X4: $o] : ( member_o @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_207_Collect__mem__eq,axiom,
! [A2: set_Dedekind_preal] :
( ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] : ( member6871284927547481791_preal @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_208_Collect__mem__eq,axiom,
! [A2: set_set_rat] :
( ( collect_set_rat
@ ^ [X4: set_rat] : ( member_set_rat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_209_Collect__mem__eq,axiom,
! [A2: set_rat] :
( ( collect_rat
@ ^ [X4: rat] : ( member_rat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_210_Collect__cong,axiom,
! [P: set_rat > $o,Q2: set_rat > $o] :
( ! [X3: set_rat] :
( ( P @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_set_rat @ P )
= ( collect_set_rat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_211_Collect__cong,axiom,
! [P: rat > $o,Q2: rat > $o] :
( ! [X3: rat] :
( ( P @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_rat @ P )
= ( collect_rat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_212_UNION__empty__conv_I2_J,axiom,
! [B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) )
= bot_bot_set_rat )
= ( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_rat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_213_UNION__empty__conv_I2_J,axiom,
! [B2: set_rat > set_rat,A2: set_set_rat] :
( ( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ A2 ) )
= bot_bot_set_rat )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_rat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_214_UNION__empty__conv_I2_J,axiom,
! [B2: rat > set_rat,A2: set_rat] :
( ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) )
= bot_bot_set_rat )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_rat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_215_UNION__empty__conv_I2_J,axiom,
! [B2: set_rat > set_set_rat,A2: set_set_rat] :
( ( ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ A2 ) )
= bot_bot_set_set_rat )
= ( ! [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_set_rat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_216_UNION__empty__conv_I2_J,axiom,
! [B2: rat > set_set_rat,A2: set_rat] :
( ( ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ A2 ) )
= bot_bot_set_set_rat )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_set_rat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_217_UNION__empty__conv_I2_J,axiom,
! [B2: rat > set_o,A2: set_rat] :
( ( ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) )
= bot_bot_set_o )
= ( ! [X4: rat] :
( ( member_rat @ X4 @ A2 )
=> ( ( B2 @ X4 )
= bot_bot_set_o ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_218_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_219_zero__reorient,axiom,
! [X: rat] :
( ( zero_zero_rat = X )
= ( X = zero_zero_rat ) ) ).
% zero_reorient
thf(fact_220_Sup_OSUP__cong,axiom,
! [A2: set_rat,B2: set_rat,C: rat > set_o,D: rat > set_o,Sup: set_set_o > set_o] :
( ( A2 = B2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_rat_set_o @ C @ A2 ) )
= ( Sup @ ( image_rat_set_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_221_Sup_OSUP__cong,axiom,
! [A2: set_rat,B2: set_rat,C: rat > set_set_rat,D: rat > set_set_rat,Sup: set_set_set_rat > set_set_rat] :
( ( A2 = B2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_5169559123115255331et_rat @ C @ A2 ) )
= ( Sup @ ( image_5169559123115255331et_rat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_222_Sup_OSUP__cong,axiom,
! [A2: set_rat,B2: set_rat,C: rat > set_rat,D: rat > set_rat,Sup: set_set_rat > set_rat] :
( ( A2 = B2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_rat_set_rat @ C @ A2 ) )
= ( Sup @ ( image_rat_set_rat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_223_Sup_OSUP__cong,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_rat > set_set_rat,D: set_rat > set_set_rat,Sup: set_set_set_rat > set_set_rat] :
( ( A2 = B2 )
=> ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_9010471772881432025et_rat @ C @ A2 ) )
= ( Sup @ ( image_9010471772881432025et_rat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_224_Sup_OSUP__cong,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_rat > set_rat,D: set_rat > set_rat,Sup: set_set_rat > set_rat] :
( ( A2 = B2 )
=> ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_3939399684171694371et_rat @ C @ A2 ) )
= ( Sup @ ( image_3939399684171694371et_rat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_225_Sup_OSUP__cong,axiom,
! [A2: set_Dedekind_preal,B2: set_Dedekind_preal,C: dedekind_preal > set_rat,D: dedekind_preal > set_rat,Sup: set_set_rat > set_rat] :
( ( A2 = B2 )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_2231269897757205599et_rat @ C @ A2 ) )
= ( Sup @ ( image_2231269897757205599et_rat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_226_Inf_OINF__cong,axiom,
! [A2: set_rat,B2: set_rat,C: rat > set_o,D: rat > set_o,Inf: set_set_o > set_o] :
( ( A2 = B2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_rat_set_o @ C @ A2 ) )
= ( Inf @ ( image_rat_set_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_227_Inf_OINF__cong,axiom,
! [A2: set_rat,B2: set_rat,C: rat > set_set_rat,D: rat > set_set_rat,Inf: set_set_set_rat > set_set_rat] :
( ( A2 = B2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_5169559123115255331et_rat @ C @ A2 ) )
= ( Inf @ ( image_5169559123115255331et_rat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_228_Inf_OINF__cong,axiom,
! [A2: set_rat,B2: set_rat,C: rat > set_rat,D: rat > set_rat,Inf: set_set_rat > set_rat] :
( ( A2 = B2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_rat_set_rat @ C @ A2 ) )
= ( Inf @ ( image_rat_set_rat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_229_Inf_OINF__cong,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_rat > set_set_rat,D: set_rat > set_set_rat,Inf: set_set_set_rat > set_set_rat] :
( ( A2 = B2 )
=> ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_9010471772881432025et_rat @ C @ A2 ) )
= ( Inf @ ( image_9010471772881432025et_rat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_230_Inf_OINF__cong,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_rat > set_rat,D: set_rat > set_rat,Inf: set_set_rat > set_rat] :
( ( A2 = B2 )
=> ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_3939399684171694371et_rat @ C @ A2 ) )
= ( Inf @ ( image_3939399684171694371et_rat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_231_Inf_OINF__cong,axiom,
! [A2: set_Dedekind_preal,B2: set_Dedekind_preal,C: dedekind_preal > set_rat,D: dedekind_preal > set_rat,Inf: set_set_rat > set_rat] :
( ( A2 = B2 )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_2231269897757205599et_rat @ C @ A2 ) )
= ( Inf @ ( image_2231269897757205599et_rat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_232_UnionE,axiom,
! [A2: dedekind_preal,C: set_se4284277844656040150_preal] :
( ( member6871284927547481791_preal @ A2 @ ( comple800427960086186211_preal @ C ) )
=> ~ ! [X5: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ A2 @ X5 )
=> ~ ( member2000626104285586591_preal @ X5 @ C ) ) ) ).
% UnionE
thf(fact_233_UnionE,axiom,
! [A2: rat,C: set_set_rat] :
( ( member_rat @ A2 @ ( comple3890839924845867745et_rat @ C ) )
=> ~ ! [X5: set_rat] :
( ( member_rat @ A2 @ X5 )
=> ~ ( member_set_rat @ X5 @ C ) ) ) ).
% UnionE
thf(fact_234_UnionE,axiom,
! [A2: set_rat,C: set_set_set_rat] :
( ( member_set_rat @ A2 @ ( comple3392050375588816791et_rat @ C ) )
=> ~ ! [X5: set_set_rat] :
( ( member_set_rat @ A2 @ X5 )
=> ~ ( member_set_set_rat @ X5 @ C ) ) ) ).
% UnionE
thf(fact_235_UnionE,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) )
=> ~ ! [X5: set_o] :
( ( member_o @ A2 @ X5 )
=> ~ ( member_set_o @ X5 @ C ) ) ) ).
% UnionE
thf(fact_236_Sup_OSUP__identity__eq,axiom,
! [Sup: set_set_rat > set_rat,A2: set_set_rat] :
( ( Sup
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : X4
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_237_Inf_OINF__identity__eq,axiom,
! [Inf: set_set_rat > set_rat,A2: set_set_rat] :
( ( Inf
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : X4
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_238_SUP__cong,axiom,
! [A2: set_rat,B2: set_rat,C: rat > $o,D: rat > $o] :
( ( A2 = B2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_rat_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_239_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_240_SUP__cong,axiom,
! [A2: set_Dedekind_preal,B2: set_Dedekind_preal,C: dedekind_preal > $o,D: dedekind_preal > $o] :
( ( A2 = B2 )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_241_SUP__cong,axiom,
! [A2: set_rat,B2: set_rat,C: rat > set_rat,D: rat > set_rat] :
( ( A2 = B2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C @ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_242_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_rat,D: $o > set_rat] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ C @ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_o_set_rat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_243_SUP__cong,axiom,
! [A2: set_Dedekind_preal,B2: set_Dedekind_preal,C: dedekind_preal > set_rat,D: dedekind_preal > set_rat] :
( ( A2 = B2 )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_244_SUP__cong,axiom,
! [A2: set_set_rat,B2: set_set_rat,C: set_rat > $o,D: set_rat > $o] :
( ( A2 = B2 )
=> ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_rat_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_set_rat_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_245_SUP__cong,axiom,
! [A2: set_rat,B2: set_rat,C: rat > set_o,D: rat > set_o] :
( ( A2 = B2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_rat_set_o @ C @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_rat_set_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_246_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_o,D: $o > set_o] :
( ( A2 = B2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_247_SUP__cong,axiom,
! [A2: set_Dedekind_preal,B2: set_Dedekind_preal,C: dedekind_preal > set_o,D: dedekind_preal > set_o] :
( ( A2 = B2 )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ B2 )
=> ( ( C @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ C @ A2 ) )
= ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_248_SUP__commute,axiom,
! [F: dedekind_preal > dedekind_preal > set_rat,B2: set_Dedekind_preal,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [I2: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [J: dedekind_preal] :
( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [I2: dedekind_preal] : ( F @ I2 @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_249_SUP__commute,axiom,
! [F: dedekind_preal > rat > set_rat,B2: set_rat,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [I2: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [J: rat] :
( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [I2: dedekind_preal] : ( F @ I2 @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_250_SUP__commute,axiom,
! [F: rat > dedekind_preal > set_rat,B2: set_Dedekind_preal,A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [I2: rat] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [J: dedekind_preal] :
( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [I2: rat] : ( F @ I2 @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_251_SUP__commute,axiom,
! [F: rat > rat > set_rat,B2: set_rat,A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [I2: rat] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [J: rat] :
( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [I2: rat] : ( F @ I2 @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_252_SUP__commute,axiom,
! [F: rat > rat > set_o,B2: set_rat,A2: set_rat] :
( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [I2: rat] : ( comple90263536869209701_set_o @ ( image_rat_set_o @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [J: rat] :
( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [I2: rat] : ( F @ I2 @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_253_SUP__commute,axiom,
! [F: dedekind_preal > set_rat > set_rat,B2: set_set_rat,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [I2: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [J: set_rat] :
( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [I2: dedekind_preal] : ( F @ I2 @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_254_SUP__commute,axiom,
! [F: set_rat > dedekind_preal > set_rat,B2: set_Dedekind_preal,A2: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [I2: set_rat] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [J: dedekind_preal] :
( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [I2: set_rat] : ( F @ I2 @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_255_SUP__commute,axiom,
! [F: set_rat > rat > set_rat,B2: set_rat,A2: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [I2: set_rat] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [J: rat] :
( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [I2: set_rat] : ( F @ I2 @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_256_SUP__commute,axiom,
! [F: rat > set_rat > set_rat,B2: set_set_rat,A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [I2: rat] : ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [J: set_rat] :
( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [I2: rat] : ( F @ I2 @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_257_SUP__commute,axiom,
! [F: rat > rat > set_set_rat,B2: set_rat,A2: set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [I2: rat] : ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ ( F @ I2 ) @ B2 ) )
@ A2 ) )
= ( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [J: rat] :
( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [I2: rat] : ( F @ I2 @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_258_image__Union,axiom,
! [F: rat > rat,S2: set_set_rat] :
( ( image_rat_rat @ F @ ( comple3890839924845867745et_rat @ S2 ) )
= ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ ( image_rat_rat @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_259_image__Union,axiom,
! [F: rat > $o,S2: set_set_rat] :
( ( image_rat_o @ F @ ( comple3890839924845867745et_rat @ S2 ) )
= ( comple90263536869209701_set_o @ ( image_set_rat_set_o @ ( image_rat_o @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_260_image__Union,axiom,
! [F: $o > rat,S2: set_set_o] :
( ( image_o_rat @ F @ ( comple90263536869209701_set_o @ S2 ) )
= ( comple3890839924845867745et_rat @ ( image_set_o_set_rat @ ( image_o_rat @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_261_image__Union,axiom,
! [F: $o > $o,S2: set_set_o] :
( ( image_o_o @ F @ ( comple90263536869209701_set_o @ S2 ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o @ ( image_o_o @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_262_image__Union,axiom,
! [F: dedekind_preal > set_rat,S2: set_se4284277844656040150_preal] :
( ( image_2231269897757205599et_rat @ F @ ( comple800427960086186211_preal @ S2 ) )
= ( comple3392050375588816791et_rat @ ( image_1970760655392924405et_rat @ ( image_2231269897757205599et_rat @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_263_image__Union,axiom,
! [F: rat > set_o,S2: set_set_rat] :
( ( image_rat_set_o @ F @ ( comple3890839924845867745et_rat @ S2 ) )
= ( comple4436988014476444997_set_o @ ( image_5893626272549079683_set_o @ ( image_rat_set_o @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_264_image__Union,axiom,
! [F: rat > set_rat,S2: set_set_rat] :
( ( image_rat_set_rat @ F @ ( comple3890839924845867745et_rat @ S2 ) )
= ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ ( image_rat_set_rat @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_265_image__Union,axiom,
! [F: set_rat > rat,S2: set_set_set_rat] :
( ( image_set_rat_rat @ F @ ( comple3392050375588816791et_rat @ S2 ) )
= ( comple3890839924845867745et_rat @ ( image_5585803834317650393et_rat @ ( image_set_rat_rat @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_266_image__Union,axiom,
! [F: set_rat > $o,S2: set_set_set_rat] :
( ( image_set_rat_o @ F @ ( comple3392050375588816791et_rat @ S2 ) )
= ( comple90263536869209701_set_o @ ( image_9166368412354806125_set_o @ ( image_set_rat_o @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_267_image__Union,axiom,
! [F: $o > set_rat,S2: set_set_o] :
( ( image_o_set_rat @ F @ ( comple90263536869209701_set_o @ S2 ) )
= ( comple3392050375588816791et_rat @ ( image_1318631078669633143et_rat @ ( image_o_set_rat @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_268_greaterThan__def,axiom,
( set_or6174011595382531368et_rat
= ( ^ [L: set_rat] : ( collect_set_rat @ ( ord_less_set_rat @ L ) ) ) ) ).
% greaterThan_def
thf(fact_269_greaterThan__def,axiom,
( set_or4526639430071246588_preal
= ( ^ [L: dedekind_preal] : ( collec1132657498972982273_preal @ ( ord_le5708704896291381698_preal @ L ) ) ) ) ).
% greaterThan_def
thf(fact_270_greaterThan__def,axiom,
( set_or575021546402375026an_rat
= ( ^ [L: rat] : ( collect_rat @ ( ord_less_rat @ L ) ) ) ) ).
% greaterThan_def
thf(fact_271_UN__UN__flatten,axiom,
! [C: dedekind_preal > set_rat,B2: dedekind_preal > set_Dedekind_preal,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B2 @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y2: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_272_UN__UN__flatten,axiom,
! [C: dedekind_preal > set_rat,B2: rat > set_Dedekind_preal,A2: set_rat] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B2 @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [Y2: rat] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_273_UN__UN__flatten,axiom,
! [C: rat > set_rat,B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y2: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_274_UN__UN__flatten,axiom,
! [C: rat > set_rat,B2: rat > set_rat,A2: set_rat] :
( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [Y2: rat] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_275_UN__UN__flatten,axiom,
! [C: $o > set_rat,B2: dedekind_preal > set_o,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ C @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [Y2: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_o_set_rat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_276_UN__UN__flatten,axiom,
! [C: $o > set_rat,B2: rat > set_o,A2: set_rat] :
( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ C @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [Y2: rat] : ( comple3890839924845867745et_rat @ ( image_o_set_rat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_277_UN__UN__flatten,axiom,
! [C: rat > set_o,B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( comple90263536869209701_set_o @ ( image_rat_set_o @ C @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [Y2: dedekind_preal] : ( comple90263536869209701_set_o @ ( image_rat_set_o @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_278_UN__UN__flatten,axiom,
! [C: rat > set_o,B2: rat > set_rat,A2: set_rat] :
( ( comple90263536869209701_set_o @ ( image_rat_set_o @ C @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [Y2: rat] : ( comple90263536869209701_set_o @ ( image_rat_set_o @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_279_UN__UN__flatten,axiom,
! [C: $o > set_o,B2: rat > set_o,A2: set_rat] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [Y2: rat] : ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_280_UN__UN__flatten,axiom,
! [C: dedekind_preal > set_rat,B2: set_rat > set_Dedekind_preal,A2: set_set_rat] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ ( comple800427960086186211_preal @ ( image_7647003310076334625_preal @ B2 @ A2 ) ) ) )
= ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [Y2: set_rat] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_281_UN__E,axiom,
! [B: dedekind_preal,B2: rat > set_Dedekind_preal,A2: set_rat] :
( ( member6871284927547481791_preal @ B @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B2 @ A2 ) ) )
=> ~ ! [X3: rat] :
( ( member_rat @ X3 @ A2 )
=> ~ ( member6871284927547481791_preal @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_282_UN__E,axiom,
! [B: dedekind_preal,B2: $o > set_Dedekind_preal,A2: set_o] :
( ( member6871284927547481791_preal @ B @ ( comple800427960086186211_preal @ ( image_2856262260070713571_preal @ B2 @ A2 ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member6871284927547481791_preal @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_283_UN__E,axiom,
! [B: dedekind_preal,B2: dedekind_preal > set_Dedekind_preal,A2: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ B @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B2 @ A2 ) ) )
=> ~ ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A2 )
=> ~ ( member6871284927547481791_preal @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_284_UN__E,axiom,
! [B: rat,B2: rat > set_rat,A2: set_rat] :
( ( member_rat @ B @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) )
=> ~ ! [X3: rat] :
( ( member_rat @ X3 @ A2 )
=> ~ ( member_rat @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_285_UN__E,axiom,
! [B: rat,B2: $o > set_rat,A2: set_o] :
( ( member_rat @ B @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ A2 ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member_rat @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_286_UN__E,axiom,
! [B: rat,B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( member_rat @ B @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) )
=> ~ ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A2 )
=> ~ ( member_rat @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_287_UN__E,axiom,
! [B: $o,B2: rat > set_o,A2: set_rat] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) )
=> ~ ! [X3: rat] :
( ( member_rat @ X3 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_288_UN__E,axiom,
! [B: $o,B2: $o > set_o,A2: set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_289_UN__E,axiom,
! [B: $o,B2: dedekind_preal > set_o,A2: set_Dedekind_preal] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ A2 ) ) )
=> ~ ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_290_UN__E,axiom,
! [B: dedekind_preal,B2: set_rat > set_Dedekind_preal,A2: set_set_rat] :
( ( member6871284927547481791_preal @ B @ ( comple800427960086186211_preal @ ( image_7647003310076334625_preal @ B2 @ A2 ) ) )
=> ~ ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A2 )
=> ~ ( member6871284927547481791_preal @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_291_UN__extend__simps_I8_J,axiom,
! [B2: dedekind_preal > set_rat,A2: set_se4284277844656040150_preal] :
( ( comple3890839924845867745et_rat
@ ( image_4042146516684777023et_rat
@ ^ [Y2: set_Dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ ( comple800427960086186211_preal @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_292_UN__extend__simps_I8_J,axiom,
! [B2: rat > set_rat,A2: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [Y2: set_rat] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ ( comple3890839924845867745et_rat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_293_UN__extend__simps_I8_J,axiom,
! [B2: set_rat > set_rat,A2: set_set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_5585803834317650393et_rat
@ ^ [Y2: set_set_rat] : ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ ( comple3392050375588816791et_rat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_294_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_rat,A2: set_set_o] :
( ( comple3890839924845867745et_rat
@ ( image_set_o_set_rat
@ ^ [Y2: set_o] : ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_295_UN__extend__simps_I8_J,axiom,
! [B2: rat > set_set_rat,A2: set_set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_9010471772881432025et_rat
@ ^ [Y2: set_rat] : ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ ( comple3890839924845867745et_rat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_296_UN__extend__simps_I8_J,axiom,
! [B2: set_rat > set_set_rat,A2: set_set_set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_1224367690689810063et_rat
@ ^ [Y2: set_set_rat] : ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ ( comple3392050375588816791et_rat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_297_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_set_rat,A2: set_set_o] :
( ( comple3392050375588816791et_rat
@ ( image_1318631078669633143et_rat
@ ^ [Y2: set_o] : ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_298_UN__extend__simps_I8_J,axiom,
! [B2: rat > set_o,A2: set_set_rat] :
( ( comple90263536869209701_set_o
@ ( image_set_rat_set_o
@ ^ [Y2: set_rat] : ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ Y2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ ( comple3890839924845867745et_rat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_299_UN__extend__simps_I8_J,axiom,
! [B2: set_rat > set_o,A2: set_set_set_rat] :
( ( comple90263536869209701_set_o
@ ( image_9166368412354806125_set_o
@ ^ [Y2: set_set_rat] : ( comple90263536869209701_set_o @ ( image_set_rat_set_o @ B2 @ Y2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_set_rat_set_o @ B2 @ ( comple3392050375588816791et_rat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_300_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_o,A2: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o
@ ^ [Y2: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ Y2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_301_UN__extend__simps_I9_J,axiom,
! [C: dedekind_preal > set_rat,B2: dedekind_preal > set_Dedekind_preal,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ ( B2 @ X4 ) ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_302_UN__extend__simps_I9_J,axiom,
! [C: dedekind_preal > set_rat,B2: rat > set_Dedekind_preal,A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ ( B2 @ X4 ) ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_303_UN__extend__simps_I9_J,axiom,
! [C: rat > set_rat,B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C @ ( B2 @ X4 ) ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_304_UN__extend__simps_I9_J,axiom,
! [C: rat > set_rat,B2: rat > set_rat,A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C @ ( B2 @ X4 ) ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ C @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_305_UN__extend__simps_I9_J,axiom,
! [C: $o > set_rat,B2: dedekind_preal > set_o,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( comple3890839924845867745et_rat @ ( image_o_set_rat @ C @ ( B2 @ X4 ) ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_o_set_rat @ C @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_306_UN__extend__simps_I9_J,axiom,
! [C: $o > set_rat,B2: rat > set_o,A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( comple3890839924845867745et_rat @ ( image_o_set_rat @ C @ ( B2 @ X4 ) ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_o_set_rat @ C @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_307_UN__extend__simps_I9_J,axiom,
! [C: rat > set_o,B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( comple90263536869209701_set_o @ ( image_rat_set_o @ C @ ( B2 @ X4 ) ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_rat_set_o @ C @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_308_UN__extend__simps_I9_J,axiom,
! [C: rat > set_o,B2: rat > set_rat,A2: set_rat] :
( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( comple90263536869209701_set_o @ ( image_rat_set_o @ C @ ( B2 @ X4 ) ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_rat_set_o @ C @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_309_UN__extend__simps_I9_J,axiom,
! [C: $o > set_o,B2: rat > set_o,A2: set_rat] :
( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( B2 @ X4 ) ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ C @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_310_UN__extend__simps_I9_J,axiom,
! [C: dedekind_preal > set_rat,B2: set_rat > set_Dedekind_preal,A2: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ ( B2 @ X4 ) ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ C @ ( comple800427960086186211_preal @ ( image_7647003310076334625_preal @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_311_Rep__preal__exists__bound,axiom,
! [X2: dedekind_preal] :
? [X3: rat] :
( ( ord_less_rat @ zero_zero_rat @ X3 )
& ~ ( member_rat @ X3 @ ( dedekind_Rep_preal @ X2 ) ) ) ).
% Rep_preal_exists_bound
thf(fact_312_not__in__Rep__preal__ub,axiom,
! [X: rat,Xa: dedekind_preal,Y: rat] :
( ~ ( member_rat @ X @ ( dedekind_Rep_preal @ Xa ) )
=> ( ( member_rat @ Y @ ( dedekind_Rep_preal @ Xa ) )
=> ( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_rat @ Y @ X ) ) ) ) ).
% not_in_Rep_preal_ub
thf(fact_313_SUP__lessD,axiom,
! [F: rat > $o,A2: set_rat,Y: $o,I: rat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A2 ) ) @ Y )
=> ( ( member_rat @ I @ A2 )
=> ( ord_less_o @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_314_SUP__lessD,axiom,
! [F: $o > $o,A2: set_o,Y: $o,I: $o] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ Y )
=> ( ( member_o @ I @ A2 )
=> ( ord_less_o @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_315_SUP__lessD,axiom,
! [F: dedekind_preal > $o,A2: set_Dedekind_preal,Y: $o,I: dedekind_preal] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A2 ) ) @ Y )
=> ( ( member6871284927547481791_preal @ I @ A2 )
=> ( ord_less_o @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_316_SUP__lessD,axiom,
! [F: rat > set_rat,A2: set_rat,Y: set_rat,I: rat] :
( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A2 ) ) @ Y )
=> ( ( member_rat @ I @ A2 )
=> ( ord_less_set_rat @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_317_SUP__lessD,axiom,
! [F: $o > set_rat,A2: set_o,Y: set_rat,I: $o] :
( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A2 ) ) @ Y )
=> ( ( member_o @ I @ A2 )
=> ( ord_less_set_rat @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_318_SUP__lessD,axiom,
! [F: dedekind_preal > set_rat,A2: set_Dedekind_preal,Y: set_rat,I: dedekind_preal] :
( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) ) @ Y )
=> ( ( member6871284927547481791_preal @ I @ A2 )
=> ( ord_less_set_rat @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_319_SUP__lessD,axiom,
! [F: set_rat > $o,A2: set_set_rat,Y: $o,I: set_rat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ A2 ) ) @ Y )
=> ( ( member_set_rat @ I @ A2 )
=> ( ord_less_o @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_320_SUP__lessD,axiom,
! [F: rat > set_o,A2: set_rat,Y: set_o,I: rat] :
( ( ord_less_set_o @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ A2 ) ) @ Y )
=> ( ( member_rat @ I @ A2 )
=> ( ord_less_set_o @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_321_SUP__lessD,axiom,
! [F: $o > set_o,A2: set_o,Y: set_o,I: $o] :
( ( ord_less_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ Y )
=> ( ( member_o @ I @ A2 )
=> ( ord_less_set_o @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_322_SUP__lessD,axiom,
! [F: dedekind_preal > set_o,A2: set_Dedekind_preal,Y: set_o,I: dedekind_preal] :
( ( ord_less_set_o @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ A2 ) ) @ Y )
=> ( ( member6871284927547481791_preal @ I @ A2 )
=> ( ord_less_set_o @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_323_image__UN,axiom,
! [F: rat > rat,B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( image_rat_rat @ F @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( image_rat_rat @ F @ ( B2 @ X4 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_324_image__UN,axiom,
! [F: rat > rat,B2: rat > set_rat,A2: set_rat] :
( ( image_rat_rat @ F @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( image_rat_rat @ F @ ( B2 @ X4 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_325_image__UN,axiom,
! [F: rat > $o,B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( image_rat_o @ F @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( image_rat_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_326_image__UN,axiom,
! [F: rat > $o,B2: rat > set_rat,A2: set_rat] :
( ( image_rat_o @ F @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( image_rat_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_327_image__UN,axiom,
! [F: $o > rat,B2: dedekind_preal > set_o,A2: set_Dedekind_preal] :
( ( image_o_rat @ F @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ A2 ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( image_o_rat @ F @ ( B2 @ X4 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_328_image__UN,axiom,
! [F: $o > rat,B2: rat > set_o,A2: set_rat] :
( ( image_o_rat @ F @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( image_o_rat @ F @ ( B2 @ X4 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_329_image__UN,axiom,
! [F: $o > $o,B2: rat > set_o,A2: set_rat] :
( ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( image_o_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_330_image__UN,axiom,
! [F: dedekind_preal > set_rat,B2: rat > set_Dedekind_preal,A2: set_rat] :
( ( image_2231269897757205599et_rat @ F @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B2 @ A2 ) ) )
= ( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [X4: rat] : ( image_2231269897757205599et_rat @ F @ ( B2 @ X4 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_331_image__UN,axiom,
! [F: rat > set_o,B2: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( image_rat_set_o @ F @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_9033230029842370631_set_o
@ ^ [X4: dedekind_preal] : ( image_rat_set_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_332_image__UN,axiom,
! [F: rat > set_o,B2: rat > set_rat,A2: set_rat] :
( ( image_rat_set_o @ F @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) )
= ( comple4436988014476444997_set_o
@ ( image_rat_set_set_o
@ ^ [X4: rat] : ( image_rat_set_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_333_UN__extend__simps_I10_J,axiom,
! [B2: dedekind_preal > set_rat,F: dedekind_preal > dedekind_preal,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [A4: dedekind_preal] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ ( image_4420143364984939013_preal @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_334_UN__extend__simps_I10_J,axiom,
! [B2: rat > set_rat,F: dedekind_preal > rat,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [A4: dedekind_preal] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ ( image_6572656775217987753al_rat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_335_UN__extend__simps_I10_J,axiom,
! [B2: dedekind_preal > set_rat,F: rat > dedekind_preal,A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [A4: rat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ ( image_7112546572676366711_preal @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_336_UN__extend__simps_I10_J,axiom,
! [B2: rat > set_rat,F: rat > rat,A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [A4: rat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ ( image_rat_rat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_337_UN__extend__simps_I10_J,axiom,
! [B2: rat > set_o,F: rat > rat,A2: set_rat] :
( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [A4: rat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ ( image_rat_rat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_338_UN__extend__simps_I10_J,axiom,
! [B2: set_rat > set_rat,F: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [A4: dedekind_preal] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ ( image_2231269897757205599et_rat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_339_UN__extend__simps_I10_J,axiom,
! [B2: dedekind_preal > set_rat,F: set_rat > dedekind_preal,A2: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [A4: set_rat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ ( image_6579680541524808769_preal @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_340_UN__extend__simps_I10_J,axiom,
! [B2: rat > set_rat,F: set_rat > rat,A2: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [A4: set_rat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ ( image_set_rat_rat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_341_UN__extend__simps_I10_J,axiom,
! [B2: set_o > set_rat,F: rat > set_o,A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [A4: rat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_set_o_set_rat @ B2 @ ( image_rat_set_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_342_UN__extend__simps_I10_J,axiom,
! [B2: set_rat > set_rat,F: rat > set_rat,A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [A4: rat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ ( image_rat_set_rat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_343_cSUP__const,axiom,
! [A2: set_Dedekind_preal,C2: $o] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [X4: dedekind_preal] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_344_cSUP__const,axiom,
! [A2: set_rat,C2: $o] :
( ( A2 != bot_bot_set_rat )
=> ( ( complete_Sup_Sup_o
@ ( image_rat_o
@ ^ [X4: rat] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_345_cSUP__const,axiom,
! [A2: set_o,C2: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X4: $o] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_346_cSUP__const,axiom,
! [A2: set_Dedekind_preal,C2: set_rat] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_347_cSUP__const,axiom,
! [A2: set_rat,C2: set_rat] :
( ( A2 != bot_bot_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_348_cSUP__const,axiom,
! [A2: set_o,C2: set_rat] :
( ( A2 != bot_bot_set_o )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_349_cSUP__const,axiom,
! [A2: set_Dedekind_preal,C2: set_o] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_350_cSUP__const,axiom,
! [A2: set_rat,C2: set_o] :
( ( A2 != bot_bot_set_rat )
=> ( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_351_cSUP__const,axiom,
! [A2: set_o,C2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_352_cSUP__const,axiom,
! [A2: set_set_rat,C2: set_rat] :
( ( A2 != bot_bot_set_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : C2
@ A2 ) )
= C2 ) ) ).
% cSUP_const
thf(fact_353_image__is__empty,axiom,
! [F: dedekind_preal > dedekind_preal,A2: set_Dedekind_preal] :
( ( ( image_4420143364984939013_preal @ F @ A2 )
= bot_bo4848840305443107682_preal )
= ( A2 = bot_bo4848840305443107682_preal ) ) ).
% image_is_empty
thf(fact_354_image__is__empty,axiom,
! [F: rat > dedekind_preal,A2: set_rat] :
( ( ( image_7112546572676366711_preal @ F @ A2 )
= bot_bo4848840305443107682_preal )
= ( A2 = bot_bot_set_rat ) ) ).
% image_is_empty
thf(fact_355_image__is__empty,axiom,
! [F: $o > dedekind_preal,A2: set_o] :
( ( ( image_3026614506929738499_preal @ F @ A2 )
= bot_bo4848840305443107682_preal )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_356_image__is__empty,axiom,
! [F: dedekind_preal > rat,A2: set_Dedekind_preal] :
( ( ( image_6572656775217987753al_rat @ F @ A2 )
= bot_bot_set_rat )
= ( A2 = bot_bo4848840305443107682_preal ) ) ).
% image_is_empty
thf(fact_357_image__is__empty,axiom,
! [F: rat > rat,A2: set_rat] :
( ( ( image_rat_rat @ F @ A2 )
= bot_bot_set_rat )
= ( A2 = bot_bot_set_rat ) ) ).
% image_is_empty
thf(fact_358_image__is__empty,axiom,
! [F: $o > rat,A2: set_o] :
( ( ( image_o_rat @ F @ A2 )
= bot_bot_set_rat )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_359_image__is__empty,axiom,
! [F: dedekind_preal > $o,A2: set_Dedekind_preal] :
( ( ( image_248103505173020295real_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bo4848840305443107682_preal ) ) ).
% image_is_empty
thf(fact_360_image__is__empty,axiom,
! [F: rat > $o,A2: set_rat] :
( ( ( image_rat_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_rat ) ) ).
% image_is_empty
thf(fact_361_image__is__empty,axiom,
! [F: $o > $o,A2: set_o] :
( ( ( image_o_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_362_image__is__empty,axiom,
! [F: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( ( image_2231269897757205599et_rat @ F @ A2 )
= bot_bot_set_set_rat )
= ( A2 = bot_bo4848840305443107682_preal ) ) ).
% image_is_empty
thf(fact_363_empty__is__image,axiom,
! [F: dedekind_preal > dedekind_preal,A2: set_Dedekind_preal] :
( ( bot_bo4848840305443107682_preal
= ( image_4420143364984939013_preal @ F @ A2 ) )
= ( A2 = bot_bo4848840305443107682_preal ) ) ).
% empty_is_image
thf(fact_364_empty__is__image,axiom,
! [F: rat > dedekind_preal,A2: set_rat] :
( ( bot_bo4848840305443107682_preal
= ( image_7112546572676366711_preal @ F @ A2 ) )
= ( A2 = bot_bot_set_rat ) ) ).
% empty_is_image
thf(fact_365_empty__is__image,axiom,
! [F: $o > dedekind_preal,A2: set_o] :
( ( bot_bo4848840305443107682_preal
= ( image_3026614506929738499_preal @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_366_empty__is__image,axiom,
! [F: dedekind_preal > rat,A2: set_Dedekind_preal] :
( ( bot_bot_set_rat
= ( image_6572656775217987753al_rat @ F @ A2 ) )
= ( A2 = bot_bo4848840305443107682_preal ) ) ).
% empty_is_image
thf(fact_367_empty__is__image,axiom,
! [F: rat > rat,A2: set_rat] :
( ( bot_bot_set_rat
= ( image_rat_rat @ F @ A2 ) )
= ( A2 = bot_bot_set_rat ) ) ).
% empty_is_image
thf(fact_368_empty__is__image,axiom,
! [F: $o > rat,A2: set_o] :
( ( bot_bot_set_rat
= ( image_o_rat @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_369_empty__is__image,axiom,
! [F: dedekind_preal > $o,A2: set_Dedekind_preal] :
( ( bot_bot_set_o
= ( image_248103505173020295real_o @ F @ A2 ) )
= ( A2 = bot_bo4848840305443107682_preal ) ) ).
% empty_is_image
thf(fact_370_empty__is__image,axiom,
! [F: rat > $o,A2: set_rat] :
( ( bot_bot_set_o
= ( image_rat_o @ F @ A2 ) )
= ( A2 = bot_bot_set_rat ) ) ).
% empty_is_image
thf(fact_371_empty__is__image,axiom,
! [F: $o > $o,A2: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_372_empty__is__image,axiom,
! [F: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( bot_bot_set_set_rat
= ( image_2231269897757205599et_rat @ F @ A2 ) )
= ( A2 = bot_bo4848840305443107682_preal ) ) ).
% empty_is_image
thf(fact_373_image__empty,axiom,
! [F: dedekind_preal > dedekind_preal] :
( ( image_4420143364984939013_preal @ F @ bot_bo4848840305443107682_preal )
= bot_bo4848840305443107682_preal ) ).
% image_empty
thf(fact_374_image__empty,axiom,
! [F: dedekind_preal > rat] :
( ( image_6572656775217987753al_rat @ F @ bot_bo4848840305443107682_preal )
= bot_bot_set_rat ) ).
% image_empty
thf(fact_375_image__empty,axiom,
! [F: dedekind_preal > $o] :
( ( image_248103505173020295real_o @ F @ bot_bo4848840305443107682_preal )
= bot_bot_set_o ) ).
% image_empty
thf(fact_376_image__empty,axiom,
! [F: rat > dedekind_preal] :
( ( image_7112546572676366711_preal @ F @ bot_bot_set_rat )
= bot_bo4848840305443107682_preal ) ).
% image_empty
thf(fact_377_image__empty,axiom,
! [F: rat > rat] :
( ( image_rat_rat @ F @ bot_bot_set_rat )
= bot_bot_set_rat ) ).
% image_empty
thf(fact_378_image__empty,axiom,
! [F: rat > $o] :
( ( image_rat_o @ F @ bot_bot_set_rat )
= bot_bot_set_o ) ).
% image_empty
thf(fact_379_image__empty,axiom,
! [F: $o > dedekind_preal] :
( ( image_3026614506929738499_preal @ F @ bot_bot_set_o )
= bot_bo4848840305443107682_preal ) ).
% image_empty
thf(fact_380_image__empty,axiom,
! [F: $o > rat] :
( ( image_o_rat @ F @ bot_bot_set_o )
= bot_bot_set_rat ) ).
% image_empty
thf(fact_381_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_382_image__empty,axiom,
! [F: dedekind_preal > set_rat] :
( ( image_2231269897757205599et_rat @ F @ bot_bo4848840305443107682_preal )
= bot_bot_set_set_rat ) ).
% image_empty
thf(fact_383_image__ident,axiom,
! [Y3: set_set_rat] :
( ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : X4
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_384_Dedekind__Real_Ocut__def,axiom,
( dedekind_cut
= ( ^ [A3: set_rat] :
( ( ord_less_set_rat @ bot_bot_set_rat @ A3 )
& ( ord_less_set_rat @ A3 @ ( set_or575021546402375026an_rat @ zero_zero_rat ) )
& ! [X4: rat] :
( ( member_rat @ X4 @ A3 )
=> ( ! [Z: rat] :
( ( ( ord_less_rat @ zero_zero_rat @ Z )
& ( ord_less_rat @ Z @ X4 ) )
=> ( member_rat @ Z @ A3 ) )
& ? [Y2: rat] :
( ( member_rat @ Y2 @ A3 )
& ( ord_less_rat @ X4 @ Y2 ) ) ) ) ) ) ) ).
% Dedekind_Real.cut_def
thf(fact_385_empty__iff,axiom,
! [C2: set_rat] :
~ ( member_set_rat @ C2 @ bot_bot_set_set_rat ) ).
% empty_iff
thf(fact_386_empty__iff,axiom,
! [C2: dedekind_preal] :
~ ( member6871284927547481791_preal @ C2 @ bot_bo4848840305443107682_preal ) ).
% empty_iff
thf(fact_387_empty__iff,axiom,
! [C2: rat] :
~ ( member_rat @ C2 @ bot_bot_set_rat ) ).
% empty_iff
thf(fact_388_empty__iff,axiom,
! [C2: $o] :
~ ( member_o @ C2 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_389_all__not__in__conv,axiom,
! [A2: set_set_rat] :
( ( ! [X4: set_rat] :
~ ( member_set_rat @ X4 @ A2 ) )
= ( A2 = bot_bot_set_set_rat ) ) ).
% all_not_in_conv
thf(fact_390_all__not__in__conv,axiom,
! [A2: set_Dedekind_preal] :
( ( ! [X4: dedekind_preal] :
~ ( member6871284927547481791_preal @ X4 @ A2 ) )
= ( A2 = bot_bo4848840305443107682_preal ) ) ).
% all_not_in_conv
thf(fact_391_all__not__in__conv,axiom,
! [A2: set_rat] :
( ( ! [X4: rat] :
~ ( member_rat @ X4 @ A2 ) )
= ( A2 = bot_bot_set_rat ) ) ).
% all_not_in_conv
thf(fact_392_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X4: $o] :
~ ( member_o @ X4 @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_393_Collect__empty__eq,axiom,
! [P: set_rat > $o] :
( ( ( collect_set_rat @ P )
= bot_bot_set_set_rat )
= ( ! [X4: set_rat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_394_Collect__empty__eq,axiom,
! [P: dedekind_preal > $o] :
( ( ( collec1132657498972982273_preal @ P )
= bot_bo4848840305443107682_preal )
= ( ! [X4: dedekind_preal] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_395_Collect__empty__eq,axiom,
! [P: rat > $o] :
( ( ( collect_rat @ P )
= bot_bot_set_rat )
= ( ! [X4: rat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_396_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X4: $o] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_397_empty__Collect__eq,axiom,
! [P: set_rat > $o] :
( ( bot_bot_set_set_rat
= ( collect_set_rat @ P ) )
= ( ! [X4: set_rat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_398_empty__Collect__eq,axiom,
! [P: dedekind_preal > $o] :
( ( bot_bo4848840305443107682_preal
= ( collec1132657498972982273_preal @ P ) )
= ( ! [X4: dedekind_preal] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_399_empty__Collect__eq,axiom,
! [P: rat > $o] :
( ( bot_bot_set_rat
= ( collect_rat @ P ) )
= ( ! [X4: rat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_400_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X4: $o] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_401_image__eqI,axiom,
! [B: rat,F: rat > rat,X: rat,A2: set_rat] :
( ( B
= ( F @ X ) )
=> ( ( member_rat @ X @ A2 )
=> ( member_rat @ B @ ( image_rat_rat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_402_image__eqI,axiom,
! [B: $o,F: rat > $o,X: rat,A2: set_rat] :
( ( B
= ( F @ X ) )
=> ( ( member_rat @ X @ A2 )
=> ( member_o @ B @ ( image_rat_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_403_image__eqI,axiom,
! [B: dedekind_preal,F: rat > dedekind_preal,X: rat,A2: set_rat] :
( ( B
= ( F @ X ) )
=> ( ( member_rat @ X @ A2 )
=> ( member6871284927547481791_preal @ B @ ( image_7112546572676366711_preal @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_404_image__eqI,axiom,
! [B: rat,F: $o > rat,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_rat @ B @ ( image_o_rat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_405_image__eqI,axiom,
! [B: $o,F: $o > $o,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_406_image__eqI,axiom,
! [B: dedekind_preal,F: $o > dedekind_preal,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member6871284927547481791_preal @ B @ ( image_3026614506929738499_preal @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_407_image__eqI,axiom,
! [B: rat,F: dedekind_preal > rat,X: dedekind_preal,A2: set_Dedekind_preal] :
( ( B
= ( F @ X ) )
=> ( ( member6871284927547481791_preal @ X @ A2 )
=> ( member_rat @ B @ ( image_6572656775217987753al_rat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_408_image__eqI,axiom,
! [B: $o,F: dedekind_preal > $o,X: dedekind_preal,A2: set_Dedekind_preal] :
( ( B
= ( F @ X ) )
=> ( ( member6871284927547481791_preal @ X @ A2 )
=> ( member_o @ B @ ( image_248103505173020295real_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_409_image__eqI,axiom,
! [B: dedekind_preal,F: dedekind_preal > dedekind_preal,X: dedekind_preal,A2: set_Dedekind_preal] :
( ( B
= ( F @ X ) )
=> ( ( member6871284927547481791_preal @ X @ A2 )
=> ( member6871284927547481791_preal @ B @ ( image_4420143364984939013_preal @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_410_image__eqI,axiom,
! [B: set_o,F: rat > set_o,X: rat,A2: set_rat] :
( ( B
= ( F @ X ) )
=> ( ( member_rat @ X @ A2 )
=> ( member_set_o @ B @ ( image_rat_set_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_411_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_412_preal__Ex__mem,axiom,
! [A2: set_rat] :
( ( dedekind_cut @ A2 )
=> ? [X3: rat] : ( member_rat @ X3 @ A2 ) ) ).
% preal_Ex_mem
thf(fact_413_bot__set__def,axiom,
( bot_bot_set_set_rat
= ( collect_set_rat @ bot_bot_set_rat_o ) ) ).
% bot_set_def
thf(fact_414_bot__set__def,axiom,
( bot_bo4848840305443107682_preal
= ( collec1132657498972982273_preal @ bot_bo4815710026557797371real_o ) ) ).
% bot_set_def
thf(fact_415_bot__set__def,axiom,
( bot_bot_set_rat
= ( collect_rat @ bot_bot_rat_o ) ) ).
% bot_set_def
thf(fact_416_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_417_preal__exists__greater,axiom,
! [A2: set_rat,Y: rat] :
( ( dedekind_cut @ A2 )
=> ( ( member_rat @ Y @ A2 )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ( ord_less_rat @ Y @ X3 ) ) ) ) ).
% preal_exists_greater
thf(fact_418_Rep__preal,axiom,
! [X: dedekind_preal] : ( member_set_rat @ ( dedekind_Rep_preal @ X ) @ ( collect_set_rat @ dedekind_cut ) ) ).
% Rep_preal
thf(fact_419_Rep__preal__cases,axiom,
! [Y: set_rat] :
( ( member_set_rat @ Y @ ( collect_set_rat @ dedekind_cut ) )
=> ~ ! [X3: dedekind_preal] :
( Y
!= ( dedekind_Rep_preal @ X3 ) ) ) ).
% Rep_preal_cases
thf(fact_420_Rep__preal__induct,axiom,
! [Y: set_rat,P: set_rat > $o] :
( ( member_set_rat @ Y @ ( collect_set_rat @ dedekind_cut ) )
=> ( ! [X3: dedekind_preal] : ( P @ ( dedekind_Rep_preal @ X3 ) )
=> ( P @ Y ) ) ) ).
% Rep_preal_induct
thf(fact_421_cut__Rep__preal,axiom,
! [X: dedekind_preal] : ( dedekind_cut @ ( dedekind_Rep_preal @ X ) ) ).
% cut_Rep_preal
thf(fact_422_preal__imp__pos,axiom,
! [A2: set_rat,R2: rat] :
( ( dedekind_cut @ A2 )
=> ( ( member_rat @ R2 @ A2 )
=> ( ord_less_rat @ zero_zero_rat @ R2 ) ) ) ).
% preal_imp_pos
thf(fact_423_preal__nonempty,axiom,
! [A2: set_rat] :
( ( dedekind_cut @ A2 )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ( ord_less_rat @ zero_zero_rat @ X3 ) ) ) ).
% preal_nonempty
thf(fact_424_not__in__preal__ub,axiom,
! [A2: set_rat,X: rat,Y: rat] :
( ( dedekind_cut @ A2 )
=> ( ~ ( member_rat @ X @ A2 )
=> ( ( member_rat @ Y @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_rat @ Y @ X ) ) ) ) ) ).
% not_in_preal_ub
thf(fact_425_preal__exists__bound,axiom,
! [A2: set_rat] :
( ( dedekind_cut @ A2 )
=> ? [X3: rat] :
( ( ord_less_rat @ zero_zero_rat @ X3 )
& ~ ( member_rat @ X3 @ A2 ) ) ) ).
% preal_exists_bound
thf(fact_426_preal__downwards__closed,axiom,
! [A2: set_rat,Y: rat,Z2: rat] :
( ( dedekind_cut @ A2 )
=> ( ( member_rat @ Y @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_rat @ Z2 @ Y )
=> ( member_rat @ Z2 @ A2 ) ) ) ) ) ).
% preal_downwards_closed
thf(fact_427_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_428_imageI,axiom,
! [X: rat,A2: set_rat,F: rat > rat] :
( ( member_rat @ X @ A2 )
=> ( member_rat @ ( F @ X ) @ ( image_rat_rat @ F @ A2 ) ) ) ).
% imageI
thf(fact_429_imageI,axiom,
! [X: rat,A2: set_rat,F: rat > $o] :
( ( member_rat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_rat_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_430_imageI,axiom,
! [X: rat,A2: set_rat,F: rat > dedekind_preal] :
( ( member_rat @ X @ A2 )
=> ( member6871284927547481791_preal @ ( F @ X ) @ ( image_7112546572676366711_preal @ F @ A2 ) ) ) ).
% imageI
thf(fact_431_imageI,axiom,
! [X: $o,A2: set_o,F: $o > rat] :
( ( member_o @ X @ A2 )
=> ( member_rat @ ( F @ X ) @ ( image_o_rat @ F @ A2 ) ) ) ).
% imageI
thf(fact_432_imageI,axiom,
! [X: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_o_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_433_imageI,axiom,
! [X: $o,A2: set_o,F: $o > dedekind_preal] :
( ( member_o @ X @ A2 )
=> ( member6871284927547481791_preal @ ( F @ X ) @ ( image_3026614506929738499_preal @ F @ A2 ) ) ) ).
% imageI
thf(fact_434_imageI,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,F: dedekind_preal > rat] :
( ( member6871284927547481791_preal @ X @ A2 )
=> ( member_rat @ ( F @ X ) @ ( image_6572656775217987753al_rat @ F @ A2 ) ) ) ).
% imageI
thf(fact_435_imageI,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,F: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_248103505173020295real_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_436_imageI,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,F: dedekind_preal > dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A2 )
=> ( member6871284927547481791_preal @ ( F @ X ) @ ( image_4420143364984939013_preal @ F @ A2 ) ) ) ).
% imageI
thf(fact_437_imageI,axiom,
! [X: rat,A2: set_rat,F: rat > set_o] :
( ( member_rat @ X @ A2 )
=> ( member_set_o @ ( F @ X ) @ ( image_rat_set_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_438_image__iff,axiom,
! [Z2: set_set_rat,F: set_rat > set_set_rat,A2: set_set_rat] :
( ( member_set_set_rat @ Z2 @ ( image_9010471772881432025et_rat @ F @ A2 ) )
= ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_439_image__iff,axiom,
! [Z2: set_o,F: rat > set_o,A2: set_rat] :
( ( member_set_o @ Z2 @ ( image_rat_set_o @ F @ A2 ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_440_image__iff,axiom,
! [Z2: set_set_rat,F: rat > set_set_rat,A2: set_rat] :
( ( member_set_set_rat @ Z2 @ ( image_5169559123115255331et_rat @ F @ A2 ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_441_image__iff,axiom,
! [Z2: set_rat,F: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( member_set_rat @ Z2 @ ( image_2231269897757205599et_rat @ F @ A2 ) )
= ( ? [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_442_image__iff,axiom,
! [Z2: set_rat,F: set_rat > set_rat,A2: set_set_rat] :
( ( member_set_rat @ Z2 @ ( image_3939399684171694371et_rat @ F @ A2 ) )
= ( ? [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_443_image__iff,axiom,
! [Z2: set_rat,F: rat > set_rat,A2: set_rat] :
( ( member_set_rat @ Z2 @ ( image_rat_set_rat @ F @ A2 ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_444_bex__imageD,axiom,
! [F: dedekind_preal > set_rat,A2: set_Dedekind_preal,P: set_rat > $o] :
( ? [X6: set_rat] :
( ( member_set_rat @ X6 @ ( image_2231269897757205599et_rat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_445_bex__imageD,axiom,
! [F: set_rat > set_set_rat,A2: set_set_rat,P: set_set_rat > $o] :
( ? [X6: set_set_rat] :
( ( member_set_set_rat @ X6 @ ( image_9010471772881432025et_rat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: set_rat] :
( ( member_set_rat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_446_bex__imageD,axiom,
! [F: set_rat > set_rat,A2: set_set_rat,P: set_rat > $o] :
( ? [X6: set_rat] :
( ( member_set_rat @ X6 @ ( image_3939399684171694371et_rat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: set_rat] :
( ( member_set_rat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_447_bex__imageD,axiom,
! [F: rat > set_o,A2: set_rat,P: set_o > $o] :
( ? [X6: set_o] :
( ( member_set_o @ X6 @ ( image_rat_set_o @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_448_bex__imageD,axiom,
! [F: rat > set_set_rat,A2: set_rat,P: set_set_rat > $o] :
( ? [X6: set_set_rat] :
( ( member_set_set_rat @ X6 @ ( image_5169559123115255331et_rat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_449_bex__imageD,axiom,
! [F: rat > set_rat,A2: set_rat,P: set_rat > $o] :
( ? [X6: set_rat] :
( ( member_set_rat @ X6 @ ( image_rat_set_rat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_450_image__cong,axiom,
! [M: set_rat,N: set_rat,F: rat > set_o,G: rat > set_o] :
( ( M = N )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_rat_set_o @ F @ M )
= ( image_rat_set_o @ G @ N ) ) ) ) ).
% image_cong
thf(fact_451_image__cong,axiom,
! [M: set_rat,N: set_rat,F: rat > set_set_rat,G: rat > set_set_rat] :
( ( M = N )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_5169559123115255331et_rat @ F @ M )
= ( image_5169559123115255331et_rat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_452_image__cong,axiom,
! [M: set_rat,N: set_rat,F: rat > set_rat,G: rat > set_rat] :
( ( M = N )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_rat_set_rat @ F @ M )
= ( image_rat_set_rat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_453_image__cong,axiom,
! [M: set_set_rat,N: set_set_rat,F: set_rat > set_set_rat,G: set_rat > set_set_rat] :
( ( M = N )
=> ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_9010471772881432025et_rat @ F @ M )
= ( image_9010471772881432025et_rat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_454_image__cong,axiom,
! [M: set_set_rat,N: set_set_rat,F: set_rat > set_rat,G: set_rat > set_rat] :
( ( M = N )
=> ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_3939399684171694371et_rat @ F @ M )
= ( image_3939399684171694371et_rat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_455_image__cong,axiom,
! [M: set_Dedekind_preal,N: set_Dedekind_preal,F: dedekind_preal > set_rat,G: dedekind_preal > set_rat] :
( ( M = N )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_2231269897757205599et_rat @ F @ M )
= ( image_2231269897757205599et_rat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_456_ball__imageD,axiom,
! [F: dedekind_preal > set_rat,A2: set_Dedekind_preal,P: set_rat > $o] :
( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ ( image_2231269897757205599et_rat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: dedekind_preal] :
( ( member6871284927547481791_preal @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_457_ball__imageD,axiom,
! [F: set_rat > set_set_rat,A2: set_set_rat,P: set_set_rat > $o] :
( ! [X3: set_set_rat] :
( ( member_set_set_rat @ X3 @ ( image_9010471772881432025et_rat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: set_rat] :
( ( member_set_rat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_458_ball__imageD,axiom,
! [F: set_rat > set_rat,A2: set_set_rat,P: set_rat > $o] :
( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ ( image_3939399684171694371et_rat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: set_rat] :
( ( member_set_rat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_459_ball__imageD,axiom,
! [F: rat > set_o,A2: set_rat,P: set_o > $o] :
( ! [X3: set_o] :
( ( member_set_o @ X3 @ ( image_rat_set_o @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: rat] :
( ( member_rat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_460_ball__imageD,axiom,
! [F: rat > set_set_rat,A2: set_rat,P: set_set_rat > $o] :
( ! [X3: set_set_rat] :
( ( member_set_set_rat @ X3 @ ( image_5169559123115255331et_rat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: rat] :
( ( member_rat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_461_ball__imageD,axiom,
! [F: rat > set_rat,A2: set_rat,P: set_rat > $o] :
( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ ( image_rat_set_rat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X6: rat] :
( ( member_rat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_462_rev__image__eqI,axiom,
! [X: rat,A2: set_rat,B: rat,F: rat > rat] :
( ( member_rat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_rat_rat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_463_rev__image__eqI,axiom,
! [X: rat,A2: set_rat,B: $o,F: rat > $o] :
( ( member_rat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_rat_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_464_rev__image__eqI,axiom,
! [X: rat,A2: set_rat,B: dedekind_preal,F: rat > dedekind_preal] :
( ( member_rat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member6871284927547481791_preal @ B @ ( image_7112546572676366711_preal @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_465_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: rat,F: $o > rat] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_o_rat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_466_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: $o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_467_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: dedekind_preal,F: $o > dedekind_preal] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member6871284927547481791_preal @ B @ ( image_3026614506929738499_preal @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_468_rev__image__eqI,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,B: rat,F: dedekind_preal > rat] :
( ( member6871284927547481791_preal @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_rat @ B @ ( image_6572656775217987753al_rat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_469_rev__image__eqI,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,B: $o,F: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_248103505173020295real_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_470_rev__image__eqI,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member6871284927547481791_preal @ B @ ( image_4420143364984939013_preal @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_471_rev__image__eqI,axiom,
! [X: rat,A2: set_rat,B: set_o,F: rat > set_o] :
( ( member_rat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_o @ B @ ( image_rat_set_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_472_ex__in__conv,axiom,
! [A2: set_set_rat] :
( ( ? [X4: set_rat] : ( member_set_rat @ X4 @ A2 ) )
= ( A2 != bot_bot_set_set_rat ) ) ).
% ex_in_conv
thf(fact_473_ex__in__conv,axiom,
! [A2: set_Dedekind_preal] :
( ( ? [X4: dedekind_preal] : ( member6871284927547481791_preal @ X4 @ A2 ) )
= ( A2 != bot_bo4848840305443107682_preal ) ) ).
% ex_in_conv
thf(fact_474_ex__in__conv,axiom,
! [A2: set_rat] :
( ( ? [X4: rat] : ( member_rat @ X4 @ A2 ) )
= ( A2 != bot_bot_set_rat ) ) ).
% ex_in_conv
thf(fact_475_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X4: $o] : ( member_o @ X4 @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_476_equals0I,axiom,
! [A2: set_set_rat] :
( ! [Y4: set_rat] :
~ ( member_set_rat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_set_rat ) ) ).
% equals0I
thf(fact_477_equals0I,axiom,
! [A2: set_Dedekind_preal] :
( ! [Y4: dedekind_preal] :
~ ( member6871284927547481791_preal @ Y4 @ A2 )
=> ( A2 = bot_bo4848840305443107682_preal ) ) ).
% equals0I
thf(fact_478_equals0I,axiom,
! [A2: set_rat] :
( ! [Y4: rat] :
~ ( member_rat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_rat ) ) ).
% equals0I
thf(fact_479_equals0I,axiom,
! [A2: set_o] :
( ! [Y4: $o] :
~ ( member_o @ Y4 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_480_equals0D,axiom,
! [A2: set_set_rat,A: set_rat] :
( ( A2 = bot_bot_set_set_rat )
=> ~ ( member_set_rat @ A @ A2 ) ) ).
% equals0D
thf(fact_481_equals0D,axiom,
! [A2: set_Dedekind_preal,A: dedekind_preal] :
( ( A2 = bot_bo4848840305443107682_preal )
=> ~ ( member6871284927547481791_preal @ A @ A2 ) ) ).
% equals0D
thf(fact_482_equals0D,axiom,
! [A2: set_rat,A: rat] :
( ( A2 = bot_bot_set_rat )
=> ~ ( member_rat @ A @ A2 ) ) ).
% equals0D
thf(fact_483_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_484_emptyE,axiom,
! [A: set_rat] :
~ ( member_set_rat @ A @ bot_bot_set_set_rat ) ).
% emptyE
thf(fact_485_emptyE,axiom,
! [A: dedekind_preal] :
~ ( member6871284927547481791_preal @ A @ bot_bo4848840305443107682_preal ) ).
% emptyE
thf(fact_486_emptyE,axiom,
! [A: rat] :
~ ( member_rat @ A @ bot_bot_set_rat ) ).
% emptyE
thf(fact_487_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_488_psubsetD,axiom,
! [A2: set_o,B2: set_o,C2: $o] :
( ( ord_less_set_o @ A2 @ B2 )
=> ( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_489_psubsetD,axiom,
! [A2: set_set_rat,B2: set_set_rat,C2: set_rat] :
( ( ord_less_set_set_rat @ A2 @ B2 )
=> ( ( member_set_rat @ C2 @ A2 )
=> ( member_set_rat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_490_psubsetD,axiom,
! [A2: set_Dedekind_preal,B2: set_Dedekind_preal,C2: dedekind_preal] :
( ( ord_le1802228187270208418_preal @ A2 @ B2 )
=> ( ( member6871284927547481791_preal @ C2 @ A2 )
=> ( member6871284927547481791_preal @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_491_psubsetD,axiom,
! [A2: set_rat,B2: set_rat,C2: rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( member_rat @ C2 @ A2 )
=> ( member_rat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_492_psubset__trans,axiom,
! [A2: set_rat,B2: set_rat,C: set_rat] :
( ( ord_less_set_rat @ A2 @ B2 )
=> ( ( ord_less_set_rat @ B2 @ C )
=> ( ord_less_set_rat @ A2 @ C ) ) ) ).
% psubset_trans
thf(fact_493_imageE,axiom,
! [B: rat,F: rat > rat,A2: set_rat] :
( ( member_rat @ B @ ( image_rat_rat @ F @ A2 ) )
=> ~ ! [X3: rat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_rat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_494_imageE,axiom,
! [B: rat,F: $o > rat,A2: set_o] :
( ( member_rat @ B @ ( image_o_rat @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_495_imageE,axiom,
! [B: rat,F: dedekind_preal > rat,A2: set_Dedekind_preal] :
( ( member_rat @ B @ ( image_6572656775217987753al_rat @ F @ A2 ) )
=> ~ ! [X3: dedekind_preal] :
( ( B
= ( F @ X3 ) )
=> ~ ( member6871284927547481791_preal @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_496_imageE,axiom,
! [B: $o,F: rat > $o,A2: set_rat] :
( ( member_o @ B @ ( image_rat_o @ F @ A2 ) )
=> ~ ! [X3: rat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_rat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_497_imageE,axiom,
! [B: $o,F: $o > $o,A2: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_498_imageE,axiom,
! [B: $o,F: dedekind_preal > $o,A2: set_Dedekind_preal] :
( ( member_o @ B @ ( image_248103505173020295real_o @ F @ A2 ) )
=> ~ ! [X3: dedekind_preal] :
( ( B
= ( F @ X3 ) )
=> ~ ( member6871284927547481791_preal @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_499_imageE,axiom,
! [B: dedekind_preal,F: rat > dedekind_preal,A2: set_rat] :
( ( member6871284927547481791_preal @ B @ ( image_7112546572676366711_preal @ F @ A2 ) )
=> ~ ! [X3: rat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_rat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_500_imageE,axiom,
! [B: dedekind_preal,F: $o > dedekind_preal,A2: set_o] :
( ( member6871284927547481791_preal @ B @ ( image_3026614506929738499_preal @ F @ A2 ) )
=> ~ ! [X3: $o] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_501_imageE,axiom,
! [B: dedekind_preal,F: dedekind_preal > dedekind_preal,A2: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ B @ ( image_4420143364984939013_preal @ F @ A2 ) )
=> ~ ! [X3: dedekind_preal] :
( ( B
= ( F @ X3 ) )
=> ~ ( member6871284927547481791_preal @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_502_imageE,axiom,
! [B: set_o,F: rat > set_o,A2: set_rat] :
( ( member_set_o @ B @ ( image_rat_set_o @ F @ A2 ) )
=> ~ ! [X3: rat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_rat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_503_image__image,axiom,
! [F: dedekind_preal > set_rat,G: dedekind_preal > dedekind_preal,A2: set_Dedekind_preal] :
( ( image_2231269897757205599et_rat @ F @ ( image_4420143364984939013_preal @ G @ A2 ) )
= ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( F @ ( G @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_504_image__image,axiom,
! [F: dedekind_preal > set_rat,G: rat > dedekind_preal,A2: set_rat] :
( ( image_2231269897757205599et_rat @ F @ ( image_7112546572676366711_preal @ G @ A2 ) )
= ( image_rat_set_rat
@ ^ [X4: rat] : ( F @ ( G @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_505_image__image,axiom,
! [F: rat > set_o,G: rat > rat,A2: set_rat] :
( ( image_rat_set_o @ F @ ( image_rat_rat @ G @ A2 ) )
= ( image_rat_set_o
@ ^ [X4: rat] : ( F @ ( G @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_506_image__image,axiom,
! [F: rat > set_rat,G: dedekind_preal > rat,A2: set_Dedekind_preal] :
( ( image_rat_set_rat @ F @ ( image_6572656775217987753al_rat @ G @ A2 ) )
= ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( F @ ( G @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_507_image__image,axiom,
! [F: rat > set_rat,G: rat > rat,A2: set_rat] :
( ( image_rat_set_rat @ F @ ( image_rat_rat @ G @ A2 ) )
= ( image_rat_set_rat
@ ^ [X4: rat] : ( F @ ( G @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_508_image__image,axiom,
! [F: set_o > set_o,G: rat > set_o,A2: set_rat] :
( ( image_set_o_set_o @ F @ ( image_rat_set_o @ G @ A2 ) )
= ( image_rat_set_o
@ ^ [X4: rat] : ( F @ ( G @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_509_image__image,axiom,
! [F: set_o > set_rat,G: rat > set_o,A2: set_rat] :
( ( image_set_o_set_rat @ F @ ( image_rat_set_o @ G @ A2 ) )
= ( image_rat_set_rat
@ ^ [X4: rat] : ( F @ ( G @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_510_image__image,axiom,
! [F: set_rat > set_o,G: rat > set_rat,A2: set_rat] :
( ( image_set_rat_set_o @ F @ ( image_rat_set_rat @ G @ A2 ) )
= ( image_rat_set_o
@ ^ [X4: rat] : ( F @ ( G @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_511_image__image,axiom,
! [F: dedekind_preal > set_rat,G: set_rat > dedekind_preal,A2: set_set_rat] :
( ( image_2231269897757205599et_rat @ F @ ( image_6579680541524808769_preal @ G @ A2 ) )
= ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( F @ ( G @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_512_image__image,axiom,
! [F: set_rat > set_rat,G: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( image_3939399684171694371et_rat @ F @ ( image_2231269897757205599et_rat @ G @ A2 ) )
= ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( F @ ( G @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_513_Compr__image__eq,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_o_o @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_514_Compr__image__eq,axiom,
! [F: dedekind_preal > $o,A2: set_Dedekind_preal,P: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_248103505173020295real_o @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_248103505173020295real_o @ F
@ ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_515_Compr__image__eq,axiom,
! [F: $o > dedekind_preal,A2: set_o,P: dedekind_preal > $o] :
( ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ ( image_3026614506929738499_preal @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_3026614506929738499_preal @ F
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_516_Compr__image__eq,axiom,
! [F: dedekind_preal > dedekind_preal,A2: set_Dedekind_preal,P: dedekind_preal > $o] :
( ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ ( image_4420143364984939013_preal @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_4420143364984939013_preal @ F
@ ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_517_Compr__image__eq,axiom,
! [F: rat > $o,A2: set_rat,P: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_rat_o @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_rat_o @ F
@ ( collect_rat
@ ^ [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_518_Compr__image__eq,axiom,
! [F: rat > dedekind_preal,A2: set_rat,P: dedekind_preal > $o] :
( ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ ( image_7112546572676366711_preal @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_7112546572676366711_preal @ F
@ ( collect_rat
@ ^ [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_519_Compr__image__eq,axiom,
! [F: $o > rat,A2: set_o,P: rat > $o] :
( ( collect_rat
@ ^ [X4: rat] :
( ( member_rat @ X4 @ ( image_o_rat @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_o_rat @ F
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_520_Compr__image__eq,axiom,
! [F: dedekind_preal > rat,A2: set_Dedekind_preal,P: rat > $o] :
( ( collect_rat
@ ^ [X4: rat] :
( ( member_rat @ X4 @ ( image_6572656775217987753al_rat @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_6572656775217987753al_rat @ F
@ ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_521_Compr__image__eq,axiom,
! [F: rat > rat,A2: set_rat,P: rat > $o] :
( ( collect_rat
@ ^ [X4: rat] :
( ( member_rat @ X4 @ ( image_rat_rat @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_rat_rat @ F
@ ( collect_rat
@ ^ [X4: rat] :
( ( member_rat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_522_Compr__image__eq,axiom,
! [F: set_rat > $o,A2: set_set_rat,P: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_set_rat_o @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_set_rat_o @ F
@ ( collect_set_rat
@ ^ [X4: set_rat] :
( ( member_set_rat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_523_empty__def,axiom,
( bot_bot_set_set_rat
= ( collect_set_rat
@ ^ [X4: set_rat] : $false ) ) ).
% empty_def
thf(fact_524_empty__def,axiom,
( bot_bo4848840305443107682_preal
= ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] : $false ) ) ).
% empty_def
thf(fact_525_empty__def,axiom,
( bot_bot_set_rat
= ( collect_rat
@ ^ [X4: rat] : $false ) ) ).
% empty_def
thf(fact_526_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X4: $o] : $false ) ) ).
% empty_def
thf(fact_527_less__set__def,axiom,
( ord_less_set_o
= ( ^ [A3: set_o,B3: set_o] :
( ord_less_o_o
@ ^ [X4: $o] : ( member_o @ X4 @ A3 )
@ ^ [X4: $o] : ( member_o @ X4 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_528_less__set__def,axiom,
( ord_less_set_set_rat
= ( ^ [A3: set_set_rat,B3: set_set_rat] :
( ord_less_set_rat_o
@ ^ [X4: set_rat] : ( member_set_rat @ X4 @ A3 )
@ ^ [X4: set_rat] : ( member_set_rat @ X4 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_529_less__set__def,axiom,
( ord_le1802228187270208418_preal
= ( ^ [A3: set_Dedekind_preal,B3: set_Dedekind_preal] :
( ord_le1914443497901267899real_o
@ ^ [X4: dedekind_preal] : ( member6871284927547481791_preal @ X4 @ A3 )
@ ^ [X4: dedekind_preal] : ( member6871284927547481791_preal @ X4 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_530_less__set__def,axiom,
( ord_less_set_rat
= ( ^ [A3: set_rat,B3: set_rat] :
( ord_less_rat_o
@ ^ [X4: rat] : ( member_rat @ X4 @ A3 )
@ ^ [X4: rat] : ( member_rat @ X4 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_531_not__psubset__empty,axiom,
! [A2: set_Dedekind_preal] :
~ ( ord_le1802228187270208418_preal @ A2 @ bot_bo4848840305443107682_preal ) ).
% not_psubset_empty
thf(fact_532_not__psubset__empty,axiom,
! [A2: set_o] :
~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).
% not_psubset_empty
thf(fact_533_not__psubset__empty,axiom,
! [A2: set_rat] :
~ ( ord_less_set_rat @ A2 @ bot_bot_set_rat ) ).
% not_psubset_empty
thf(fact_534_mem__diff__set,axiom,
! [R2: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S3 )
=> ( dedekind_cut @ ( dedekind_diff_set @ ( dedekind_Rep_preal @ S3 ) @ ( dedekind_Rep_preal @ R2 ) ) ) ) ).
% mem_diff_set
thf(fact_535_UN__constant__eq,axiom,
! [A: rat,A2: set_rat,F: rat > set_rat,C2: set_rat] :
( ( member_rat @ A @ A2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_536_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_rat,C2: set_rat] :
( ( member_o @ A @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_537_UN__constant__eq,axiom,
! [A: dedekind_preal,A2: set_Dedekind_preal,F: dedekind_preal > set_rat,C2: set_rat] :
( ( member6871284927547481791_preal @ A @ A2 )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_538_UN__constant__eq,axiom,
! [A: rat,A2: set_rat,F: rat > set_o,C2: set_o] :
( ( member_rat @ A @ A2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_539_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_o,C2: set_o] :
( ( member_o @ A @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_540_UN__constant__eq,axiom,
! [A: dedekind_preal,A2: set_Dedekind_preal,F: dedekind_preal > set_o,C2: set_o] :
( ( member6871284927547481791_preal @ A @ A2 )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_541_UN__constant__eq,axiom,
! [A: set_rat,A2: set_set_rat,F: set_rat > set_rat,C2: set_rat] :
( ( member_set_rat @ A @ A2 )
=> ( ! [X3: set_rat] :
( ( member_set_rat @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_542_UN__constant__eq,axiom,
! [A: rat,A2: set_rat,F: rat > set_set_rat,C2: set_set_rat] :
( ( member_rat @ A @ A2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_543_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_set_rat,C2: set_set_rat] :
( ( member_o @ A @ A2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_544_UN__constant__eq,axiom,
! [A: dedekind_preal,A2: set_Dedekind_preal,F: dedekind_preal > set_set_rat,C2: set_set_rat] :
( ( member6871284927547481791_preal @ A @ A2 )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( comple3392050375588816791et_rat @ ( image_8268678932589273365et_rat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_545_SUP__UN__eq,axiom,
! [R2: dedekind_preal > set_rat,S2: set_Dedekind_preal] :
( ( comple4580332206425622756_rat_o
@ ( image_8369558857164962022_rat_o
@ ^ [I2: dedekind_preal,X4: rat] : ( member_rat @ X4 @ ( R2 @ I2 ) )
@ S2 ) )
= ( ^ [X4: rat] : ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ R2 @ S2 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_546_SUP__UN__eq,axiom,
! [R2: set_rat > set_rat,S2: set_set_rat] :
( ( comple4580332206425622756_rat_o
@ ( image_set_rat_rat_o
@ ^ [I2: set_rat,X4: rat] : ( member_rat @ X4 @ ( R2 @ I2 ) )
@ S2 ) )
= ( ^ [X4: rat] : ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ R2 @ S2 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_547_SUP__UN__eq,axiom,
! [R2: rat > set_rat,S2: set_rat] :
( ( comple4580332206425622756_rat_o
@ ( image_rat_rat_o
@ ^ [I2: rat,X4: rat] : ( member_rat @ X4 @ ( R2 @ I2 ) )
@ S2 ) )
= ( ^ [X4: rat] : ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ R2 @ S2 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_548_SUP__UN__eq,axiom,
! [R2: set_rat > set_set_rat,S2: set_set_rat] :
( ( comple3057484590954014638_rat_o
@ ( image_8508488739866243052_rat_o
@ ^ [I2: set_rat,X4: set_rat] : ( member_set_rat @ X4 @ ( R2 @ I2 ) )
@ S2 ) )
= ( ^ [X4: set_rat] : ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ R2 @ S2 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_549_SUP__UN__eq,axiom,
! [R2: rat > set_set_rat,S2: set_rat] :
( ( comple3057484590954014638_rat_o
@ ( image_rat_set_rat_o
@ ^ [I2: rat,X4: set_rat] : ( member_set_rat @ X4 @ ( R2 @ I2 ) )
@ S2 ) )
= ( ^ [X4: set_rat] : ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ R2 @ S2 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_550_SUP__UN__eq,axiom,
! [R2: rat > set_o,S2: set_rat] :
( ( complete_Sup_Sup_o_o
@ ( image_rat_o_o
@ ^ [I2: rat,X4: $o] : ( member_o @ X4 @ ( R2 @ I2 ) )
@ S2 ) )
= ( ^ [X4: $o] : ( member_o @ X4 @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ R2 @ S2 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_551_bot_Oextremum__strict,axiom,
! [A: set_Dedekind_preal] :
~ ( ord_le1802228187270208418_preal @ A @ bot_bo4848840305443107682_preal ) ).
% bot.extremum_strict
thf(fact_552_bot_Oextremum__strict,axiom,
! [A: set_o] :
~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).
% bot.extremum_strict
thf(fact_553_bot_Oextremum__strict,axiom,
! [A: set_rat] :
~ ( ord_less_set_rat @ A @ bot_bot_set_rat ) ).
% bot.extremum_strict
thf(fact_554_bot_Onot__eq__extremum,axiom,
! [A: set_Dedekind_preal] :
( ( A != bot_bo4848840305443107682_preal )
= ( ord_le1802228187270208418_preal @ bot_bo4848840305443107682_preal @ A ) ) ).
% bot.not_eq_extremum
thf(fact_555_bot_Onot__eq__extremum,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
= ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).
% bot.not_eq_extremum
thf(fact_556_bot_Onot__eq__extremum,axiom,
! [A: set_rat] :
( ( A != bot_bot_set_rat )
= ( ord_less_set_rat @ bot_bot_set_rat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_557_less__numeral__extra_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% less_numeral_extra(3)
thf(fact_558_field__lbound__gt__zero,axiom,
! [D1: rat,D2: rat] :
( ( ord_less_rat @ zero_zero_rat @ D1 )
=> ( ( ord_less_rat @ zero_zero_rat @ D2 )
=> ? [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
& ( ord_less_rat @ E @ D1 )
& ( ord_less_rat @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_559_psup__def,axiom,
( dedekind_psup
= ( ^ [P2: set_Dedekind_preal] : ( dedekind_Abs_preal @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ dedekind_Rep_preal @ P2 ) ) ) ) ) ).
% psup_def
thf(fact_560_Abs__preal__inverse,axiom,
! [Y: set_rat] :
( ( member_set_rat @ Y @ ( collect_set_rat @ dedekind_cut ) )
=> ( ( dedekind_Rep_preal @ ( dedekind_Abs_preal @ Y ) )
= Y ) ) ).
% Abs_preal_inverse
thf(fact_561_Abs__preal__inject,axiom,
! [X: set_rat,Y: set_rat] :
( ( member_set_rat @ X @ ( collect_set_rat @ dedekind_cut ) )
=> ( ( member_set_rat @ Y @ ( collect_set_rat @ dedekind_cut ) )
=> ( ( ( dedekind_Abs_preal @ X )
= ( dedekind_Abs_preal @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_preal_inject
thf(fact_562_Rep__preal__inverse,axiom,
! [X: dedekind_preal] :
( ( dedekind_Abs_preal @ ( dedekind_Rep_preal @ X ) )
= X ) ).
% Rep_preal_inverse
thf(fact_563_Abs__preal__cases,axiom,
! [X: dedekind_preal] :
~ ! [Y4: set_rat] :
( ( X
= ( dedekind_Abs_preal @ Y4 ) )
=> ~ ( member_set_rat @ Y4 @ ( collect_set_rat @ dedekind_cut ) ) ) ).
% Abs_preal_cases
thf(fact_564_preal_OAbs__preal__induct,axiom,
! [P: dedekind_preal > $o,X: dedekind_preal] :
( ! [Y4: set_rat] :
( ( member_set_rat @ Y4 @ ( collect_set_rat @ dedekind_cut ) )
=> ( P @ ( dedekind_Abs_preal @ Y4 ) ) )
=> ( P @ X ) ) ).
% preal.Abs_preal_induct
thf(fact_565_Dedekind__Real_OAbs__preal__induct,axiom,
! [P: dedekind_preal > $o,X: dedekind_preal] :
( ! [X3: set_rat] :
( ( dedekind_cut @ X3 )
=> ( P @ ( dedekind_Abs_preal @ X3 ) ) )
=> ( P @ X ) ) ).
% Dedekind_Real.Abs_preal_induct
thf(fact_566_order__less__imp__not__less,axiom,
! [X: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X @ Y )
=> ~ ( ord_less_set_rat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_567_order__less__imp__not__less,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_568_order__less__imp__not__less,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ~ ( ord_le5708704896291381698_preal @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_569_order__less__imp__not__eq2,axiom,
! [X: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_570_order__less__imp__not__eq2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_571_order__less__imp__not__eq2,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_572_order__less__imp__not__eq,axiom,
! [X: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_573_order__less__imp__not__eq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_574_order__less__imp__not__eq,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_575_linorder__less__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
| ( X = Y )
| ( ord_less_rat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_576_linorder__less__linear,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
| ( X = Y )
| ( ord_le5708704896291381698_preal @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_577_order__less__imp__triv,axiom,
! [X: set_rat,Y: set_rat,P: $o] :
( ( ord_less_set_rat @ X @ Y )
=> ( ( ord_less_set_rat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_578_order__less__imp__triv,axiom,
! [X: rat,Y: rat,P: $o] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_579_order__less__imp__triv,axiom,
! [X: dedekind_preal,Y: dedekind_preal,P: $o] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ( ord_le5708704896291381698_preal @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_580_order__less__not__sym,axiom,
! [X: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X @ Y )
=> ~ ( ord_less_set_rat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_581_order__less__not__sym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_582_order__less__not__sym,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ~ ( ord_le5708704896291381698_preal @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_583_order__less__subst2,axiom,
! [A: set_rat,B: set_rat,F: set_rat > set_rat,C2: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ord_less_set_rat @ ( F @ B ) @ C2 )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
=> ( ord_less_set_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_584_order__less__subst2,axiom,
! [A: set_rat,B: set_rat,F: set_rat > rat,C2: rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_585_order__less__subst2,axiom,
! [A: set_rat,B: set_rat,F: set_rat > dedekind_preal,C2: dedekind_preal] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C2 )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_586_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > set_rat,C2: set_rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_set_rat @ ( F @ B ) @ C2 )
=> ( ! [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_less_set_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_587_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_588_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > dedekind_preal,C2: dedekind_preal] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C2 )
=> ( ! [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_589_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > set_rat,C2: set_rat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_set_rat @ ( F @ B ) @ C2 )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y4 )
=> ( ord_less_set_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_590_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > rat,C2: rat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C2 )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y4 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_591_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C2 )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_592_order__less__subst1,axiom,
! [A: set_rat,F: set_rat > set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_set_rat @ A @ ( F @ B ) )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
=> ( ord_less_set_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_593_order__less__subst1,axiom,
! [A: set_rat,F: rat > set_rat,B: rat,C2: rat] :
( ( ord_less_set_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_less_set_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_594_order__less__subst1,axiom,
! [A: set_rat,F: dedekind_preal > set_rat,B: dedekind_preal,C2: dedekind_preal] :
( ( ord_less_set_rat @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C2 )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y4 )
=> ( ord_less_set_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_595_order__less__subst1,axiom,
! [A: rat,F: set_rat > rat,B: set_rat,C2: set_rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_596_order__less__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_597_order__less__subst1,axiom,
! [A: rat,F: dedekind_preal > rat,B: dedekind_preal,C2: dedekind_preal] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C2 )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y4 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_598_order__less__subst1,axiom,
! [A: dedekind_preal,F: set_rat > dedekind_preal,B: set_rat,C2: set_rat] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_599_order__less__subst1,axiom,
! [A: dedekind_preal,F: rat > dedekind_preal,B: rat,C2: rat] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_600_order__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C2 )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_601_order__less__irrefl,axiom,
! [X: set_rat] :
~ ( ord_less_set_rat @ X @ X ) ).
% order_less_irrefl
thf(fact_602_order__less__irrefl,axiom,
! [X: rat] :
~ ( ord_less_rat @ X @ X ) ).
% order_less_irrefl
thf(fact_603_order__less__irrefl,axiom,
! [X: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ X @ X ) ).
% order_less_irrefl
thf(fact_604_ord__less__eq__subst,axiom,
! [A: set_rat,B: set_rat,F: set_rat > set_rat,C2: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
=> ( ord_less_set_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_605_ord__less__eq__subst,axiom,
! [A: set_rat,B: set_rat,F: set_rat > rat,C2: rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_606_ord__less__eq__subst,axiom,
! [A: set_rat,B: set_rat,F: set_rat > dedekind_preal,C2: dedekind_preal] :
( ( ord_less_set_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_607_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > set_rat,C2: set_rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_less_set_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_608_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_609_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > dedekind_preal,C2: dedekind_preal] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_610_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > set_rat,C2: set_rat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y4 )
=> ( ord_less_set_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_611_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > rat,C2: rat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y4 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_612_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_613_ord__eq__less__subst,axiom,
! [A: set_rat,F: set_rat > set_rat,B: set_rat,C2: set_rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
=> ( ord_less_set_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_614_ord__eq__less__subst,axiom,
! [A: rat,F: set_rat > rat,B: set_rat,C2: set_rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_615_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: set_rat > dedekind_preal,B: set_rat,C2: set_rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( ord_less_set_rat @ X3 @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_616_ord__eq__less__subst,axiom,
! [A: set_rat,F: rat > set_rat,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_less_set_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_617_ord__eq__less__subst,axiom,
! [A: rat,F: rat > rat,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_618_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: rat > dedekind_preal,B: rat,C2: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ! [X3: rat,Y4: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_619_ord__eq__less__subst,axiom,
! [A: set_rat,F: dedekind_preal > set_rat,B: dedekind_preal,C2: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C2 )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y4 )
=> ( ord_less_set_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_620_ord__eq__less__subst,axiom,
! [A: rat,F: dedekind_preal > rat,B: dedekind_preal,C2: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C2 )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y4 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_rat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_621_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C2: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C2 )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y4 )
=> ( ord_le5708704896291381698_preal @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_622_order__less__trans,axiom,
! [X: set_rat,Y: set_rat,Z2: set_rat] :
( ( ord_less_set_rat @ X @ Y )
=> ( ( ord_less_set_rat @ Y @ Z2 )
=> ( ord_less_set_rat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_623_order__less__trans,axiom,
! [X: rat,Y: rat,Z2: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ Z2 )
=> ( ord_less_rat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_624_order__less__trans,axiom,
! [X: dedekind_preal,Y: dedekind_preal,Z2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ( ord_le5708704896291381698_preal @ Y @ Z2 )
=> ( ord_le5708704896291381698_preal @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_625_order__less__asym_H,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ~ ( ord_less_set_rat @ B @ A ) ) ).
% order_less_asym'
thf(fact_626_order__less__asym_H,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order_less_asym'
thf(fact_627_order__less__asym_H,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A ) ) ).
% order_less_asym'
thf(fact_628_linorder__neq__iff,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
= ( ( ord_less_rat @ X @ Y )
| ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_629_linorder__neq__iff,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( X != Y )
= ( ( ord_le5708704896291381698_preal @ X @ Y )
| ( ord_le5708704896291381698_preal @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_630_order__less__asym,axiom,
! [X: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X @ Y )
=> ~ ( ord_less_set_rat @ Y @ X ) ) ).
% order_less_asym
thf(fact_631_order__less__asym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_asym
thf(fact_632_order__less__asym,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ~ ( ord_le5708704896291381698_preal @ Y @ X ) ) ).
% order_less_asym
thf(fact_633_linorder__neqE,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
=> ( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_634_linorder__neqE,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( X != Y )
=> ( ~ ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_635_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_rat,A: set_rat] :
( ( ord_less_set_rat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_636_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_637_dual__order_Ostrict__implies__not__eq,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_638_order_Ostrict__implies__not__eq,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_639_order_Ostrict__implies__not__eq,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_640_order_Ostrict__implies__not__eq,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_641_dual__order_Ostrict__trans,axiom,
! [B: set_rat,A: set_rat,C2: set_rat] :
( ( ord_less_set_rat @ B @ A )
=> ( ( ord_less_set_rat @ C2 @ B )
=> ( ord_less_set_rat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_642_dual__order_Ostrict__trans,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C2 @ B )
=> ( ord_less_rat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_643_dual__order_Ostrict__trans,axiom,
! [B: dedekind_preal,A: dedekind_preal,C2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ( ord_le5708704896291381698_preal @ C2 @ B )
=> ( ord_le5708704896291381698_preal @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_644_not__less__iff__gr__or__eq,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X @ Y ) )
= ( ( ord_less_rat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_645_not__less__iff__gr__or__eq,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X @ Y ) )
= ( ( ord_le5708704896291381698_preal @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_646_order_Ostrict__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 ) ) ) ).
% order.strict_trans
thf(fact_647_order_Ostrict__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_648_order_Ostrict__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C2 )
=> ( ord_le5708704896291381698_preal @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_649_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A: rat,B: rat] :
( ! [A5: rat,B4: rat] :
( ( ord_less_rat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: rat] : ( P @ A5 @ A5 )
=> ( ! [A5: rat,B4: rat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_650_linorder__less__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A: dedekind_preal,B: dedekind_preal] :
( ! [A5: dedekind_preal,B4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: dedekind_preal] : ( P @ A5 @ A5 )
=> ( ! [A5: dedekind_preal,B4: dedekind_preal] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_651_dual__order_Oirrefl,axiom,
! [A: set_rat] :
~ ( ord_less_set_rat @ A @ A ) ).
% dual_order.irrefl
thf(fact_652_dual__order_Oirrefl,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% dual_order.irrefl
thf(fact_653_dual__order_Oirrefl,axiom,
! [A: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A @ A ) ).
% dual_order.irrefl
thf(fact_654_dual__order_Oasym,axiom,
! [B: set_rat,A: set_rat] :
( ( ord_less_set_rat @ B @ A )
=> ~ ( ord_less_set_rat @ A @ B ) ) ).
% dual_order.asym
thf(fact_655_dual__order_Oasym,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ~ ( ord_less_rat @ A @ B ) ) ).
% dual_order.asym
thf(fact_656_dual__order_Oasym,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ~ ( ord_le5708704896291381698_preal @ A @ B ) ) ).
% dual_order.asym
thf(fact_657_linorder__cases,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_658_linorder__cases,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ( X != Y )
=> ( ord_le5708704896291381698_preal @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_659_antisym__conv3,axiom,
! [Y: rat,X: rat] :
( ~ ( ord_less_rat @ Y @ X )
=> ( ( ~ ( ord_less_rat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_660_antisym__conv3,axiom,
! [Y: dedekind_preal,X: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ Y @ X )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_661_ord__less__eq__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_662_ord__less__eq__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_663_ord__less__eq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( B = C2 )
=> ( ord_le5708704896291381698_preal @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_664_ord__eq__less__trans,axiom,
! [A: set_rat,B: set_rat,C2: set_rat] :
( ( A = B )
=> ( ( ord_less_set_rat @ B @ C2 )
=> ( ord_less_set_rat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_665_ord__eq__less__trans,axiom,
! [A: rat,B: rat,C2: rat] :
( ( A = B )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_666_ord__eq__less__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C2: dedekind_preal] :
( ( A = B )
=> ( ( ord_le5708704896291381698_preal @ B @ C2 )
=> ( ord_le5708704896291381698_preal @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_667_order_Oasym,axiom,
! [A: set_rat,B: set_rat] :
( ( ord_less_set_rat @ A @ B )
=> ~ ( ord_less_set_rat @ B @ A ) ) ).
% order.asym
thf(fact_668_order_Oasym,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order.asym
thf(fact_669_order_Oasym,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A ) ) ).
% order.asym
thf(fact_670_less__imp__neq,axiom,
! [X: set_rat,Y: set_rat] :
( ( ord_less_set_rat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_671_less__imp__neq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_672_less__imp__neq,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_673_dense,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ? [Z3: rat] :
( ( ord_less_rat @ X @ Z3 )
& ( ord_less_rat @ Z3 @ Y ) ) ) ).
% dense
thf(fact_674_gt__ex,axiom,
! [X: rat] :
? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% gt_ex
thf(fact_675_lt__ex,axiom,
! [X: rat] :
? [Y4: rat] : ( ord_less_rat @ Y4 @ X ) ).
% lt_ex
thf(fact_676_bot__empty__eq,axiom,
( bot_bot_set_rat_o
= ( ^ [X4: set_rat] : ( member_set_rat @ X4 @ bot_bot_set_set_rat ) ) ) ).
% bot_empty_eq
thf(fact_677_bot__empty__eq,axiom,
( bot_bo4815710026557797371real_o
= ( ^ [X4: dedekind_preal] : ( member6871284927547481791_preal @ X4 @ bot_bo4848840305443107682_preal ) ) ) ).
% bot_empty_eq
thf(fact_678_bot__empty__eq,axiom,
( bot_bot_rat_o
= ( ^ [X4: rat] : ( member_rat @ X4 @ bot_bot_set_rat ) ) ) ).
% bot_empty_eq
thf(fact_679_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X4: $o] : ( member_o @ X4 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_680_Sup__SUP__eq,axiom,
( comple7848071142344908218real_o
= ( ^ [S4: set_Dedekind_preal_o,X4: dedekind_preal] : ( member6871284927547481791_preal @ X4 @ ( comple800427960086186211_preal @ ( image_8702985839003149832_preal @ collec1132657498972982273_preal @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_681_Sup__SUP__eq,axiom,
( comple4580332206425622756_rat_o
= ( ^ [S4: set_rat_o,X4: rat] : ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ ( image_rat_o_set_rat @ collect_rat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_682_Sup__SUP__eq,axiom,
( comple3057484590954014638_rat_o
= ( ^ [S4: set_set_rat_o,X4: set_rat] : ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ ( image_3676206661936088128et_rat @ collect_set_rat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_683_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S4: set_o_o,X4: $o] : ( member_o @ X4 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_684_SUP__Sup__eq,axiom,
! [S2: set_se4284277844656040150_preal] :
( ( comple7848071142344908218real_o
@ ( image_7211531428116543256real_o
@ ^ [I2: set_Dedekind_preal,X4: dedekind_preal] : ( member6871284927547481791_preal @ X4 @ I2 )
@ S2 ) )
= ( ^ [X4: dedekind_preal] : ( member6871284927547481791_preal @ X4 @ ( comple800427960086186211_preal @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_685_SUP__Sup__eq,axiom,
! [S2: set_set_rat] :
( ( comple4580332206425622756_rat_o
@ ( image_set_rat_rat_o
@ ^ [I2: set_rat,X4: rat] : ( member_rat @ X4 @ I2 )
@ S2 ) )
= ( ^ [X4: rat] : ( member_rat @ X4 @ ( comple3890839924845867745et_rat @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_686_SUP__Sup__eq,axiom,
! [S2: set_set_set_rat] :
( ( comple3057484590954014638_rat_o
@ ( image_5922895781192582838_rat_o
@ ^ [I2: set_set_rat,X4: set_rat] : ( member_set_rat @ X4 @ I2 )
@ S2 ) )
= ( ^ [X4: set_rat] : ( member_set_rat @ X4 @ ( comple3392050375588816791et_rat @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_687_SUP__Sup__eq,axiom,
! [S2: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I2: set_o,X4: $o] : ( member_o @ X4 @ I2 )
@ S2 ) )
= ( ^ [X4: $o] : ( member_o @ X4 @ ( comple90263536869209701_set_o @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_688_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_689_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_690_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_691_Collect__empty__eq__bot,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( P = bot_bot_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_692_cSUP__UNION,axiom,
! [A2: set_Dedekind_preal,B2: dedekind_preal > set_Dedekind_preal,F: dedekind_preal > $o] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A2 )
=> ( ( B2 @ X3 )
!= bot_bo4848840305443107682_preal ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( image_248103505173020295real_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B2 @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [X4: dedekind_preal] : ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( B2 @ X4 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_693_cSUP__UNION,axiom,
! [A2: set_rat,B2: rat > set_Dedekind_preal,F: dedekind_preal > $o] :
( ( A2 != bot_bot_set_rat )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ A2 )
=> ( ( B2 @ X3 )
!= bot_bo4848840305443107682_preal ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( image_248103505173020295real_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B2 @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_rat_o
@ ^ [X4: rat] : ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( B2 @ X4 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_694_cSUP__UNION,axiom,
! [A2: set_o,B2: $o > set_Dedekind_preal,F: dedekind_preal > $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( B2 @ X3 )
!= bot_bo4848840305443107682_preal ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( image_248103505173020295real_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( comple800427960086186211_preal @ ( image_2856262260070713571_preal @ B2 @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X4: $o] : ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( B2 @ X4 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_695_cSUP__UNION,axiom,
! [A2: set_Dedekind_preal,B2: dedekind_preal > set_rat,F: rat > $o] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A2 )
=> ( ( B2 @ X3 )
!= bot_bot_set_rat ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( image_rat_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [X4: dedekind_preal] : ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( B2 @ X4 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_696_cSUP__UNION,axiom,
! [A2: set_rat,B2: rat > set_rat,F: rat > $o] :
( ( A2 != bot_bot_set_rat )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ A2 )
=> ( ( B2 @ X3 )
!= bot_bot_set_rat ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( image_rat_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_rat_o
@ ^ [X4: rat] : ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( B2 @ X4 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_697_cSUP__UNION,axiom,
! [A2: set_o,B2: $o > set_rat,F: rat > $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( B2 @ X3 )
!= bot_bot_set_rat ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( image_rat_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X4: $o] : ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( B2 @ X4 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_698_cSUP__UNION,axiom,
! [A2: set_Dedekind_preal,B2: dedekind_preal > set_o,F: $o > $o] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A2 )
=> ( ( B2 @ X3 )
!= bot_bot_set_o ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( image_o_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [X4: dedekind_preal] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( B2 @ X4 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_699_cSUP__UNION,axiom,
! [A2: set_rat,B2: rat > set_o,F: $o > $o] :
( ( A2 != bot_bot_set_rat )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ A2 )
=> ( ( B2 @ X3 )
!= bot_bot_set_o ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( image_o_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_rat_o
@ ^ [X4: rat] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( B2 @ X4 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_700_cSUP__UNION,axiom,
! [A2: set_o,B2: $o > set_o,F: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( B2 @ X3 )
!= bot_bot_set_o ) )
=> ( ( condit5488710616941104124bove_o
@ ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( image_o_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X4: $o] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( B2 @ X4 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_701_cSUP__UNION,axiom,
! [A2: set_Dedekind_preal,B2: dedekind_preal > set_Dedekind_preal,F: dedekind_preal > set_o] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ! [X3: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A2 )
=> ( ( B2 @ X3 )
!= bot_bo4848840305443107682_preal ) )
=> ( ( condit9194965722646069980_set_o
@ ( comple4436988014476444997_set_o
@ ( image_9033230029842370631_set_o
@ ^ [X4: dedekind_preal] : ( image_8163909360452110695_set_o @ F @ ( B2 @ X4 ) )
@ A2 ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B2 @ A2 ) ) ) )
= ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ ( B2 @ X4 ) ) )
@ A2 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_702_the__elem__image__unique,axiom,
! [A2: set_set_rat,F: set_rat > set_set_rat,X: set_rat] :
( ( A2 != bot_bot_set_set_rat )
=> ( ! [Y4: set_rat] :
( ( member_set_rat @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X ) ) )
=> ( ( the_elem_set_set_rat @ ( image_9010471772881432025et_rat @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_703_the__elem__image__unique,axiom,
! [A2: set_set_rat,F: set_rat > set_rat,X: set_rat] :
( ( A2 != bot_bot_set_set_rat )
=> ( ! [Y4: set_rat] :
( ( member_set_rat @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X ) ) )
=> ( ( the_elem_set_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_704_the__elem__image__unique,axiom,
! [A2: set_Dedekind_preal,F: dedekind_preal > set_rat,X: dedekind_preal] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ! [Y4: dedekind_preal] :
( ( member6871284927547481791_preal @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X ) ) )
=> ( ( the_elem_set_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_705_the__elem__image__unique,axiom,
! [A2: set_rat,F: rat > set_o,X: rat] :
( ( A2 != bot_bot_set_rat )
=> ( ! [Y4: rat] :
( ( member_rat @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X ) ) )
=> ( ( the_elem_set_o @ ( image_rat_set_o @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_706_the__elem__image__unique,axiom,
! [A2: set_rat,F: rat > set_set_rat,X: rat] :
( ( A2 != bot_bot_set_rat )
=> ( ! [Y4: rat] :
( ( member_rat @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X ) ) )
=> ( ( the_elem_set_set_rat @ ( image_5169559123115255331et_rat @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_707_the__elem__image__unique,axiom,
! [A2: set_rat,F: rat > set_rat,X: rat] :
( ( A2 != bot_bot_set_rat )
=> ( ! [Y4: rat] :
( ( member_rat @ Y4 @ A2 )
=> ( ( F @ Y4 )
= ( F @ X ) ) )
=> ( ( the_elem_set_rat @ ( image_rat_set_rat @ F @ A2 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_708_Set_Ois__empty__def,axiom,
( is_emp2142545888348893928_preal
= ( ^ [A3: set_Dedekind_preal] : ( A3 = bot_bo4848840305443107682_preal ) ) ) ).
% Set.is_empty_def
thf(fact_709_Set_Ois__empty__def,axiom,
( is_empty_rat
= ( ^ [A3: set_rat] : ( A3 = bot_bot_set_rat ) ) ) ).
% Set.is_empty_def
thf(fact_710_Set_Ois__empty__def,axiom,
( is_empty_o
= ( ^ [A3: set_o] : ( A3 = bot_bot_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_711_mem__add__set,axiom,
! [A2: set_rat,B2: set_rat] :
( ( dedekind_cut @ A2 )
=> ( ( dedekind_cut @ B2 )
=> ( dedekind_cut @ ( dedekind_add_set @ A2 @ B2 ) ) ) ) ).
% mem_add_set
thf(fact_712_mem__mult__set,axiom,
! [A2: set_rat,B2: set_rat] :
( ( dedekind_cut @ A2 )
=> ( ( dedekind_cut @ B2 )
=> ( dedekind_cut @ ( dedekind_mult_set @ A2 @ B2 ) ) ) ) ).
% mem_mult_set
thf(fact_713_mem__inverse__set,axiom,
! [A2: set_rat] :
( ( dedekind_cut @ A2 )
=> ( dedekind_cut @ ( dedekind_inverse_set @ A2 ) ) ) ).
% mem_inverse_set
thf(fact_714_UNION__singleton__eq__range,axiom,
! [F: set_rat > set_set_rat,A2: set_set_rat] :
( ( comple5990833217840624973et_rat
@ ( image_6398182499340170127et_rat
@ ^ [X4: set_rat] : ( insert_set_set_rat @ ( F @ X4 ) @ bot_bo6619408370577057422et_rat )
@ A2 ) )
= ( image_9010471772881432025et_rat @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_715_UNION__singleton__eq__range,axiom,
! [F: rat > set_o,A2: set_rat] :
( ( comple4436988014476444997_set_o
@ ( image_rat_set_set_o
@ ^ [X4: rat] : ( insert_set_o @ ( F @ X4 ) @ bot_bot_set_set_o )
@ A2 ) )
= ( image_rat_set_o @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_716_UNION__singleton__eq__range,axiom,
! [F: rat > set_set_rat,A2: set_rat] :
( ( comple5990833217840624973et_rat
@ ( image_8006369670470206425et_rat
@ ^ [X4: rat] : ( insert_set_set_rat @ ( F @ X4 ) @ bot_bo6619408370577057422et_rat )
@ A2 ) )
= ( image_5169559123115255331et_rat @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_717_UNION__singleton__eq__range,axiom,
! [F: dedekind_preal > rat,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( insert_rat @ ( F @ X4 ) @ bot_bot_set_rat )
@ A2 ) )
= ( image_6572656775217987753al_rat @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_718_UNION__singleton__eq__range,axiom,
! [F: set_rat > rat,A2: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( insert_rat @ ( F @ X4 ) @ bot_bot_set_rat )
@ A2 ) )
= ( image_set_rat_rat @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_719_UNION__singleton__eq__range,axiom,
! [F: rat > rat,A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( insert_rat @ ( F @ X4 ) @ bot_bot_set_rat )
@ A2 ) )
= ( image_rat_rat @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_720_UNION__singleton__eq__range,axiom,
! [F: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( comple3392050375588816791et_rat
@ ( image_8268678932589273365et_rat
@ ^ [X4: dedekind_preal] : ( insert_set_rat @ ( F @ X4 ) @ bot_bot_set_set_rat )
@ A2 ) )
= ( image_2231269897757205599et_rat @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_721_UNION__singleton__eq__range,axiom,
! [F: set_rat > set_rat,A2: set_set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_9010471772881432025et_rat
@ ^ [X4: set_rat] : ( insert_set_rat @ ( F @ X4 ) @ bot_bot_set_set_rat )
@ A2 ) )
= ( image_3939399684171694371et_rat @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_722_UNION__singleton__eq__range,axiom,
! [F: rat > set_rat,A2: set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [X4: rat] : ( insert_set_rat @ ( F @ X4 ) @ bot_bot_set_set_rat )
@ A2 ) )
= ( image_rat_set_rat @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_723_UNION__singleton__eq__range,axiom,
! [F: rat > $o,A2: set_rat] :
( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( insert_o @ ( F @ X4 ) @ bot_bot_set_o )
@ A2 ) )
= ( image_rat_o @ F @ A2 ) ) ).
% UNION_singleton_eq_range
thf(fact_724_insertCI,axiom,
! [A: rat,B2: set_rat,B: rat] :
( ( ~ ( member_rat @ A @ B2 )
=> ( A = B ) )
=> ( member_rat @ A @ ( insert_rat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_725_insertCI,axiom,
! [A: $o,B2: set_o,B: $o] :
( ( ~ ( member_o @ A @ B2 )
=> ( A = B ) )
=> ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% insertCI
thf(fact_726_insertCI,axiom,
! [A: set_rat,B2: set_set_rat,B: set_rat] :
( ( ~ ( member_set_rat @ A @ B2 )
=> ( A = B ) )
=> ( member_set_rat @ A @ ( insert_set_rat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_727_insertCI,axiom,
! [A: dedekind_preal,B2: set_Dedekind_preal,B: dedekind_preal] :
( ( ~ ( member6871284927547481791_preal @ A @ B2 )
=> ( A = B ) )
=> ( member6871284927547481791_preal @ A @ ( insert4264952892349296358_preal @ B @ B2 ) ) ) ).
% insertCI
thf(fact_728_insert__iff,axiom,
! [A: rat,B: rat,A2: set_rat] :
( ( member_rat @ A @ ( insert_rat @ B @ A2 ) )
= ( ( A = B )
| ( member_rat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_729_insert__iff,axiom,
! [A: $o,B: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A2 ) )
= ( ( A = B )
| ( member_o @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_730_insert__iff,axiom,
! [A: set_rat,B: set_rat,A2: set_set_rat] :
( ( member_set_rat @ A @ ( insert_set_rat @ B @ A2 ) )
= ( ( A = B )
| ( member_set_rat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_731_insert__iff,axiom,
! [A: dedekind_preal,B: dedekind_preal,A2: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ A @ ( insert4264952892349296358_preal @ B @ A2 ) )
= ( ( A = B )
| ( member6871284927547481791_preal @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_732_insert__absorb2,axiom,
! [X: $o,A2: set_o] :
( ( insert_o @ X @ ( insert_o @ X @ A2 ) )
= ( insert_o @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_733_image__insert,axiom,
! [F: dedekind_preal > set_rat,A: dedekind_preal,B2: set_Dedekind_preal] :
( ( image_2231269897757205599et_rat @ F @ ( insert4264952892349296358_preal @ A @ B2 ) )
= ( insert_set_rat @ ( F @ A ) @ ( image_2231269897757205599et_rat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_734_image__insert,axiom,
! [F: set_rat > set_set_rat,A: set_rat,B2: set_set_rat] :
( ( image_9010471772881432025et_rat @ F @ ( insert_set_rat @ A @ B2 ) )
= ( insert_set_set_rat @ ( F @ A ) @ ( image_9010471772881432025et_rat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_735_image__insert,axiom,
! [F: set_rat > set_rat,A: set_rat,B2: set_set_rat] :
( ( image_3939399684171694371et_rat @ F @ ( insert_set_rat @ A @ B2 ) )
= ( insert_set_rat @ ( F @ A ) @ ( image_3939399684171694371et_rat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_736_image__insert,axiom,
! [F: rat > set_o,A: rat,B2: set_rat] :
( ( image_rat_set_o @ F @ ( insert_rat @ A @ B2 ) )
= ( insert_set_o @ ( F @ A ) @ ( image_rat_set_o @ F @ B2 ) ) ) ).
% image_insert
thf(fact_737_image__insert,axiom,
! [F: rat > set_set_rat,A: rat,B2: set_rat] :
( ( image_5169559123115255331et_rat @ F @ ( insert_rat @ A @ B2 ) )
= ( insert_set_set_rat @ ( F @ A ) @ ( image_5169559123115255331et_rat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_738_image__insert,axiom,
! [F: rat > set_rat,A: rat,B2: set_rat] :
( ( image_rat_set_rat @ F @ ( insert_rat @ A @ B2 ) )
= ( insert_set_rat @ ( F @ A ) @ ( image_rat_set_rat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_739_image__insert,axiom,
! [F: $o > $o,A: $o,B2: set_o] :
( ( image_o_o @ F @ ( insert_o @ A @ B2 ) )
= ( insert_o @ ( F @ A ) @ ( image_o_o @ F @ B2 ) ) ) ).
% image_insert
thf(fact_740_insert__image,axiom,
! [X: rat,A2: set_rat,F: rat > set_o] :
( ( member_rat @ X @ A2 )
=> ( ( insert_set_o @ ( F @ X ) @ ( image_rat_set_o @ F @ A2 ) )
= ( image_rat_set_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_741_insert__image,axiom,
! [X: rat,A2: set_rat,F: rat > set_set_rat] :
( ( member_rat @ X @ A2 )
=> ( ( insert_set_set_rat @ ( F @ X ) @ ( image_5169559123115255331et_rat @ F @ A2 ) )
= ( image_5169559123115255331et_rat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_742_insert__image,axiom,
! [X: rat,A2: set_rat,F: rat > set_rat] :
( ( member_rat @ X @ A2 )
=> ( ( insert_set_rat @ ( F @ X ) @ ( image_rat_set_rat @ F @ A2 ) )
= ( image_rat_set_rat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_743_insert__image,axiom,
! [X: rat,A2: set_rat,F: rat > $o] :
( ( member_rat @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_rat_o @ F @ A2 ) )
= ( image_rat_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_744_insert__image,axiom,
! [X: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_o_o @ F @ A2 ) )
= ( image_o_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_745_insert__image,axiom,
! [X: set_rat,A2: set_set_rat,F: set_rat > set_set_rat] :
( ( member_set_rat @ X @ A2 )
=> ( ( insert_set_set_rat @ ( F @ X ) @ ( image_9010471772881432025et_rat @ F @ A2 ) )
= ( image_9010471772881432025et_rat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_746_insert__image,axiom,
! [X: set_rat,A2: set_set_rat,F: set_rat > set_rat] :
( ( member_set_rat @ X @ A2 )
=> ( ( insert_set_rat @ ( F @ X ) @ ( image_3939399684171694371et_rat @ F @ A2 ) )
= ( image_3939399684171694371et_rat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_747_insert__image,axiom,
! [X: set_rat,A2: set_set_rat,F: set_rat > $o] :
( ( member_set_rat @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_set_rat_o @ F @ A2 ) )
= ( image_set_rat_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_748_insert__image,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,F: dedekind_preal > set_rat] :
( ( member6871284927547481791_preal @ X @ A2 )
=> ( ( insert_set_rat @ ( F @ X ) @ ( image_2231269897757205599et_rat @ F @ A2 ) )
= ( image_2231269897757205599et_rat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_749_insert__image,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,F: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_248103505173020295real_o @ F @ A2 ) )
= ( image_248103505173020295real_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_750_singletonI,axiom,
! [A: set_rat] : ( member_set_rat @ A @ ( insert_set_rat @ A @ bot_bot_set_set_rat ) ) ).
% singletonI
thf(fact_751_singletonI,axiom,
! [A: dedekind_preal] : ( member6871284927547481791_preal @ A @ ( insert4264952892349296358_preal @ A @ bot_bo4848840305443107682_preal ) ) ).
% singletonI
thf(fact_752_singletonI,axiom,
! [A: rat] : ( member_rat @ A @ ( insert_rat @ A @ bot_bot_set_rat ) ) ).
% singletonI
thf(fact_753_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_754_bdd__above__empty,axiom,
condit1512905060068503930_preal @ bot_bo4848840305443107682_preal ).
% bdd_above_empty
thf(fact_755_bdd__above__empty,axiom,
condit1579696412822616692ve_rat @ bot_bot_set_rat ).
% bdd_above_empty
thf(fact_756_bdd__above__empty,axiom,
condit5488710616941104124bove_o @ bot_bot_set_o ).
% bdd_above_empty
thf(fact_757_bdd__above__insert,axiom,
! [A: $o,A2: set_o] :
( ( condit5488710616941104124bove_o @ ( insert_o @ A @ A2 ) )
= ( condit5488710616941104124bove_o @ A2 ) ) ).
% bdd_above_insert
thf(fact_758_singleton__conv2,axiom,
! [A: set_rat] :
( ( collect_set_rat
@ ( ^ [Y5: set_rat,Z4: set_rat] : ( Y5 = Z4 )
@ A ) )
= ( insert_set_rat @ A @ bot_bot_set_set_rat ) ) ).
% singleton_conv2
thf(fact_759_singleton__conv2,axiom,
! [A: dedekind_preal] :
( ( collec1132657498972982273_preal
@ ( ^ [Y5: dedekind_preal,Z4: dedekind_preal] : ( Y5 = Z4 )
@ A ) )
= ( insert4264952892349296358_preal @ A @ bot_bo4848840305443107682_preal ) ) ).
% singleton_conv2
thf(fact_760_singleton__conv2,axiom,
! [A: rat] :
( ( collect_rat
@ ( ^ [Y5: rat,Z4: rat] : ( Y5 = Z4 )
@ A ) )
= ( insert_rat @ A @ bot_bot_set_rat ) ) ).
% singleton_conv2
thf(fact_761_singleton__conv2,axiom,
! [A: $o] :
( ( collect_o
@ ( ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
@ A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_762_singleton__conv,axiom,
! [A: set_rat] :
( ( collect_set_rat
@ ^ [X4: set_rat] : ( X4 = A ) )
= ( insert_set_rat @ A @ bot_bot_set_set_rat ) ) ).
% singleton_conv
thf(fact_763_singleton__conv,axiom,
! [A: dedekind_preal] :
( ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] : ( X4 = A ) )
= ( insert4264952892349296358_preal @ A @ bot_bo4848840305443107682_preal ) ) ).
% singleton_conv
thf(fact_764_singleton__conv,axiom,
! [A: rat] :
( ( collect_rat
@ ^ [X4: rat] : ( X4 = A ) )
= ( insert_rat @ A @ bot_bot_set_rat ) ) ).
% singleton_conv
thf(fact_765_singleton__conv,axiom,
! [A: $o] :
( ( collect_o
@ ^ [X4: $o] : ( X4 = A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_766_cSup__singleton,axiom,
! [X: set_rat] :
( ( comple3890839924845867745et_rat @ ( insert_set_rat @ X @ bot_bot_set_set_rat ) )
= X ) ).
% cSup_singleton
thf(fact_767_cSup__singleton,axiom,
! [X: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% cSup_singleton
thf(fact_768_cSup__singleton,axiom,
! [X: set_set_rat] :
( ( comple3392050375588816791et_rat @ ( insert_set_set_rat @ X @ bot_bo6619408370577057422et_rat ) )
= X ) ).
% cSup_singleton
thf(fact_769_cSup__singleton,axiom,
! [X: set_o] :
( ( comple90263536869209701_set_o @ ( insert_set_o @ X @ bot_bot_set_set_o ) )
= X ) ).
% cSup_singleton
thf(fact_770_the__elem__eq,axiom,
! [X: dedekind_preal] :
( ( the_el245288408674363275_preal @ ( insert4264952892349296358_preal @ X @ bot_bo4848840305443107682_preal ) )
= X ) ).
% the_elem_eq
thf(fact_771_the__elem__eq,axiom,
! [X: rat] :
( ( the_elem_rat @ ( insert_rat @ X @ bot_bot_set_rat ) )
= X ) ).
% the_elem_eq
thf(fact_772_the__elem__eq,axiom,
! [X: $o] :
( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% the_elem_eq
thf(fact_773_UN__simps_I1_J,axiom,
! [C: set_Dedekind_preal,A: dedekind_preal,B2: dedekind_preal > set_Dedekind_preal] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [X4: dedekind_preal] : ( insert4264952892349296358_preal @ A @ ( B2 @ X4 ) )
@ C ) )
= bot_bo4848840305443107682_preal ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [X4: dedekind_preal] : ( insert4264952892349296358_preal @ A @ ( B2 @ X4 ) )
@ C ) )
= ( insert4264952892349296358_preal @ A @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_774_UN__simps_I1_J,axiom,
! [C: set_rat,A: dedekind_preal,B2: rat > set_Dedekind_preal] :
( ( ( C = bot_bot_set_rat )
=> ( ( comple800427960086186211_preal
@ ( image_1093189101101737815_preal
@ ^ [X4: rat] : ( insert4264952892349296358_preal @ A @ ( B2 @ X4 ) )
@ C ) )
= bot_bo4848840305443107682_preal ) )
& ( ( C != bot_bot_set_rat )
=> ( ( comple800427960086186211_preal
@ ( image_1093189101101737815_preal
@ ^ [X4: rat] : ( insert4264952892349296358_preal @ A @ ( B2 @ X4 ) )
@ C ) )
= ( insert4264952892349296358_preal @ A @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_775_UN__simps_I1_J,axiom,
! [C: set_o,A: dedekind_preal,B2: $o > set_Dedekind_preal] :
( ( ( C = bot_bot_set_o )
=> ( ( comple800427960086186211_preal
@ ( image_2856262260070713571_preal
@ ^ [X4: $o] : ( insert4264952892349296358_preal @ A @ ( B2 @ X4 ) )
@ C ) )
= bot_bo4848840305443107682_preal ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple800427960086186211_preal
@ ( image_2856262260070713571_preal
@ ^ [X4: $o] : ( insert4264952892349296358_preal @ A @ ( B2 @ X4 ) )
@ C ) )
= ( insert4264952892349296358_preal @ A @ ( comple800427960086186211_preal @ ( image_2856262260070713571_preal @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_776_UN__simps_I1_J,axiom,
! [C: set_Dedekind_preal,A: rat,B2: dedekind_preal > set_rat] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_rat ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ C ) )
= ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_777_UN__simps_I1_J,axiom,
! [C: set_rat,A: rat,B2: rat > set_rat] :
( ( ( C = bot_bot_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_rat ) )
& ( ( C != bot_bot_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ C ) )
= ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_778_UN__simps_I1_J,axiom,
! [C: set_o,A: rat,B2: $o > set_rat] :
( ( ( C = bot_bot_set_o )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_rat ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ C ) )
= ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_779_UN__simps_I1_J,axiom,
! [C: set_Dedekind_preal,A: $o,B2: dedekind_preal > set_o] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( insert_o @ A @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( insert_o @ A @ ( B2 @ X4 ) )
@ C ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_780_UN__simps_I1_J,axiom,
! [C: set_rat,A: $o,B2: rat > set_o] :
( ( ( C = bot_bot_set_rat )
=> ( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( insert_o @ A @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_rat )
=> ( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( insert_o @ A @ ( B2 @ X4 ) )
@ C ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_781_UN__simps_I1_J,axiom,
! [C: set_o,A: $o,B2: $o > set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ A @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ A @ ( B2 @ X4 ) )
@ C ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_782_UN__simps_I1_J,axiom,
! [C: set_set_rat,A: rat,B2: set_rat > set_rat] :
( ( ( C = bot_bot_set_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_rat ) )
& ( ( C != bot_bot_set_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ C ) )
= ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_783_UN__singleton,axiom,
! [A2: set_Dedekind_preal] :
( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [X4: dedekind_preal] : ( insert4264952892349296358_preal @ X4 @ bot_bo4848840305443107682_preal )
@ A2 ) )
= A2 ) ).
% UN_singleton
thf(fact_784_UN__singleton,axiom,
! [A2: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( insert_rat @ X4 @ bot_bot_set_rat )
@ A2 ) )
= A2 ) ).
% UN_singleton
thf(fact_785_UN__singleton,axiom,
! [A2: set_set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_9010471772881432025et_rat
@ ^ [X4: set_rat] : ( insert_set_rat @ X4 @ bot_bot_set_set_rat )
@ A2 ) )
= A2 ) ).
% UN_singleton
thf(fact_786_UN__singleton,axiom,
! [A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ X4 @ bot_bot_set_o )
@ A2 ) )
= A2 ) ).
% UN_singleton
thf(fact_787_insert__compr,axiom,
( insert_o
= ( ^ [A4: $o,B3: set_o] :
( collect_o
@ ^ [X4: $o] :
( ( X4 = A4 )
| ( member_o @ X4 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_788_insert__compr,axiom,
( insert4264952892349296358_preal
= ( ^ [A4: dedekind_preal,B3: set_Dedekind_preal] :
( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] :
( ( X4 = A4 )
| ( member6871284927547481791_preal @ X4 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_789_insert__compr,axiom,
( insert_set_rat
= ( ^ [A4: set_rat,B3: set_set_rat] :
( collect_set_rat
@ ^ [X4: set_rat] :
( ( X4 = A4 )
| ( member_set_rat @ X4 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_790_insert__compr,axiom,
( insert_rat
= ( ^ [A4: rat,B3: set_rat] :
( collect_rat
@ ^ [X4: rat] :
( ( X4 = A4 )
| ( member_rat @ X4 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_791_insert__Collect,axiom,
! [A: $o,P: $o > $o] :
( ( insert_o @ A @ ( collect_o @ P ) )
= ( collect_o
@ ^ [U: $o] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_792_insert__Collect,axiom,
! [A: set_rat,P: set_rat > $o] :
( ( insert_set_rat @ A @ ( collect_set_rat @ P ) )
= ( collect_set_rat
@ ^ [U: set_rat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_793_insert__Collect,axiom,
! [A: rat,P: rat > $o] :
( ( insert_rat @ A @ ( collect_rat @ P ) )
= ( collect_rat
@ ^ [U: rat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_794_insertE,axiom,
! [A: rat,B: rat,A2: set_rat] :
( ( member_rat @ A @ ( insert_rat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_rat @ A @ A2 ) ) ) ).
% insertE
thf(fact_795_insertE,axiom,
! [A: $o,B: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A2 ) )
=> ( ( A = ~ B )
=> ( member_o @ A @ A2 ) ) ) ).
% insertE
thf(fact_796_insertE,axiom,
! [A: set_rat,B: set_rat,A2: set_set_rat] :
( ( member_set_rat @ A @ ( insert_set_rat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_rat @ A @ A2 ) ) ) ).
% insertE
thf(fact_797_insertE,axiom,
! [A: dedekind_preal,B: dedekind_preal,A2: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ A @ ( insert4264952892349296358_preal @ B @ A2 ) )
=> ( ( A != B )
=> ( member6871284927547481791_preal @ A @ A2 ) ) ) ).
% insertE
thf(fact_798_insertI1,axiom,
! [A: rat,B2: set_rat] : ( member_rat @ A @ ( insert_rat @ A @ B2 ) ) ).
% insertI1
thf(fact_799_insertI1,axiom,
! [A: $o,B2: set_o] : ( member_o @ A @ ( insert_o @ A @ B2 ) ) ).
% insertI1
thf(fact_800_insertI1,axiom,
! [A: set_rat,B2: set_set_rat] : ( member_set_rat @ A @ ( insert_set_rat @ A @ B2 ) ) ).
% insertI1
thf(fact_801_insertI1,axiom,
! [A: dedekind_preal,B2: set_Dedekind_preal] : ( member6871284927547481791_preal @ A @ ( insert4264952892349296358_preal @ A @ B2 ) ) ).
% insertI1
thf(fact_802_insertI2,axiom,
! [A: rat,B2: set_rat,B: rat] :
( ( member_rat @ A @ B2 )
=> ( member_rat @ A @ ( insert_rat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_803_insertI2,axiom,
! [A: $o,B2: set_o,B: $o] :
( ( member_o @ A @ B2 )
=> ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% insertI2
thf(fact_804_insertI2,axiom,
! [A: set_rat,B2: set_set_rat,B: set_rat] :
( ( member_set_rat @ A @ B2 )
=> ( member_set_rat @ A @ ( insert_set_rat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_805_insertI2,axiom,
! [A: dedekind_preal,B2: set_Dedekind_preal,B: dedekind_preal] :
( ( member6871284927547481791_preal @ A @ B2 )
=> ( member6871284927547481791_preal @ A @ ( insert4264952892349296358_preal @ B @ B2 ) ) ) ).
% insertI2
thf(fact_806_Set_Oset__insert,axiom,
! [X: rat,A2: set_rat] :
( ( member_rat @ X @ A2 )
=> ~ ! [B5: set_rat] :
( ( A2
= ( insert_rat @ X @ B5 ) )
=> ( member_rat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_807_Set_Oset__insert,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ~ ! [B5: set_o] :
( ( A2
= ( insert_o @ X @ B5 ) )
=> ( member_o @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_808_Set_Oset__insert,axiom,
! [X: set_rat,A2: set_set_rat] :
( ( member_set_rat @ X @ A2 )
=> ~ ! [B5: set_set_rat] :
( ( A2
= ( insert_set_rat @ X @ B5 ) )
=> ( member_set_rat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_809_Set_Oset__insert,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A2 )
=> ~ ! [B5: set_Dedekind_preal] :
( ( A2
= ( insert4264952892349296358_preal @ X @ B5 ) )
=> ( member6871284927547481791_preal @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_810_insert__ident,axiom,
! [X: rat,A2: set_rat,B2: set_rat] :
( ~ ( member_rat @ X @ A2 )
=> ( ~ ( member_rat @ X @ B2 )
=> ( ( ( insert_rat @ X @ A2 )
= ( insert_rat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_811_insert__ident,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ~ ( member_o @ X @ B2 )
=> ( ( ( insert_o @ X @ A2 )
= ( insert_o @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_812_insert__ident,axiom,
! [X: set_rat,A2: set_set_rat,B2: set_set_rat] :
( ~ ( member_set_rat @ X @ A2 )
=> ( ~ ( member_set_rat @ X @ B2 )
=> ( ( ( insert_set_rat @ X @ A2 )
= ( insert_set_rat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_813_insert__ident,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,B2: set_Dedekind_preal] :
( ~ ( member6871284927547481791_preal @ X @ A2 )
=> ( ~ ( member6871284927547481791_preal @ X @ B2 )
=> ( ( ( insert4264952892349296358_preal @ X @ A2 )
= ( insert4264952892349296358_preal @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_814_insert__absorb,axiom,
! [A: rat,A2: set_rat] :
( ( member_rat @ A @ A2 )
=> ( ( insert_rat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_815_insert__absorb,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_816_insert__absorb,axiom,
! [A: set_rat,A2: set_set_rat] :
( ( member_set_rat @ A @ A2 )
=> ( ( insert_set_rat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_817_insert__absorb,axiom,
! [A: dedekind_preal,A2: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ A @ A2 )
=> ( ( insert4264952892349296358_preal @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_818_insert__eq__iff,axiom,
! [A: rat,A2: set_rat,B: rat,B2: set_rat] :
( ~ ( member_rat @ A @ A2 )
=> ( ~ ( member_rat @ B @ B2 )
=> ( ( ( insert_rat @ A @ A2 )
= ( insert_rat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_rat] :
( ( A2
= ( insert_rat @ B @ C3 ) )
& ~ ( member_rat @ B @ C3 )
& ( B2
= ( insert_rat @ A @ C3 ) )
& ~ ( member_rat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_819_insert__eq__iff,axiom,
! [A: $o,A2: set_o,B: $o,B2: set_o] :
( ~ ( member_o @ A @ A2 )
=> ( ~ ( member_o @ B @ B2 )
=> ( ( ( insert_o @ A @ A2 )
= ( insert_o @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A = ~ B )
=> ? [C3: set_o] :
( ( A2
= ( insert_o @ B @ C3 ) )
& ~ ( member_o @ B @ C3 )
& ( B2
= ( insert_o @ A @ C3 ) )
& ~ ( member_o @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_820_insert__eq__iff,axiom,
! [A: set_rat,A2: set_set_rat,B: set_rat,B2: set_set_rat] :
( ~ ( member_set_rat @ A @ A2 )
=> ( ~ ( member_set_rat @ B @ B2 )
=> ( ( ( insert_set_rat @ A @ A2 )
= ( insert_set_rat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_set_rat] :
( ( A2
= ( insert_set_rat @ B @ C3 ) )
& ~ ( member_set_rat @ B @ C3 )
& ( B2
= ( insert_set_rat @ A @ C3 ) )
& ~ ( member_set_rat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_821_insert__eq__iff,axiom,
! [A: dedekind_preal,A2: set_Dedekind_preal,B: dedekind_preal,B2: set_Dedekind_preal] :
( ~ ( member6871284927547481791_preal @ A @ A2 )
=> ( ~ ( member6871284927547481791_preal @ B @ B2 )
=> ( ( ( insert4264952892349296358_preal @ A @ A2 )
= ( insert4264952892349296358_preal @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_Dedekind_preal] :
( ( A2
= ( insert4264952892349296358_preal @ B @ C3 ) )
& ~ ( member6871284927547481791_preal @ B @ C3 )
& ( B2
= ( insert4264952892349296358_preal @ A @ C3 ) )
& ~ ( member6871284927547481791_preal @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_822_insert__commute,axiom,
! [X: $o,Y: $o,A2: set_o] :
( ( insert_o @ X @ ( insert_o @ Y @ A2 ) )
= ( insert_o @ Y @ ( insert_o @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_823_mk__disjoint__insert,axiom,
! [A: rat,A2: set_rat] :
( ( member_rat @ A @ A2 )
=> ? [B5: set_rat] :
( ( A2
= ( insert_rat @ A @ B5 ) )
& ~ ( member_rat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_824_mk__disjoint__insert,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ? [B5: set_o] :
( ( A2
= ( insert_o @ A @ B5 ) )
& ~ ( member_o @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_825_mk__disjoint__insert,axiom,
! [A: set_rat,A2: set_set_rat] :
( ( member_set_rat @ A @ A2 )
=> ? [B5: set_set_rat] :
( ( A2
= ( insert_set_rat @ A @ B5 ) )
& ~ ( member_set_rat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_826_mk__disjoint__insert,axiom,
! [A: dedekind_preal,A2: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ A @ A2 )
=> ? [B5: set_Dedekind_preal] :
( ( A2
= ( insert4264952892349296358_preal @ A @ B5 ) )
& ~ ( member6871284927547481791_preal @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_827_singleton__inject,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ( insert4264952892349296358_preal @ A @ bot_bo4848840305443107682_preal )
= ( insert4264952892349296358_preal @ B @ bot_bo4848840305443107682_preal ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_828_singleton__inject,axiom,
! [A: rat,B: rat] :
( ( ( insert_rat @ A @ bot_bot_set_rat )
= ( insert_rat @ B @ bot_bot_set_rat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_829_singleton__inject,axiom,
! [A: $o,B: $o] :
( ( ( insert_o @ A @ bot_bot_set_o )
= ( insert_o @ B @ bot_bot_set_o ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_830_insert__not__empty,axiom,
! [A: dedekind_preal,A2: set_Dedekind_preal] :
( ( insert4264952892349296358_preal @ A @ A2 )
!= bot_bo4848840305443107682_preal ) ).
% insert_not_empty
thf(fact_831_insert__not__empty,axiom,
! [A: rat,A2: set_rat] :
( ( insert_rat @ A @ A2 )
!= bot_bot_set_rat ) ).
% insert_not_empty
thf(fact_832_insert__not__empty,axiom,
! [A: $o,A2: set_o] :
( ( insert_o @ A @ A2 )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_833_doubleton__eq__iff,axiom,
! [A: dedekind_preal,B: dedekind_preal,C2: dedekind_preal,D3: dedekind_preal] :
( ( ( insert4264952892349296358_preal @ A @ ( insert4264952892349296358_preal @ B @ bot_bo4848840305443107682_preal ) )
= ( insert4264952892349296358_preal @ C2 @ ( insert4264952892349296358_preal @ D3 @ bot_bo4848840305443107682_preal ) ) )
= ( ( ( A = C2 )
& ( B = D3 ) )
| ( ( A = D3 )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_834_doubleton__eq__iff,axiom,
! [A: rat,B: rat,C2: rat,D3: rat] :
( ( ( insert_rat @ A @ ( insert_rat @ B @ bot_bot_set_rat ) )
= ( insert_rat @ C2 @ ( insert_rat @ D3 @ bot_bot_set_rat ) ) )
= ( ( ( A = C2 )
& ( B = D3 ) )
| ( ( A = D3 )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_835_doubleton__eq__iff,axiom,
! [A: $o,B: $o,C2: $o,D3: $o] :
( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
= ( insert_o @ C2 @ ( insert_o @ D3 @ bot_bot_set_o ) ) )
= ( ( ( A = C2 )
& ( B = D3 ) )
| ( ( A = D3 )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_836_singleton__iff,axiom,
! [B: set_rat,A: set_rat] :
( ( member_set_rat @ B @ ( insert_set_rat @ A @ bot_bot_set_set_rat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_837_singleton__iff,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( member6871284927547481791_preal @ B @ ( insert4264952892349296358_preal @ A @ bot_bo4848840305443107682_preal ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_838_singleton__iff,axiom,
! [B: rat,A: rat] :
( ( member_rat @ B @ ( insert_rat @ A @ bot_bot_set_rat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_839_singleton__iff,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_840_singletonD,axiom,
! [B: set_rat,A: set_rat] :
( ( member_set_rat @ B @ ( insert_set_rat @ A @ bot_bot_set_set_rat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_841_singletonD,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( member6871284927547481791_preal @ B @ ( insert4264952892349296358_preal @ A @ bot_bo4848840305443107682_preal ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_842_singletonD,axiom,
! [B: rat,A: rat] :
( ( member_rat @ B @ ( insert_rat @ A @ bot_bot_set_rat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_843_singletonD,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_844_Collect__conv__if2,axiom,
! [P: set_rat > $o,A: set_rat] :
( ( ( P @ A )
=> ( ( collect_set_rat
@ ^ [X4: set_rat] :
( ( A = X4 )
& ( P @ X4 ) ) )
= ( insert_set_rat @ A @ bot_bot_set_set_rat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_rat
@ ^ [X4: set_rat] :
( ( A = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_set_rat ) ) ) ).
% Collect_conv_if2
thf(fact_845_Collect__conv__if2,axiom,
! [P: dedekind_preal > $o,A: dedekind_preal] :
( ( ( P @ A )
=> ( ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] :
( ( A = X4 )
& ( P @ X4 ) ) )
= ( insert4264952892349296358_preal @ A @ bot_bo4848840305443107682_preal ) ) )
& ( ~ ( P @ A )
=> ( ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] :
( ( A = X4 )
& ( P @ X4 ) ) )
= bot_bo4848840305443107682_preal ) ) ) ).
% Collect_conv_if2
thf(fact_846_Collect__conv__if2,axiom,
! [P: rat > $o,A: rat] :
( ( ( P @ A )
=> ( ( collect_rat
@ ^ [X4: rat] :
( ( A = X4 )
& ( P @ X4 ) ) )
= ( insert_rat @ A @ bot_bot_set_rat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_rat
@ ^ [X4: rat] :
( ( A = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_rat ) ) ) ).
% Collect_conv_if2
thf(fact_847_Collect__conv__if2,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X4: $o] :
( ( A = X4 )
& ( P @ X4 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X4: $o] :
( ( A = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_848_Collect__conv__if,axiom,
! [P: set_rat > $o,A: set_rat] :
( ( ( P @ A )
=> ( ( collect_set_rat
@ ^ [X4: set_rat] :
( ( X4 = A )
& ( P @ X4 ) ) )
= ( insert_set_rat @ A @ bot_bot_set_set_rat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_rat
@ ^ [X4: set_rat] :
( ( X4 = A )
& ( P @ X4 ) ) )
= bot_bot_set_set_rat ) ) ) ).
% Collect_conv_if
thf(fact_849_Collect__conv__if,axiom,
! [P: dedekind_preal > $o,A: dedekind_preal] :
( ( ( P @ A )
=> ( ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] :
( ( X4 = A )
& ( P @ X4 ) ) )
= ( insert4264952892349296358_preal @ A @ bot_bo4848840305443107682_preal ) ) )
& ( ~ ( P @ A )
=> ( ( collec1132657498972982273_preal
@ ^ [X4: dedekind_preal] :
( ( X4 = A )
& ( P @ X4 ) ) )
= bot_bo4848840305443107682_preal ) ) ) ).
% Collect_conv_if
thf(fact_850_Collect__conv__if,axiom,
! [P: rat > $o,A: rat] :
( ( ( P @ A )
=> ( ( collect_rat
@ ^ [X4: rat] :
( ( X4 = A )
& ( P @ X4 ) ) )
= ( insert_rat @ A @ bot_bot_set_rat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_rat
@ ^ [X4: rat] :
( ( X4 = A )
& ( P @ X4 ) ) )
= bot_bot_set_rat ) ) ) ).
% Collect_conv_if
thf(fact_851_Collect__conv__if,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X4: $o] :
( ( X4 = A )
& ( P @ X4 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X4: $o] :
( ( X4 = A )
& ( P @ X4 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_852_image__constant,axiom,
! [X: rat,A2: set_rat,C2: dedekind_preal] :
( ( member_rat @ X @ A2 )
=> ( ( image_7112546572676366711_preal
@ ^ [X4: rat] : C2
@ A2 )
= ( insert4264952892349296358_preal @ C2 @ bot_bo4848840305443107682_preal ) ) ) ).
% image_constant
thf(fact_853_image__constant,axiom,
! [X: $o,A2: set_o,C2: dedekind_preal] :
( ( member_o @ X @ A2 )
=> ( ( image_3026614506929738499_preal
@ ^ [X4: $o] : C2
@ A2 )
= ( insert4264952892349296358_preal @ C2 @ bot_bo4848840305443107682_preal ) ) ) ).
% image_constant
thf(fact_854_image__constant,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,C2: dedekind_preal] :
( ( member6871284927547481791_preal @ X @ A2 )
=> ( ( image_4420143364984939013_preal
@ ^ [X4: dedekind_preal] : C2
@ A2 )
= ( insert4264952892349296358_preal @ C2 @ bot_bo4848840305443107682_preal ) ) ) ).
% image_constant
thf(fact_855_image__constant,axiom,
! [X: rat,A2: set_rat,C2: rat] :
( ( member_rat @ X @ A2 )
=> ( ( image_rat_rat
@ ^ [X4: rat] : C2
@ A2 )
= ( insert_rat @ C2 @ bot_bot_set_rat ) ) ) ).
% image_constant
thf(fact_856_image__constant,axiom,
! [X: $o,A2: set_o,C2: rat] :
( ( member_o @ X @ A2 )
=> ( ( image_o_rat
@ ^ [X4: $o] : C2
@ A2 )
= ( insert_rat @ C2 @ bot_bot_set_rat ) ) ) ).
% image_constant
thf(fact_857_image__constant,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,C2: rat] :
( ( member6871284927547481791_preal @ X @ A2 )
=> ( ( image_6572656775217987753al_rat
@ ^ [X4: dedekind_preal] : C2
@ A2 )
= ( insert_rat @ C2 @ bot_bot_set_rat ) ) ) ).
% image_constant
thf(fact_858_image__constant,axiom,
! [X: rat,A2: set_rat,C2: $o] :
( ( member_rat @ X @ A2 )
=> ( ( image_rat_o
@ ^ [X4: rat] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_859_image__constant,axiom,
! [X: $o,A2: set_o,C2: $o] :
( ( member_o @ X @ A2 )
=> ( ( image_o_o
@ ^ [X4: $o] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_860_image__constant,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,C2: $o] :
( ( member6871284927547481791_preal @ X @ A2 )
=> ( ( image_248103505173020295real_o
@ ^ [X4: dedekind_preal] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_861_image__constant,axiom,
! [X: rat,A2: set_rat,C2: set_o] :
( ( member_rat @ X @ A2 )
=> ( ( image_rat_set_o
@ ^ [X4: rat] : C2
@ A2 )
= ( insert_set_o @ C2 @ bot_bot_set_set_o ) ) ) ).
% image_constant
thf(fact_862_image__constant__conv,axiom,
! [A2: set_Dedekind_preal,C2: dedekind_preal] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( image_4420143364984939013_preal
@ ^ [X4: dedekind_preal] : C2
@ A2 )
= bot_bo4848840305443107682_preal ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( image_4420143364984939013_preal
@ ^ [X4: dedekind_preal] : C2
@ A2 )
= ( insert4264952892349296358_preal @ C2 @ bot_bo4848840305443107682_preal ) ) ) ) ).
% image_constant_conv
thf(fact_863_image__constant__conv,axiom,
! [A2: set_Dedekind_preal,C2: rat] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( image_6572656775217987753al_rat
@ ^ [X4: dedekind_preal] : C2
@ A2 )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( image_6572656775217987753al_rat
@ ^ [X4: dedekind_preal] : C2
@ A2 )
= ( insert_rat @ C2 @ bot_bot_set_rat ) ) ) ) ).
% image_constant_conv
thf(fact_864_image__constant__conv,axiom,
! [A2: set_Dedekind_preal,C2: $o] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( image_248103505173020295real_o
@ ^ [X4: dedekind_preal] : C2
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( image_248103505173020295real_o
@ ^ [X4: dedekind_preal] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_865_image__constant__conv,axiom,
! [A2: set_rat,C2: dedekind_preal] :
( ( ( A2 = bot_bot_set_rat )
=> ( ( image_7112546572676366711_preal
@ ^ [X4: rat] : C2
@ A2 )
= bot_bo4848840305443107682_preal ) )
& ( ( A2 != bot_bot_set_rat )
=> ( ( image_7112546572676366711_preal
@ ^ [X4: rat] : C2
@ A2 )
= ( insert4264952892349296358_preal @ C2 @ bot_bo4848840305443107682_preal ) ) ) ) ).
% image_constant_conv
thf(fact_866_image__constant__conv,axiom,
! [A2: set_rat,C2: rat] :
( ( ( A2 = bot_bot_set_rat )
=> ( ( image_rat_rat
@ ^ [X4: rat] : C2
@ A2 )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bot_set_rat )
=> ( ( image_rat_rat
@ ^ [X4: rat] : C2
@ A2 )
= ( insert_rat @ C2 @ bot_bot_set_rat ) ) ) ) ).
% image_constant_conv
thf(fact_867_image__constant__conv,axiom,
! [A2: set_rat,C2: $o] :
( ( ( A2 = bot_bot_set_rat )
=> ( ( image_rat_o
@ ^ [X4: rat] : C2
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_rat )
=> ( ( image_rat_o
@ ^ [X4: rat] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_868_image__constant__conv,axiom,
! [A2: set_o,C2: dedekind_preal] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_3026614506929738499_preal
@ ^ [X4: $o] : C2
@ A2 )
= bot_bo4848840305443107682_preal ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_3026614506929738499_preal
@ ^ [X4: $o] : C2
@ A2 )
= ( insert4264952892349296358_preal @ C2 @ bot_bo4848840305443107682_preal ) ) ) ) ).
% image_constant_conv
thf(fact_869_image__constant__conv,axiom,
! [A2: set_o,C2: rat] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_o_rat
@ ^ [X4: $o] : C2
@ A2 )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_o_rat
@ ^ [X4: $o] : C2
@ A2 )
= ( insert_rat @ C2 @ bot_bot_set_rat ) ) ) ) ).
% image_constant_conv
thf(fact_870_image__constant__conv,axiom,
! [A2: set_o,C2: $o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X4: $o] : C2
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X4: $o] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_871_image__constant__conv,axiom,
! [A2: set_Dedekind_preal,C2: set_rat] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : C2
@ A2 )
= bot_bot_set_set_rat ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : C2
@ A2 )
= ( insert_set_rat @ C2 @ bot_bot_set_set_rat ) ) ) ) ).
% image_constant_conv
thf(fact_872_UN__insert__distrib,axiom,
! [U2: rat,A2: set_rat,A: rat,B2: rat > set_rat] :
( ( member_rat @ U2 @ A2 )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ A2 ) )
= ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_873_UN__insert__distrib,axiom,
! [U2: $o,A2: set_o,A: rat,B2: $o > set_rat] :
( ( member_o @ U2 @ A2 )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ A2 ) )
= ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_874_UN__insert__distrib,axiom,
! [U2: dedekind_preal,A2: set_Dedekind_preal,A: rat,B2: dedekind_preal > set_rat] :
( ( member6871284927547481791_preal @ U2 @ A2 )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ A2 ) )
= ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_875_UN__insert__distrib,axiom,
! [U2: rat,A2: set_rat,A: $o,B2: rat > set_o] :
( ( member_rat @ U2 @ A2 )
=> ( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( insert_o @ A @ ( B2 @ X4 ) )
@ A2 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_876_UN__insert__distrib,axiom,
! [U2: $o,A2: set_o,A: $o,B2: $o > set_o] :
( ( member_o @ U2 @ A2 )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ A @ ( B2 @ X4 ) )
@ A2 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_877_UN__insert__distrib,axiom,
! [U2: dedekind_preal,A2: set_Dedekind_preal,A: $o,B2: dedekind_preal > set_o] :
( ( member6871284927547481791_preal @ U2 @ A2 )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( insert_o @ A @ ( B2 @ X4 ) )
@ A2 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_878_UN__insert__distrib,axiom,
! [U2: set_rat,A2: set_set_rat,A: rat,B2: set_rat > set_rat] :
( ( member_set_rat @ U2 @ A2 )
=> ( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ A2 ) )
= ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_879_UN__insert__distrib,axiom,
! [U2: rat,A2: set_rat,A: set_rat,B2: rat > set_set_rat] :
( ( member_rat @ U2 @ A2 )
=> ( ( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [X4: rat] : ( insert_set_rat @ A @ ( B2 @ X4 ) )
@ A2 ) )
= ( insert_set_rat @ A @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_880_UN__insert__distrib,axiom,
! [U2: $o,A2: set_o,A: set_rat,B2: $o > set_set_rat] :
( ( member_o @ U2 @ A2 )
=> ( ( comple3392050375588816791et_rat
@ ( image_o_set_set_rat
@ ^ [X4: $o] : ( insert_set_rat @ A @ ( B2 @ X4 ) )
@ A2 ) )
= ( insert_set_rat @ A @ ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_881_UN__insert__distrib,axiom,
! [U2: dedekind_preal,A2: set_Dedekind_preal,A: set_rat,B2: dedekind_preal > set_set_rat] :
( ( member6871284927547481791_preal @ U2 @ A2 )
=> ( ( comple3392050375588816791et_rat
@ ( image_8268678932589273365et_rat
@ ^ [X4: dedekind_preal] : ( insert_set_rat @ A @ ( B2 @ X4 ) )
@ A2 ) )
= ( insert_set_rat @ A @ ( comple3392050375588816791et_rat @ ( image_8268678932589273365et_rat @ B2 @ A2 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_882_cSUP__lessD,axiom,
! [F: rat > $o,A2: set_rat,Y: $o,I: rat] :
( ( condit5488710616941104124bove_o @ ( image_rat_o @ F @ A2 ) )
=> ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A2 ) ) @ Y )
=> ( ( member_rat @ I @ A2 )
=> ( ord_less_o @ ( F @ I ) @ Y ) ) ) ) ).
% cSUP_lessD
thf(fact_883_cSUP__lessD,axiom,
! [F: $o > $o,A2: set_o,Y: $o,I: $o] :
( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A2 ) )
=> ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ Y )
=> ( ( member_o @ I @ A2 )
=> ( ord_less_o @ ( F @ I ) @ Y ) ) ) ) ).
% cSUP_lessD
thf(fact_884_cSUP__lessD,axiom,
! [F: dedekind_preal > $o,A2: set_Dedekind_preal,Y: $o,I: dedekind_preal] :
( ( condit5488710616941104124bove_o @ ( image_248103505173020295real_o @ F @ A2 ) )
=> ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A2 ) ) @ Y )
=> ( ( member6871284927547481791_preal @ I @ A2 )
=> ( ord_less_o @ ( F @ I ) @ Y ) ) ) ) ).
% cSUP_lessD
thf(fact_885_cSUP__lessD,axiom,
! [F: rat > set_rat,A2: set_rat,Y: set_rat,I: rat] :
( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ F @ A2 ) )
=> ( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A2 ) ) @ Y )
=> ( ( member_rat @ I @ A2 )
=> ( ord_less_set_rat @ ( F @ I ) @ Y ) ) ) ) ).
% cSUP_lessD
thf(fact_886_cSUP__lessD,axiom,
! [F: $o > set_rat,A2: set_o,Y: set_rat,I: $o] :
( ( condit1969311730731577898et_rat @ ( image_o_set_rat @ F @ A2 ) )
=> ( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A2 ) ) @ Y )
=> ( ( member_o @ I @ A2 )
=> ( ord_less_set_rat @ ( F @ I ) @ Y ) ) ) ) ).
% cSUP_lessD
thf(fact_887_cSUP__lessD,axiom,
! [F: dedekind_preal > set_rat,A2: set_Dedekind_preal,Y: set_rat,I: dedekind_preal] :
( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) )
=> ( ( ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) ) @ Y )
=> ( ( member6871284927547481791_preal @ I @ A2 )
=> ( ord_less_set_rat @ ( F @ I ) @ Y ) ) ) ) ).
% cSUP_lessD
thf(fact_888_cSUP__lessD,axiom,
! [F: set_rat > $o,A2: set_set_rat,Y: $o,I: set_rat] :
( ( condit5488710616941104124bove_o @ ( image_set_rat_o @ F @ A2 ) )
=> ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_set_rat_o @ F @ A2 ) ) @ Y )
=> ( ( member_set_rat @ I @ A2 )
=> ( ord_less_o @ ( F @ I ) @ Y ) ) ) ) ).
% cSUP_lessD
thf(fact_889_cSUP__lessD,axiom,
! [F: rat > set_o,A2: set_rat,Y: set_o,I: rat] :
( ( condit9194965722646069980_set_o @ ( image_rat_set_o @ F @ A2 ) )
=> ( ( ord_less_set_o @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ A2 ) ) @ Y )
=> ( ( member_rat @ I @ A2 )
=> ( ord_less_set_o @ ( F @ I ) @ Y ) ) ) ) ).
% cSUP_lessD
thf(fact_890_cSUP__lessD,axiom,
! [F: $o > set_o,A2: set_o,Y: set_o,I: $o] :
( ( condit9194965722646069980_set_o @ ( image_o_set_o @ F @ A2 ) )
=> ( ( ord_less_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ Y )
=> ( ( member_o @ I @ A2 )
=> ( ord_less_set_o @ ( F @ I ) @ Y ) ) ) ) ).
% cSUP_lessD
thf(fact_891_cSUP__lessD,axiom,
! [F: dedekind_preal > set_o,A2: set_Dedekind_preal,Y: set_o,I: dedekind_preal] :
( ( condit9194965722646069980_set_o @ ( image_8163909360452110695_set_o @ F @ A2 ) )
=> ( ( ord_less_set_o @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ A2 ) ) @ Y )
=> ( ( member6871284927547481791_preal @ I @ A2 )
=> ( ord_less_set_o @ ( F @ I ) @ Y ) ) ) ) ).
% cSUP_lessD
thf(fact_892_UN__extend__simps_I1_J,axiom,
! [C: set_Dedekind_preal,A: dedekind_preal,B2: dedekind_preal > set_Dedekind_preal] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( insert4264952892349296358_preal @ A @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B2 @ C ) ) )
= ( insert4264952892349296358_preal @ A @ bot_bo4848840305443107682_preal ) ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( insert4264952892349296358_preal @ A @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B2 @ C ) ) )
= ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [X4: dedekind_preal] : ( insert4264952892349296358_preal @ A @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_893_UN__extend__simps_I1_J,axiom,
! [C: set_rat,A: dedekind_preal,B2: rat > set_Dedekind_preal] :
( ( ( C = bot_bot_set_rat )
=> ( ( insert4264952892349296358_preal @ A @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B2 @ C ) ) )
= ( insert4264952892349296358_preal @ A @ bot_bo4848840305443107682_preal ) ) )
& ( ( C != bot_bot_set_rat )
=> ( ( insert4264952892349296358_preal @ A @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B2 @ C ) ) )
= ( comple800427960086186211_preal
@ ( image_1093189101101737815_preal
@ ^ [X4: rat] : ( insert4264952892349296358_preal @ A @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_894_UN__extend__simps_I1_J,axiom,
! [C: set_o,A: dedekind_preal,B2: $o > set_Dedekind_preal] :
( ( ( C = bot_bot_set_o )
=> ( ( insert4264952892349296358_preal @ A @ ( comple800427960086186211_preal @ ( image_2856262260070713571_preal @ B2 @ C ) ) )
= ( insert4264952892349296358_preal @ A @ bot_bo4848840305443107682_preal ) ) )
& ( ( C != bot_bot_set_o )
=> ( ( insert4264952892349296358_preal @ A @ ( comple800427960086186211_preal @ ( image_2856262260070713571_preal @ B2 @ C ) ) )
= ( comple800427960086186211_preal
@ ( image_2856262260070713571_preal
@ ^ [X4: $o] : ( insert4264952892349296358_preal @ A @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_895_UN__extend__simps_I1_J,axiom,
! [C: set_Dedekind_preal,A: rat,B2: dedekind_preal > set_rat] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ C ) ) )
= ( insert_rat @ A @ bot_bot_set_rat ) ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ C ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_896_UN__extend__simps_I1_J,axiom,
! [C: set_rat,A: rat,B2: rat > set_rat] :
( ( ( C = bot_bot_set_rat )
=> ( ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ C ) ) )
= ( insert_rat @ A @ bot_bot_set_rat ) ) )
& ( ( C != bot_bot_set_rat )
=> ( ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ C ) ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_897_UN__extend__simps_I1_J,axiom,
! [C: set_o,A: rat,B2: $o > set_rat] :
( ( ( C = bot_bot_set_o )
=> ( ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ C ) ) )
= ( insert_rat @ A @ bot_bot_set_rat ) ) )
& ( ( C != bot_bot_set_o )
=> ( ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ C ) ) )
= ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_898_UN__extend__simps_I1_J,axiom,
! [C: set_Dedekind_preal,A: $o,B2: dedekind_preal > set_o] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ C ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ C ) ) )
= ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( insert_o @ A @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_899_UN__extend__simps_I1_J,axiom,
! [C: set_rat,A: $o,B2: rat > set_o] :
( ( ( C = bot_bot_set_rat )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ C ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C != bot_bot_set_rat )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ C ) ) )
= ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( insert_o @ A @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_900_UN__extend__simps_I1_J,axiom,
! [C: set_o,A: $o,B2: $o > set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ C ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C != bot_bot_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ C ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ A @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_901_UN__extend__simps_I1_J,axiom,
! [C: set_set_rat,A: rat,B2: set_rat > set_rat] :
( ( ( C = bot_bot_set_set_rat )
=> ( ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ C ) ) )
= ( insert_rat @ A @ bot_bot_set_rat ) ) )
& ( ( C != bot_bot_set_set_rat )
=> ( ( insert_rat @ A @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ C ) ) )
= ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( insert_rat @ A @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_902_ccpo__Sup__singleton,axiom,
! [X: set_rat] :
( ( comple3890839924845867745et_rat @ ( insert_set_rat @ X @ bot_bot_set_set_rat ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_903_ccpo__Sup__singleton,axiom,
! [X: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_904_ccpo__Sup__singleton,axiom,
! [X: set_set_rat] :
( ( comple3392050375588816791et_rat @ ( insert_set_set_rat @ X @ bot_bo6619408370577057422et_rat ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_905_ccpo__Sup__singleton,axiom,
! [X: set_o] :
( ( comple90263536869209701_set_o @ ( insert_set_o @ X @ bot_bot_set_set_o ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_906_is__singleton__the__elem,axiom,
( is_sin4170067188624449034_preal
= ( ^ [A3: set_Dedekind_preal] :
( A3
= ( insert4264952892349296358_preal @ ( the_el245288408674363275_preal @ A3 ) @ bot_bo4848840305443107682_preal ) ) ) ) ).
% is_singleton_the_elem
thf(fact_907_is__singleton__the__elem,axiom,
( is_singleton_rat
= ( ^ [A3: set_rat] :
( A3
= ( insert_rat @ ( the_elem_rat @ A3 ) @ bot_bot_set_rat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_908_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A3: set_o] :
( A3
= ( insert_o @ ( the_elem_o @ A3 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_909_is__singletonI,axiom,
! [X: dedekind_preal] : ( is_sin4170067188624449034_preal @ ( insert4264952892349296358_preal @ X @ bot_bo4848840305443107682_preal ) ) ).
% is_singletonI
thf(fact_910_is__singletonI,axiom,
! [X: rat] : ( is_singleton_rat @ ( insert_rat @ X @ bot_bot_set_rat ) ) ).
% is_singletonI
thf(fact_911_is__singletonI,axiom,
! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_912_cSUP__insert,axiom,
! [A2: set_Dedekind_preal,F: dedekind_preal > $o,A: dedekind_preal] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( condit5488710616941104124bove_o @ ( image_248103505173020295real_o @ F @ A2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( insert4264952892349296358_preal @ A @ A2 ) ) )
= ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_913_cSUP__insert,axiom,
! [A2: set_rat,F: rat > $o,A: rat] :
( ( A2 != bot_bot_set_rat )
=> ( ( condit5488710616941104124bove_o @ ( image_rat_o @ F @ A2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( insert_rat @ A @ A2 ) ) )
= ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_914_cSUP__insert,axiom,
! [A2: set_o,F: $o > $o,A: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( insert_o @ A @ A2 ) ) )
= ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_915_cSUP__insert,axiom,
! [A2: set_Dedekind_preal,F: dedekind_preal > set_rat,A: dedekind_preal] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ ( insert4264952892349296358_preal @ A @ A2 ) ) )
= ( sup_sup_set_rat @ ( F @ A ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_916_cSUP__insert,axiom,
! [A2: set_rat,F: rat > set_rat,A: rat] :
( ( A2 != bot_bot_set_rat )
=> ( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ F @ A2 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ ( insert_rat @ A @ A2 ) ) )
= ( sup_sup_set_rat @ ( F @ A ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_917_cSUP__insert,axiom,
! [A2: set_o,F: $o > set_rat,A: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit1969311730731577898et_rat @ ( image_o_set_rat @ F @ A2 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ ( insert_o @ A @ A2 ) ) )
= ( sup_sup_set_rat @ ( F @ A ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_918_cSUP__insert,axiom,
! [A2: set_Dedekind_preal,F: dedekind_preal > set_o,A: dedekind_preal] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( condit9194965722646069980_set_o @ ( image_8163909360452110695_set_o @ F @ A2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ ( insert4264952892349296358_preal @ A @ A2 ) ) )
= ( sup_sup_set_o @ ( F @ A ) @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_919_cSUP__insert,axiom,
! [A2: set_rat,F: rat > set_o,A: rat] :
( ( A2 != bot_bot_set_rat )
=> ( ( condit9194965722646069980_set_o @ ( image_rat_set_o @ F @ A2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ ( insert_rat @ A @ A2 ) ) )
= ( sup_sup_set_o @ ( F @ A ) @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_920_cSUP__insert,axiom,
! [A2: set_o,F: $o > set_o,A: $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit9194965722646069980_set_o @ ( image_o_set_o @ F @ A2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ ( insert_o @ A @ A2 ) ) )
= ( sup_sup_set_o @ ( F @ A ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_921_cSUP__insert,axiom,
! [A2: set_set_rat,F: set_rat > set_rat,A: set_rat] :
( ( A2 != bot_bot_set_set_rat )
=> ( ( condit1969311730731577898et_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ ( insert_set_rat @ A @ A2 ) ) )
= ( sup_sup_set_rat @ ( F @ A ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_922_is__singletonE,axiom,
! [A2: set_Dedekind_preal] :
( ( is_sin4170067188624449034_preal @ A2 )
=> ~ ! [X3: dedekind_preal] :
( A2
!= ( insert4264952892349296358_preal @ X3 @ bot_bo4848840305443107682_preal ) ) ) ).
% is_singletonE
thf(fact_923_is__singletonE,axiom,
! [A2: set_rat] :
( ( is_singleton_rat @ A2 )
=> ~ ! [X3: rat] :
( A2
!= ( insert_rat @ X3 @ bot_bot_set_rat ) ) ) ).
% is_singletonE
thf(fact_924_is__singletonE,axiom,
! [A2: set_o] :
( ( is_singleton_o @ A2 )
=> ~ ! [X3: $o] :
( A2
!= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_925_is__singleton__def,axiom,
( is_sin4170067188624449034_preal
= ( ^ [A3: set_Dedekind_preal] :
? [X4: dedekind_preal] :
( A3
= ( insert4264952892349296358_preal @ X4 @ bot_bo4848840305443107682_preal ) ) ) ) ).
% is_singleton_def
thf(fact_926_is__singleton__def,axiom,
( is_singleton_rat
= ( ^ [A3: set_rat] :
? [X4: rat] :
( A3
= ( insert_rat @ X4 @ bot_bot_set_rat ) ) ) ) ).
% is_singleton_def
thf(fact_927_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A3: set_o] :
? [X4: $o] :
( A3
= ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_928_bind__singleton__conv__image,axiom,
! [A2: set_Dedekind_preal,F: dedekind_preal > set_rat] :
( ( bind_D6606601606562128401et_rat @ A2
@ ^ [X4: dedekind_preal] : ( insert_set_rat @ ( F @ X4 ) @ bot_bot_set_set_rat ) )
= ( image_2231269897757205599et_rat @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_929_bind__singleton__conv__image,axiom,
! [A2: set_set_rat,F: set_rat > set_set_rat] :
( ( bind_s6617489967869682855et_rat @ A2
@ ^ [X4: set_rat] : ( insert_set_set_rat @ ( F @ X4 ) @ bot_bo6619408370577057422et_rat ) )
= ( image_9010471772881432025et_rat @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_930_bind__singleton__conv__image,axiom,
! [A2: set_set_rat,F: set_rat > set_rat] :
( ( bind_set_rat_set_rat @ A2
@ ^ [X4: set_rat] : ( insert_set_rat @ ( F @ X4 ) @ bot_bot_set_set_rat ) )
= ( image_3939399684171694371et_rat @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_931_bind__singleton__conv__image,axiom,
! [A2: set_rat,F: rat > set_o] :
( ( bind_rat_set_o @ A2
@ ^ [X4: rat] : ( insert_set_o @ ( F @ X4 ) @ bot_bot_set_set_o ) )
= ( image_rat_set_o @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_932_bind__singleton__conv__image,axiom,
! [A2: set_rat,F: rat > set_set_rat] :
( ( bind_rat_set_set_rat @ A2
@ ^ [X4: rat] : ( insert_set_set_rat @ ( F @ X4 ) @ bot_bo6619408370577057422et_rat ) )
= ( image_5169559123115255331et_rat @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_933_bind__singleton__conv__image,axiom,
! [A2: set_rat,F: rat > set_rat] :
( ( bind_rat_set_rat @ A2
@ ^ [X4: rat] : ( insert_set_rat @ ( F @ X4 ) @ bot_bot_set_set_rat ) )
= ( image_rat_set_rat @ F @ A2 ) ) ).
% bind_singleton_conv_image
thf(fact_934_add__right__cancel,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_935_add__left__cancel,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_936_Un__empty,axiom,
! [A2: set_Dedekind_preal,B2: set_Dedekind_preal] :
( ( ( sup_su8384800326570918602_preal @ A2 @ B2 )
= bot_bo4848840305443107682_preal )
= ( ( A2 = bot_bo4848840305443107682_preal )
& ( B2 = bot_bo4848840305443107682_preal ) ) ) ).
% Un_empty
thf(fact_937_Un__empty,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ( sup_sup_set_rat @ A2 @ B2 )
= bot_bot_set_rat )
= ( ( A2 = bot_bot_set_rat )
& ( B2 = bot_bot_set_rat ) ) ) ).
% Un_empty
thf(fact_938_Un__empty,axiom,
! [A2: set_o,B2: set_o] :
( ( ( sup_sup_set_o @ A2 @ B2 )
= bot_bot_set_o )
= ( ( A2 = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_939_Un__insert__right,axiom,
! [A2: set_o,A: $o,B2: set_o] :
( ( sup_sup_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( insert_o @ A @ ( sup_sup_set_o @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_940_Un__insert__left,axiom,
! [A: $o,B2: set_o,C: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A @ B2 ) @ C )
= ( insert_o @ A @ ( sup_sup_set_o @ B2 @ C ) ) ) ).
% Un_insert_left
thf(fact_941_Union__Un__distrib,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( comple3890839924845867745et_rat @ ( sup_sup_set_set_rat @ A2 @ B2 ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ A2 ) @ ( comple3890839924845867745et_rat @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_942_Union__Un__distrib,axiom,
! [A2: set_set_set_rat,B2: set_set_set_rat] :
( ( comple3392050375588816791et_rat @ ( sup_su3634870876412137766et_rat @ A2 @ B2 ) )
= ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ A2 ) @ ( comple3392050375588816791et_rat @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_943_Union__Un__distrib,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ( comple90263536869209701_set_o @ ( sup_sup_set_set_o @ A2 @ B2 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_944_preal__add__less__cancel__right,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S3 @ T ) )
= ( ord_le5708704896291381698_preal @ R2 @ S3 ) ) ).
% preal_add_less_cancel_right
thf(fact_945_preal__add__less__cancel__left,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S3 ) )
= ( ord_le5708704896291381698_preal @ R2 @ S3 ) ) ).
% preal_add_less_cancel_left
thf(fact_946_add__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add_0
thf(fact_947_add__cancel__right__right,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ A @ B ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
thf(fact_948_add__cancel__right__left,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ B @ A ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
thf(fact_949_add__cancel__left__right,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
thf(fact_950_add__cancel__left__left,axiom,
! [B: rat,A: rat] :
( ( ( plus_plus_rat @ B @ A )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
thf(fact_951_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A @ A ) )
= ( A = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_952_add_Oright__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.right_neutral
thf(fact_953_add__less__cancel__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_954_add__less__cancel__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_955_empty__bind,axiom,
! [F: dedekind_preal > set_Dedekind_preal] :
( ( bind_D8752870763990849491_preal @ bot_bo4848840305443107682_preal @ F )
= bot_bo4848840305443107682_preal ) ).
% empty_bind
thf(fact_956_empty__bind,axiom,
! [F: dedekind_preal > set_rat] :
( ( bind_D1371630106014942811al_rat @ bot_bo4848840305443107682_preal @ F )
= bot_bot_set_rat ) ).
% empty_bind
thf(fact_957_empty__bind,axiom,
! [F: dedekind_preal > set_o] :
( ( bind_D7840052183623059541real_o @ bot_bo4848840305443107682_preal @ F )
= bot_bot_set_o ) ).
% empty_bind
thf(fact_958_empty__bind,axiom,
! [F: rat > set_Dedekind_preal] :
( ( bind_r1911519903473321769_preal @ bot_bot_set_rat @ F )
= bot_bo4848840305443107682_preal ) ).
% empty_bind
thf(fact_959_empty__bind,axiom,
! [F: rat > set_rat] :
( ( bind_rat_rat @ bot_bot_set_rat @ F )
= bot_bot_set_rat ) ).
% empty_bind
thf(fact_960_empty__bind,axiom,
! [F: rat > set_o] :
( ( bind_rat_o @ bot_bot_set_rat @ F )
= bot_bot_set_o ) ).
% empty_bind
thf(fact_961_empty__bind,axiom,
! [F: $o > set_Dedekind_preal] :
( ( bind_o1395191148525001937_preal @ bot_bot_set_o @ F )
= bot_bo4848840305443107682_preal ) ).
% empty_bind
thf(fact_962_empty__bind,axiom,
! [F: $o > set_rat] :
( ( bind_o_rat @ bot_bot_set_o @ F )
= bot_bot_set_rat ) ).
% empty_bind
thf(fact_963_empty__bind,axiom,
! [F: $o > set_o] :
( ( bind_o_o @ bot_bot_set_o @ F )
= bot_bot_set_o ) ).
% empty_bind
thf(fact_964_UN__Un,axiom,
! [M: dedekind_preal > set_rat,A2: set_Dedekind_preal,B2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ M @ ( sup_su8384800326570918602_preal @ A2 @ B2 ) ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ M @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ M @ B2 ) ) ) ) ).
% UN_Un
thf(fact_965_UN__Un,axiom,
! [M: set_rat > set_rat,A2: set_set_rat,B2: set_set_rat] :
( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ M @ ( sup_sup_set_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ M @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ M @ B2 ) ) ) ) ).
% UN_Un
thf(fact_966_UN__Un,axiom,
! [M: rat > set_rat,A2: set_rat,B2: set_rat] :
( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ M @ ( sup_sup_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ M @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ M @ B2 ) ) ) ) ).
% UN_Un
thf(fact_967_UN__Un,axiom,
! [M: set_rat > set_set_rat,A2: set_set_rat,B2: set_set_rat] :
( ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ M @ ( sup_sup_set_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ M @ A2 ) ) @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ M @ B2 ) ) ) ) ).
% UN_Un
thf(fact_968_UN__Un,axiom,
! [M: rat > set_set_rat,A2: set_rat,B2: set_rat] :
( ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ M @ ( sup_sup_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ M @ A2 ) ) @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ M @ B2 ) ) ) ) ).
% UN_Un
thf(fact_969_UN__Un,axiom,
! [M: rat > set_o,A2: set_rat,B2: set_rat] :
( ( comple90263536869209701_set_o @ ( image_rat_set_o @ M @ ( sup_sup_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ M @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ M @ B2 ) ) ) ) ).
% UN_Un
thf(fact_970_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_971_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_972_less__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel2
thf(fact_973_less__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel1
thf(fact_974_add__less__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel2
thf(fact_975_add__less__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel1
thf(fact_976_image__add__0,axiom,
! [S2: set_rat] :
( ( image_rat_rat @ ( plus_plus_rat @ zero_zero_rat ) @ S2 )
= S2 ) ).
% image_add_0
thf(fact_977_Sup__insert,axiom,
! [A: set_rat,A2: set_set_rat] :
( ( comple3890839924845867745et_rat @ ( insert_set_rat @ A @ A2 ) )
= ( sup_sup_set_rat @ A @ ( comple3890839924845867745et_rat @ A2 ) ) ) ).
% Sup_insert
thf(fact_978_Sup__insert,axiom,
! [A: $o,A2: set_o] :
( ( complete_Sup_Sup_o @ ( insert_o @ A @ A2 ) )
= ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_insert
thf(fact_979_Sup__insert,axiom,
! [A: set_set_rat,A2: set_set_set_rat] :
( ( comple3392050375588816791et_rat @ ( insert_set_set_rat @ A @ A2 ) )
= ( sup_sup_set_set_rat @ A @ ( comple3392050375588816791et_rat @ A2 ) ) ) ).
% Sup_insert
thf(fact_980_Sup__insert,axiom,
! [A: set_o,A2: set_set_o] :
( ( comple90263536869209701_set_o @ ( insert_set_o @ A @ A2 ) )
= ( sup_sup_set_o @ A @ ( comple90263536869209701_set_o @ A2 ) ) ) ).
% Sup_insert
thf(fact_981_member__bind,axiom,
! [X: rat,A2: set_Dedekind_preal,F: dedekind_preal > set_rat] :
( ( member_rat @ X @ ( bind_D1371630106014942811al_rat @ A2 @ F ) )
= ( member_rat @ X @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) ) ) ) ).
% member_bind
thf(fact_982_member__bind,axiom,
! [X: rat,A2: set_set_rat,F: set_rat > set_rat] :
( ( member_rat @ X @ ( bind_set_rat_rat @ A2 @ F ) )
= ( member_rat @ X @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) ) ) ) ).
% member_bind
thf(fact_983_member__bind,axiom,
! [X: rat,A2: set_rat,F: rat > set_rat] :
( ( member_rat @ X @ ( bind_rat_rat @ A2 @ F ) )
= ( member_rat @ X @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A2 ) ) ) ) ).
% member_bind
thf(fact_984_member__bind,axiom,
! [X: set_rat,A2: set_set_rat,F: set_rat > set_set_rat] :
( ( member_set_rat @ X @ ( bind_set_rat_set_rat @ A2 @ F ) )
= ( member_set_rat @ X @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ F @ A2 ) ) ) ) ).
% member_bind
thf(fact_985_member__bind,axiom,
! [X: set_rat,A2: set_rat,F: rat > set_set_rat] :
( ( member_set_rat @ X @ ( bind_rat_set_rat @ A2 @ F ) )
= ( member_set_rat @ X @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ F @ A2 ) ) ) ) ).
% member_bind
thf(fact_986_member__bind,axiom,
! [X: $o,A2: set_rat,F: rat > set_o] :
( ( member_o @ X @ ( bind_rat_o @ A2 @ F ) )
= ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ A2 ) ) ) ) ).
% member_bind
thf(fact_987_UN__simps_I2_J,axiom,
! [C: set_Dedekind_preal,A2: dedekind_preal > set_Dedekind_preal,B2: set_Dedekind_preal] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [X4: dedekind_preal] : ( sup_su8384800326570918602_preal @ ( A2 @ X4 ) @ B2 )
@ C ) )
= bot_bo4848840305443107682_preal ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [X4: dedekind_preal] : ( sup_su8384800326570918602_preal @ ( A2 @ X4 ) @ B2 )
@ C ) )
= ( sup_su8384800326570918602_preal @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ A2 @ C ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_988_UN__simps_I2_J,axiom,
! [C: set_rat,A2: rat > set_Dedekind_preal,B2: set_Dedekind_preal] :
( ( ( C = bot_bot_set_rat )
=> ( ( comple800427960086186211_preal
@ ( image_1093189101101737815_preal
@ ^ [X4: rat] : ( sup_su8384800326570918602_preal @ ( A2 @ X4 ) @ B2 )
@ C ) )
= bot_bo4848840305443107682_preal ) )
& ( ( C != bot_bot_set_rat )
=> ( ( comple800427960086186211_preal
@ ( image_1093189101101737815_preal
@ ^ [X4: rat] : ( sup_su8384800326570918602_preal @ ( A2 @ X4 ) @ B2 )
@ C ) )
= ( sup_su8384800326570918602_preal @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ A2 @ C ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_989_UN__simps_I2_J,axiom,
! [C: set_o,A2: $o > set_Dedekind_preal,B2: set_Dedekind_preal] :
( ( ( C = bot_bot_set_o )
=> ( ( comple800427960086186211_preal
@ ( image_2856262260070713571_preal
@ ^ [X4: $o] : ( sup_su8384800326570918602_preal @ ( A2 @ X4 ) @ B2 )
@ C ) )
= bot_bo4848840305443107682_preal ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple800427960086186211_preal
@ ( image_2856262260070713571_preal
@ ^ [X4: $o] : ( sup_su8384800326570918602_preal @ ( A2 @ X4 ) @ B2 )
@ C ) )
= ( sup_su8384800326570918602_preal @ ( comple800427960086186211_preal @ ( image_2856262260070713571_preal @ A2 @ C ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_990_UN__simps_I2_J,axiom,
! [C: set_Dedekind_preal,A2: dedekind_preal > set_rat,B2: set_rat] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) )
= bot_bot_set_rat ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ A2 @ C ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_991_UN__simps_I2_J,axiom,
! [C: set_rat,A2: rat > set_rat,B2: set_rat] :
( ( ( C = bot_bot_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) )
= bot_bot_set_rat ) )
& ( ( C != bot_bot_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ A2 @ C ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_992_UN__simps_I2_J,axiom,
! [C: set_o,A2: $o > set_rat,B2: set_rat] :
( ( ( C = bot_bot_set_o )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) )
= bot_bot_set_rat ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ A2 @ C ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_993_UN__simps_I2_J,axiom,
! [C: set_Dedekind_preal,A2: dedekind_preal > set_o,B2: set_o] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( sup_sup_set_o @ ( A2 @ X4 ) @ B2 )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( sup_sup_set_o @ ( A2 @ X4 ) @ B2 )
@ C ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ A2 @ C ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_994_UN__simps_I2_J,axiom,
! [C: set_rat,A2: rat > set_o,B2: set_o] :
( ( ( C = bot_bot_set_rat )
=> ( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( sup_sup_set_o @ ( A2 @ X4 ) @ B2 )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_rat )
=> ( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( sup_sup_set_o @ ( A2 @ X4 ) @ B2 )
@ C ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ A2 @ C ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_995_UN__simps_I2_J,axiom,
! [C: set_o,A2: $o > set_o,B2: set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ ( A2 @ X4 ) @ B2 )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ ( A2 @ X4 ) @ B2 )
@ C ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ C ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_996_UN__simps_I2_J,axiom,
! [C: set_set_rat,A2: set_rat > set_rat,B2: set_rat] :
( ( ( C = bot_bot_set_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) )
= bot_bot_set_rat ) )
& ( ( C != bot_bot_set_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ A2 @ C ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_997_UN__simps_I3_J,axiom,
! [C: set_Dedekind_preal,A2: set_Dedekind_preal,B2: dedekind_preal > set_Dedekind_preal] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [X4: dedekind_preal] : ( sup_su8384800326570918602_preal @ A2 @ ( B2 @ X4 ) )
@ C ) )
= bot_bo4848840305443107682_preal ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( comple800427960086186211_preal
@ ( image_85258263552348133_preal
@ ^ [X4: dedekind_preal] : ( sup_su8384800326570918602_preal @ A2 @ ( B2 @ X4 ) )
@ C ) )
= ( sup_su8384800326570918602_preal @ A2 @ ( comple800427960086186211_preal @ ( image_85258263552348133_preal @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_998_UN__simps_I3_J,axiom,
! [C: set_rat,A2: set_Dedekind_preal,B2: rat > set_Dedekind_preal] :
( ( ( C = bot_bot_set_rat )
=> ( ( comple800427960086186211_preal
@ ( image_1093189101101737815_preal
@ ^ [X4: rat] : ( sup_su8384800326570918602_preal @ A2 @ ( B2 @ X4 ) )
@ C ) )
= bot_bo4848840305443107682_preal ) )
& ( ( C != bot_bot_set_rat )
=> ( ( comple800427960086186211_preal
@ ( image_1093189101101737815_preal
@ ^ [X4: rat] : ( sup_su8384800326570918602_preal @ A2 @ ( B2 @ X4 ) )
@ C ) )
= ( sup_su8384800326570918602_preal @ A2 @ ( comple800427960086186211_preal @ ( image_1093189101101737815_preal @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_999_UN__simps_I3_J,axiom,
! [C: set_o,A2: set_Dedekind_preal,B2: $o > set_Dedekind_preal] :
( ( ( C = bot_bot_set_o )
=> ( ( comple800427960086186211_preal
@ ( image_2856262260070713571_preal
@ ^ [X4: $o] : ( sup_su8384800326570918602_preal @ A2 @ ( B2 @ X4 ) )
@ C ) )
= bot_bo4848840305443107682_preal ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple800427960086186211_preal
@ ( image_2856262260070713571_preal
@ ^ [X4: $o] : ( sup_su8384800326570918602_preal @ A2 @ ( B2 @ X4 ) )
@ C ) )
= ( sup_su8384800326570918602_preal @ A2 @ ( comple800427960086186211_preal @ ( image_2856262260070713571_preal @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1000_UN__simps_I3_J,axiom,
! [C: set_Dedekind_preal,A2: set_rat,B2: dedekind_preal > set_rat] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( sup_sup_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_rat ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( sup_sup_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_rat @ A2 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1001_UN__simps_I3_J,axiom,
! [C: set_rat,A2: set_rat,B2: rat > set_rat] :
( ( ( C = bot_bot_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( sup_sup_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_rat ) )
& ( ( C != bot_bot_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( sup_sup_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_rat @ A2 @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1002_UN__simps_I3_J,axiom,
! [C: set_o,A2: set_rat,B2: $o > set_rat] :
( ( ( C = bot_bot_set_o )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( sup_sup_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_rat ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( sup_sup_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_rat @ A2 @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1003_UN__simps_I3_J,axiom,
! [C: set_Dedekind_preal,A2: set_o,B2: dedekind_preal > set_o] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( sup_sup_set_o @ A2 @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( sup_sup_set_o @ A2 @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1004_UN__simps_I3_J,axiom,
! [C: set_rat,A2: set_o,B2: rat > set_o] :
( ( ( C = bot_bot_set_rat )
=> ( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( sup_sup_set_o @ A2 @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_rat )
=> ( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( sup_sup_set_o @ A2 @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1005_UN__simps_I3_J,axiom,
! [C: set_o,A2: set_o,B2: $o > set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ A2 @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ A2 @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1006_UN__simps_I3_J,axiom,
! [C: set_set_rat,A2: set_rat,B2: set_rat > set_rat] :
( ( ( C = bot_bot_set_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( sup_sup_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) )
= bot_bot_set_rat ) )
& ( ( C != bot_bot_set_set_rat )
=> ( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( sup_sup_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_rat @ A2 @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1007_UN__insert,axiom,
! [B2: dedekind_preal > set_rat,A: dedekind_preal,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ ( insert4264952892349296358_preal @ A @ A2 ) ) )
= ( sup_sup_set_rat @ ( B2 @ A ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ A2 ) ) ) ) ).
% UN_insert
thf(fact_1008_UN__insert,axiom,
! [B2: set_rat > set_rat,A: set_rat,A2: set_set_rat] :
( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ ( insert_set_rat @ A @ A2 ) ) )
= ( sup_sup_set_rat @ ( B2 @ A ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ A2 ) ) ) ) ).
% UN_insert
thf(fact_1009_UN__insert,axiom,
! [B2: rat > set_rat,A: rat,A2: set_rat] :
( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ ( insert_rat @ A @ A2 ) ) )
= ( sup_sup_set_rat @ ( B2 @ A ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ A2 ) ) ) ) ).
% UN_insert
thf(fact_1010_UN__insert,axiom,
! [B2: $o > set_rat,A: $o,A2: set_o] :
( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ ( insert_o @ A @ A2 ) ) )
= ( sup_sup_set_rat @ ( B2 @ A ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ A2 ) ) ) ) ).
% UN_insert
thf(fact_1011_UN__insert,axiom,
! [B2: set_rat > set_set_rat,A: set_rat,A2: set_set_rat] :
( ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ ( insert_set_rat @ A @ A2 ) ) )
= ( sup_sup_set_set_rat @ ( B2 @ A ) @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ A2 ) ) ) ) ).
% UN_insert
thf(fact_1012_UN__insert,axiom,
! [B2: rat > set_set_rat,A: rat,A2: set_rat] :
( ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ ( insert_rat @ A @ A2 ) ) )
= ( sup_sup_set_set_rat @ ( B2 @ A ) @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ A2 ) ) ) ) ).
% UN_insert
thf(fact_1013_UN__insert,axiom,
! [B2: $o > set_set_rat,A: $o,A2: set_o] :
( ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ B2 @ ( insert_o @ A @ A2 ) ) )
= ( sup_sup_set_set_rat @ ( B2 @ A ) @ ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ B2 @ A2 ) ) ) ) ).
% UN_insert
thf(fact_1014_UN__insert,axiom,
! [B2: rat > set_o,A: rat,A2: set_rat] :
( ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ ( insert_rat @ A @ A2 ) ) )
= ( sup_sup_set_o @ ( B2 @ A ) @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ A2 ) ) ) ) ).
% UN_insert
thf(fact_1015_UN__insert,axiom,
! [B2: $o > set_o,A: $o,A2: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ ( insert_o @ A @ A2 ) ) )
= ( sup_sup_set_o @ ( B2 @ A ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ).
% UN_insert
thf(fact_1016_bdd__above__image__sup,axiom,
! [F: dedekind_preal > set_rat,G: dedekind_preal > set_rat,A2: set_Dedekind_preal] :
( ( condit1969311730731577898et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( sup_sup_set_rat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 ) )
= ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) )
& ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ G @ A2 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_1017_bdd__above__image__sup,axiom,
! [F: set_rat > set_set_rat,G: set_rat > set_set_rat,A2: set_set_rat] :
( ( condit6513867565548536544et_rat
@ ( image_9010471772881432025et_rat
@ ^ [X4: set_rat] : ( sup_sup_set_set_rat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 ) )
= ( ( condit6513867565548536544et_rat @ ( image_9010471772881432025et_rat @ F @ A2 ) )
& ( condit6513867565548536544et_rat @ ( image_9010471772881432025et_rat @ G @ A2 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_1018_bdd__above__image__sup,axiom,
! [F: set_rat > set_rat,G: set_rat > set_rat,A2: set_set_rat] :
( ( condit1969311730731577898et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( sup_sup_set_rat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 ) )
= ( ( condit1969311730731577898et_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) )
& ( condit1969311730731577898et_rat @ ( image_3939399684171694371et_rat @ G @ A2 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_1019_bdd__above__image__sup,axiom,
! [F: rat > set_o,G: rat > set_o,A2: set_rat] :
( ( condit9194965722646069980_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( sup_sup_set_o @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 ) )
= ( ( condit9194965722646069980_set_o @ ( image_rat_set_o @ F @ A2 ) )
& ( condit9194965722646069980_set_o @ ( image_rat_set_o @ G @ A2 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_1020_bdd__above__image__sup,axiom,
! [F: rat > set_set_rat,G: rat > set_set_rat,A2: set_rat] :
( ( condit6513867565548536544et_rat
@ ( image_5169559123115255331et_rat
@ ^ [X4: rat] : ( sup_sup_set_set_rat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 ) )
= ( ( condit6513867565548536544et_rat @ ( image_5169559123115255331et_rat @ F @ A2 ) )
& ( condit6513867565548536544et_rat @ ( image_5169559123115255331et_rat @ G @ A2 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_1021_bdd__above__image__sup,axiom,
! [F: rat > set_rat,G: rat > set_rat,A2: set_rat] :
( ( condit1969311730731577898et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( sup_sup_set_rat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A2 ) )
= ( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ F @ A2 ) )
& ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ G @ A2 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_1022_add__eq__exists,axiom,
! [A: rat,B: rat] :
? [X3: rat] :
( ( plus_plus_rat @ A @ X3 )
= B ) ).
% add_eq_exists
thf(fact_1023_add__right__imp__eq,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_1024_add__left__imp__eq,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_1025_add_Oleft__commute,axiom,
! [B: dedekind_preal,A: dedekind_preal,C2: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ B @ ( plus_p3173629198307831117_preal @ A @ C2 ) )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_1026_add_Oleft__commute,axiom,
! [B: rat,A: rat,C2: rat] :
( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C2 ) )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_1027_add_Ocommute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [A4: dedekind_preal,B6: dedekind_preal] : ( plus_p3173629198307831117_preal @ B6 @ A4 ) ) ) ).
% add.commute
thf(fact_1028_add_Ocommute,axiom,
( plus_plus_rat
= ( ^ [A4: rat,B6: rat] : ( plus_plus_rat @ B6 @ A4 ) ) ) ).
% add.commute
thf(fact_1029_add_Oright__cancel,axiom,
! [B: rat,A: rat,C2: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_1030_add_Oleft__cancel,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_1031_add_Oassoc,axiom,
! [A: dedekind_preal,B: dedekind_preal,C2: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C2 )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_1032_add_Oassoc,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_1033_group__cancel_Oadd2,axiom,
! [B2: rat,K: rat,B: rat,A: rat] :
( ( B2
= ( plus_plus_rat @ K @ B ) )
=> ( ( plus_plus_rat @ A @ B2 )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1034_group__cancel_Oadd1,axiom,
! [A2: rat,K: rat,A: rat,B: rat] :
( ( A2
= ( plus_plus_rat @ K @ A ) )
=> ( ( plus_plus_rat @ A2 @ B )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1035_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: dedekind_preal,J2: dedekind_preal,K: dedekind_preal,L2: dedekind_preal] :
( ( ( I = J2 )
& ( K = L2 ) )
=> ( ( plus_p3173629198307831117_preal @ I @ K )
= ( plus_p3173629198307831117_preal @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1036_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: rat,J2: rat,K: rat,L2: rat] :
( ( ( I = J2 )
& ( K = L2 ) )
=> ( ( plus_plus_rat @ I @ K )
= ( plus_plus_rat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1037_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: dedekind_preal,B: dedekind_preal,C2: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C2 )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1038_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1039_mem__Rep__preal__add__iff,axiom,
! [Z2: rat,R2: dedekind_preal,S3: dedekind_preal] :
( ( member_rat @ Z2 @ ( dedekind_Rep_preal @ ( plus_p3173629198307831117_preal @ R2 @ S3 ) ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ ( dedekind_Rep_preal @ R2 ) )
& ? [Y2: rat] :
( ( member_rat @ Y2 @ ( dedekind_Rep_preal @ S3 ) )
& ( Z2
= ( plus_plus_rat @ X4 @ Y2 ) ) ) ) ) ) ).
% mem_Rep_preal_add_iff
thf(fact_1040_is__num__normalize_I1_J,axiom,
! [A: rat,B: rat,C2: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_1041_Sup__union__distrib,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( comple3890839924845867745et_rat @ ( sup_sup_set_set_rat @ A2 @ B2 ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ A2 ) @ ( comple3890839924845867745et_rat @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_1042_Sup__union__distrib,axiom,
! [A2: set_o,B2: set_o] :
( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A2 @ B2 ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_1043_Sup__union__distrib,axiom,
! [A2: set_set_set_rat,B2: set_set_set_rat] :
( ( comple3392050375588816791et_rat @ ( sup_su3634870876412137766et_rat @ A2 @ B2 ) )
= ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ A2 ) @ ( comple3392050375588816791et_rat @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_1044_Sup__union__distrib,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ( comple90263536869209701_set_o @ ( sup_sup_set_set_o @ A2 @ B2 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_1045_SUP__union,axiom,
! [M: dedekind_preal > set_rat,A2: set_Dedekind_preal,B2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ M @ ( sup_su8384800326570918602_preal @ A2 @ B2 ) ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ M @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ M @ B2 ) ) ) ) ).
% SUP_union
thf(fact_1046_SUP__union,axiom,
! [M: set_rat > set_rat,A2: set_set_rat,B2: set_set_rat] :
( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ M @ ( sup_sup_set_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ M @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ M @ B2 ) ) ) ) ).
% SUP_union
thf(fact_1047_SUP__union,axiom,
! [M: rat > set_rat,A2: set_rat,B2: set_rat] :
( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ M @ ( sup_sup_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ M @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ M @ B2 ) ) ) ) ).
% SUP_union
thf(fact_1048_SUP__union,axiom,
! [M: set_rat > set_set_rat,A2: set_set_rat,B2: set_set_rat] :
( ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ M @ ( sup_sup_set_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ M @ A2 ) ) @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ M @ B2 ) ) ) ) ).
% SUP_union
thf(fact_1049_SUP__union,axiom,
! [M: rat > set_set_rat,A2: set_rat,B2: set_rat] :
( ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ M @ ( sup_sup_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ M @ A2 ) ) @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ M @ B2 ) ) ) ) ).
% SUP_union
thf(fact_1050_SUP__union,axiom,
! [M: rat > set_o,A2: set_rat,B2: set_rat] :
( ( comple90263536869209701_set_o @ ( image_rat_set_o @ M @ ( sup_sup_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ M @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ M @ B2 ) ) ) ) ).
% SUP_union
thf(fact_1051_image__Un,axiom,
! [F: dedekind_preal > set_rat,A2: set_Dedekind_preal,B2: set_Dedekind_preal] :
( ( image_2231269897757205599et_rat @ F @ ( sup_su8384800326570918602_preal @ A2 @ B2 ) )
= ( sup_sup_set_set_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) @ ( image_2231269897757205599et_rat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1052_image__Un,axiom,
! [F: set_rat > set_set_rat,A2: set_set_rat,B2: set_set_rat] :
( ( image_9010471772881432025et_rat @ F @ ( sup_sup_set_set_rat @ A2 @ B2 ) )
= ( sup_su3634870876412137766et_rat @ ( image_9010471772881432025et_rat @ F @ A2 ) @ ( image_9010471772881432025et_rat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1053_image__Un,axiom,
! [F: set_rat > set_rat,A2: set_set_rat,B2: set_set_rat] :
( ( image_3939399684171694371et_rat @ F @ ( sup_sup_set_set_rat @ A2 @ B2 ) )
= ( sup_sup_set_set_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) @ ( image_3939399684171694371et_rat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1054_image__Un,axiom,
! [F: rat > set_o,A2: set_rat,B2: set_rat] :
( ( image_rat_set_o @ F @ ( sup_sup_set_rat @ A2 @ B2 ) )
= ( sup_sup_set_set_o @ ( image_rat_set_o @ F @ A2 ) @ ( image_rat_set_o @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1055_image__Un,axiom,
! [F: rat > set_set_rat,A2: set_rat,B2: set_rat] :
( ( image_5169559123115255331et_rat @ F @ ( sup_sup_set_rat @ A2 @ B2 ) )
= ( sup_su3634870876412137766et_rat @ ( image_5169559123115255331et_rat @ F @ A2 ) @ ( image_5169559123115255331et_rat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1056_image__Un,axiom,
! [F: rat > set_rat,A2: set_rat,B2: set_rat] :
( ( image_rat_set_rat @ F @ ( sup_sup_set_rat @ A2 @ B2 ) )
= ( sup_sup_set_set_rat @ ( image_rat_set_rat @ F @ A2 ) @ ( image_rat_set_rat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1057_add_Ogroup__left__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1058_add_Ocomm__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.comm_neutral
thf(fact_1059_comm__monoid__add__class_Oadd__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1060_Un__empty__left,axiom,
! [B2: set_Dedekind_preal] :
( ( sup_su8384800326570918602_preal @ bot_bo4848840305443107682_preal @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1061_Un__empty__left,axiom,
! [B2: set_rat] :
( ( sup_sup_set_rat @ bot_bot_set_rat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1062_Un__empty__left,axiom,
! [B2: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1063_Un__empty__right,axiom,
! [A2: set_Dedekind_preal] :
( ( sup_su8384800326570918602_preal @ A2 @ bot_bo4848840305443107682_preal )
= A2 ) ).
% Un_empty_right
thf(fact_1064_Un__empty__right,axiom,
! [A2: set_rat] :
( ( sup_sup_set_rat @ A2 @ bot_bot_set_rat )
= A2 ) ).
% Un_empty_right
thf(fact_1065_Un__empty__right,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% Un_empty_right
thf(fact_1066_add__less__imp__less__right,axiom,
! [A: rat,C2: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1067_add__less__imp__less__left,axiom,
! [C2: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1068_add__strict__right__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1069_add__strict__left__mono,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1070_add__strict__mono,axiom,
! [A: rat,B: rat,C2: rat,D3: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C2 @ D3 )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_1071_add__mono__thms__linordered__field_I1_J,axiom,
! [I: rat,J2: rat,K: rat,L2: rat] :
( ( ( ord_less_rat @ I @ J2 )
& ( K = L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1072_add__mono__thms__linordered__field_I2_J,axiom,
! [I: rat,J2: rat,K: rat,L2: rat] :
( ( ( I = J2 )
& ( ord_less_rat @ K @ L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1073_add__mono__thms__linordered__field_I5_J,axiom,
! [I: rat,J2: rat,K: rat,L2: rat] :
( ( ( ord_less_rat @ I @ J2 )
& ( ord_less_rat @ K @ L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1074_Union__insert,axiom,
! [A: set_rat,B2: set_set_rat] :
( ( comple3890839924845867745et_rat @ ( insert_set_rat @ A @ B2 ) )
= ( sup_sup_set_rat @ A @ ( comple3890839924845867745et_rat @ B2 ) ) ) ).
% Union_insert
thf(fact_1075_Union__insert,axiom,
! [A: set_set_rat,B2: set_set_set_rat] :
( ( comple3392050375588816791et_rat @ ( insert_set_set_rat @ A @ B2 ) )
= ( sup_sup_set_set_rat @ A @ ( comple3392050375588816791et_rat @ B2 ) ) ) ).
% Union_insert
thf(fact_1076_Union__insert,axiom,
! [A: set_o,B2: set_set_o] :
( ( comple90263536869209701_set_o @ ( insert_set_o @ A @ B2 ) )
= ( sup_sup_set_o @ A @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Union_insert
thf(fact_1077_cSup__union__distrib,axiom,
! [A2: set_set_rat,B2: set_set_rat] :
( ( A2 != bot_bot_set_set_rat )
=> ( ( condit1969311730731577898et_rat @ A2 )
=> ( ( B2 != bot_bot_set_set_rat )
=> ( ( condit1969311730731577898et_rat @ B2 )
=> ( ( comple3890839924845867745et_rat @ ( sup_sup_set_set_rat @ A2 @ B2 ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ A2 ) @ ( comple3890839924845867745et_rat @ B2 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_1078_cSup__union__distrib,axiom,
! [A2: set_o,B2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A2 )
=> ( ( B2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ B2 )
=> ( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A2 @ B2 ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_1079_cSup__union__distrib,axiom,
! [A2: set_set_set_rat,B2: set_set_set_rat] :
( ( A2 != bot_bo6619408370577057422et_rat )
=> ( ( condit6513867565548536544et_rat @ A2 )
=> ( ( B2 != bot_bo6619408370577057422et_rat )
=> ( ( condit6513867565548536544et_rat @ B2 )
=> ( ( comple3392050375588816791et_rat @ ( sup_su3634870876412137766et_rat @ A2 @ B2 ) )
= ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ A2 ) @ ( comple3392050375588816791et_rat @ B2 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_1080_cSup__union__distrib,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ( A2 != bot_bot_set_set_o )
=> ( ( condit9194965722646069980_set_o @ A2 )
=> ( ( B2 != bot_bot_set_set_o )
=> ( ( condit9194965722646069980_set_o @ B2 )
=> ( ( comple90263536869209701_set_o @ ( sup_sup_set_set_o @ A2 @ B2 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_1081_preal__add__right__less__cancel,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S3 @ T ) )
=> ( ord_le5708704896291381698_preal @ R2 @ S3 ) ) ).
% preal_add_right_less_cancel
thf(fact_1082_preal__add__left__less__cancel,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S3 ) )
=> ( ord_le5708704896291381698_preal @ R2 @ S3 ) ) ).
% preal_add_left_less_cancel
thf(fact_1083_preal__self__less__add__left,axiom,
! [R2: dedekind_preal,S3: dedekind_preal] : ( ord_le5708704896291381698_preal @ R2 @ ( plus_p3173629198307831117_preal @ R2 @ S3 ) ) ).
% preal_self_less_add_left
thf(fact_1084_preal__add__less2__mono2,axiom,
! [R2: dedekind_preal,S3: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S3 )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S3 ) ) ) ).
% preal_add_less2_mono2
thf(fact_1085_preal__add__less2__mono1,axiom,
! [R2: dedekind_preal,S3: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S3 )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S3 @ T ) ) ) ).
% preal_add_less2_mono1
thf(fact_1086_insert__def,axiom,
( insert_o
= ( ^ [A4: $o] :
( sup_sup_set_o
@ ( collect_o
@ ^ [X4: $o] : ( X4 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1087_insert__def,axiom,
( insert_set_rat
= ( ^ [A4: set_rat] :
( sup_sup_set_set_rat
@ ( collect_set_rat
@ ^ [X4: set_rat] : ( X4 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1088_insert__def,axiom,
( insert_rat
= ( ^ [A4: rat] :
( sup_sup_set_rat
@ ( collect_rat
@ ^ [X4: rat] : ( X4 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1089_cSUP__union,axiom,
! [A2: set_Dedekind_preal,F: dedekind_preal > $o,B2: set_Dedekind_preal] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( condit5488710616941104124bove_o @ ( image_248103505173020295real_o @ F @ A2 ) )
=> ( ( B2 != bot_bo4848840305443107682_preal )
=> ( ( condit5488710616941104124bove_o @ ( image_248103505173020295real_o @ F @ B2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ ( sup_su8384800326570918602_preal @ A2 @ B2 ) ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1090_cSUP__union,axiom,
! [A2: set_rat,F: rat > $o,B2: set_rat] :
( ( A2 != bot_bot_set_rat )
=> ( ( condit5488710616941104124bove_o @ ( image_rat_o @ F @ A2 ) )
=> ( ( B2 != bot_bot_set_rat )
=> ( ( condit5488710616941104124bove_o @ ( image_rat_o @ F @ B2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_rat_o @ F @ ( sup_sup_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1091_cSUP__union,axiom,
! [A2: set_o,F: $o > $o,B2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A2 ) )
=> ( ( B2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ B2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( sup_sup_set_o @ A2 @ B2 ) ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1092_cSUP__union,axiom,
! [A2: set_Dedekind_preal,F: dedekind_preal > set_rat,B2: set_Dedekind_preal] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) )
=> ( ( B2 != bot_bo4848840305443107682_preal )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ F @ B2 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ ( sup_su8384800326570918602_preal @ A2 @ B2 ) ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1093_cSUP__union,axiom,
! [A2: set_rat,F: rat > set_rat,B2: set_rat] :
( ( A2 != bot_bot_set_rat )
=> ( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ F @ A2 ) )
=> ( ( B2 != bot_bot_set_rat )
=> ( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ F @ B2 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ ( sup_sup_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1094_cSUP__union,axiom,
! [A2: set_o,F: $o > set_rat,B2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit1969311730731577898et_rat @ ( image_o_set_rat @ F @ A2 ) )
=> ( ( B2 != bot_bot_set_o )
=> ( ( condit1969311730731577898et_rat @ ( image_o_set_rat @ F @ B2 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ ( sup_sup_set_o @ A2 @ B2 ) ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1095_cSUP__union,axiom,
! [A2: set_Dedekind_preal,F: dedekind_preal > set_o,B2: set_Dedekind_preal] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( condit9194965722646069980_set_o @ ( image_8163909360452110695_set_o @ F @ A2 ) )
=> ( ( B2 != bot_bo4848840305443107682_preal )
=> ( ( condit9194965722646069980_set_o @ ( image_8163909360452110695_set_o @ F @ B2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ ( sup_su8384800326570918602_preal @ A2 @ B2 ) ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1096_cSUP__union,axiom,
! [A2: set_rat,F: rat > set_o,B2: set_rat] :
( ( A2 != bot_bot_set_rat )
=> ( ( condit9194965722646069980_set_o @ ( image_rat_set_o @ F @ A2 ) )
=> ( ( B2 != bot_bot_set_rat )
=> ( ( condit9194965722646069980_set_o @ ( image_rat_set_o @ F @ B2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ ( sup_sup_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1097_cSUP__union,axiom,
! [A2: set_o,F: $o > set_o,B2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit9194965722646069980_set_o @ ( image_o_set_o @ F @ A2 ) )
=> ( ( B2 != bot_bot_set_o )
=> ( ( condit9194965722646069980_set_o @ ( image_o_set_o @ F @ B2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ ( sup_sup_set_o @ A2 @ B2 ) ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1098_cSUP__union,axiom,
! [A2: set_set_rat,F: set_rat > set_rat,B2: set_set_rat] :
( ( A2 != bot_bot_set_set_rat )
=> ( ( condit1969311730731577898et_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) )
=> ( ( B2 != bot_bot_set_set_rat )
=> ( ( condit1969311730731577898et_rat @ ( image_3939399684171694371et_rat @ F @ B2 ) )
=> ( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ ( sup_sup_set_set_rat @ A2 @ B2 ) ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_1099_pos__add__strict,axiom,
! [A: rat,B: rat,C2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C2 )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1100_add__pos__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1101_add__neg__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_neg
thf(fact_1102_singleton__Un__iff,axiom,
! [X: dedekind_preal,A2: set_Dedekind_preal,B2: set_Dedekind_preal] :
( ( ( insert4264952892349296358_preal @ X @ bot_bo4848840305443107682_preal )
= ( sup_su8384800326570918602_preal @ A2 @ B2 ) )
= ( ( ( A2 = bot_bo4848840305443107682_preal )
& ( B2
= ( insert4264952892349296358_preal @ X @ bot_bo4848840305443107682_preal ) ) )
| ( ( A2
= ( insert4264952892349296358_preal @ X @ bot_bo4848840305443107682_preal ) )
& ( B2 = bot_bo4848840305443107682_preal ) )
| ( ( A2
= ( insert4264952892349296358_preal @ X @ bot_bo4848840305443107682_preal ) )
& ( B2
= ( insert4264952892349296358_preal @ X @ bot_bo4848840305443107682_preal ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1103_singleton__Un__iff,axiom,
! [X: rat,A2: set_rat,B2: set_rat] :
( ( ( insert_rat @ X @ bot_bot_set_rat )
= ( sup_sup_set_rat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_rat )
& ( B2
= ( insert_rat @ X @ bot_bot_set_rat ) ) )
| ( ( A2
= ( insert_rat @ X @ bot_bot_set_rat ) )
& ( B2 = bot_bot_set_rat ) )
| ( ( A2
= ( insert_rat @ X @ bot_bot_set_rat ) )
& ( B2
= ( insert_rat @ X @ bot_bot_set_rat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1104_singleton__Un__iff,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ( ( insert_o @ X @ bot_bot_set_o )
= ( sup_sup_set_o @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_o )
& ( B2
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A2
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B2 = bot_bot_set_o ) )
| ( ( A2
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B2
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1105_Un__singleton__iff,axiom,
! [A2: set_Dedekind_preal,B2: set_Dedekind_preal,X: dedekind_preal] :
( ( ( sup_su8384800326570918602_preal @ A2 @ B2 )
= ( insert4264952892349296358_preal @ X @ bot_bo4848840305443107682_preal ) )
= ( ( ( A2 = bot_bo4848840305443107682_preal )
& ( B2
= ( insert4264952892349296358_preal @ X @ bot_bo4848840305443107682_preal ) ) )
| ( ( A2
= ( insert4264952892349296358_preal @ X @ bot_bo4848840305443107682_preal ) )
& ( B2 = bot_bo4848840305443107682_preal ) )
| ( ( A2
= ( insert4264952892349296358_preal @ X @ bot_bo4848840305443107682_preal ) )
& ( B2
= ( insert4264952892349296358_preal @ X @ bot_bo4848840305443107682_preal ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1106_Un__singleton__iff,axiom,
! [A2: set_rat,B2: set_rat,X: rat] :
( ( ( sup_sup_set_rat @ A2 @ B2 )
= ( insert_rat @ X @ bot_bot_set_rat ) )
= ( ( ( A2 = bot_bot_set_rat )
& ( B2
= ( insert_rat @ X @ bot_bot_set_rat ) ) )
| ( ( A2
= ( insert_rat @ X @ bot_bot_set_rat ) )
& ( B2 = bot_bot_set_rat ) )
| ( ( A2
= ( insert_rat @ X @ bot_bot_set_rat ) )
& ( B2
= ( insert_rat @ X @ bot_bot_set_rat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1107_Un__singleton__iff,axiom,
! [A2: set_o,B2: set_o,X: $o] :
( ( ( sup_sup_set_o @ A2 @ B2 )
= ( insert_o @ X @ bot_bot_set_o ) )
= ( ( ( A2 = bot_bot_set_o )
& ( B2
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A2
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B2 = bot_bot_set_o ) )
| ( ( A2
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B2
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1108_insert__is__Un,axiom,
( insert4264952892349296358_preal
= ( ^ [A4: dedekind_preal] : ( sup_su8384800326570918602_preal @ ( insert4264952892349296358_preal @ A4 @ bot_bo4848840305443107682_preal ) ) ) ) ).
% insert_is_Un
thf(fact_1109_insert__is__Un,axiom,
( insert_rat
= ( ^ [A4: rat] : ( sup_sup_set_rat @ ( insert_rat @ A4 @ bot_bot_set_rat ) ) ) ) ).
% insert_is_Un
thf(fact_1110_insert__is__Un,axiom,
( insert_o
= ( ^ [A4: $o] : ( sup_sup_set_o @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ).
% insert_is_Un
thf(fact_1111_bind__const,axiom,
! [A2: set_Dedekind_preal,B2: set_Dedekind_preal] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( bind_D8752870763990849491_preal @ A2
@ ^ [Uu: dedekind_preal] : B2 )
= bot_bo4848840305443107682_preal ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( bind_D8752870763990849491_preal @ A2
@ ^ [Uu: dedekind_preal] : B2 )
= B2 ) ) ) ).
% bind_const
thf(fact_1112_bind__const,axiom,
! [A2: set_Dedekind_preal,B2: set_rat] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( bind_D1371630106014942811al_rat @ A2
@ ^ [Uu: dedekind_preal] : B2 )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( bind_D1371630106014942811al_rat @ A2
@ ^ [Uu: dedekind_preal] : B2 )
= B2 ) ) ) ).
% bind_const
thf(fact_1113_bind__const,axiom,
! [A2: set_Dedekind_preal,B2: set_o] :
( ( ( A2 = bot_bo4848840305443107682_preal )
=> ( ( bind_D7840052183623059541real_o @ A2
@ ^ [Uu: dedekind_preal] : B2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( bind_D7840052183623059541real_o @ A2
@ ^ [Uu: dedekind_preal] : B2 )
= B2 ) ) ) ).
% bind_const
thf(fact_1114_bind__const,axiom,
! [A2: set_rat,B2: set_Dedekind_preal] :
( ( ( A2 = bot_bot_set_rat )
=> ( ( bind_r1911519903473321769_preal @ A2
@ ^ [Uu: rat] : B2 )
= bot_bo4848840305443107682_preal ) )
& ( ( A2 != bot_bot_set_rat )
=> ( ( bind_r1911519903473321769_preal @ A2
@ ^ [Uu: rat] : B2 )
= B2 ) ) ) ).
% bind_const
thf(fact_1115_bind__const,axiom,
! [A2: set_rat,B2: set_rat] :
( ( ( A2 = bot_bot_set_rat )
=> ( ( bind_rat_rat @ A2
@ ^ [Uu: rat] : B2 )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bot_set_rat )
=> ( ( bind_rat_rat @ A2
@ ^ [Uu: rat] : B2 )
= B2 ) ) ) ).
% bind_const
thf(fact_1116_bind__const,axiom,
! [A2: set_rat,B2: set_o] :
( ( ( A2 = bot_bot_set_rat )
=> ( ( bind_rat_o @ A2
@ ^ [Uu: rat] : B2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_rat )
=> ( ( bind_rat_o @ A2
@ ^ [Uu: rat] : B2 )
= B2 ) ) ) ).
% bind_const
thf(fact_1117_bind__const,axiom,
! [A2: set_o,B2: set_Dedekind_preal] :
( ( ( A2 = bot_bot_set_o )
=> ( ( bind_o1395191148525001937_preal @ A2
@ ^ [Uu: $o] : B2 )
= bot_bo4848840305443107682_preal ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( bind_o1395191148525001937_preal @ A2
@ ^ [Uu: $o] : B2 )
= B2 ) ) ) ).
% bind_const
thf(fact_1118_bind__const,axiom,
! [A2: set_o,B2: set_rat] :
( ( ( A2 = bot_bot_set_o )
=> ( ( bind_o_rat @ A2
@ ^ [Uu: $o] : B2 )
= bot_bot_set_rat ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( bind_o_rat @ A2
@ ^ [Uu: $o] : B2 )
= B2 ) ) ) ).
% bind_const
thf(fact_1119_bind__const,axiom,
! [A2: set_o,B2: set_o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( bind_o_o @ A2
@ ^ [Uu: $o] : B2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( bind_o_o @ A2
@ ^ [Uu: $o] : B2 )
= B2 ) ) ) ).
% bind_const
thf(fact_1120_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F: dedekind_preal > set_rat,A2: set_Dedekind_preal,G: dedekind_preal > set_rat] :
( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ A2 ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [A4: dedekind_preal] : ( sup_sup_set_rat @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_1121_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F: set_rat > set_rat,A2: set_set_rat,G: set_rat > set_rat] :
( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ G @ A2 ) ) )
= ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [A4: set_rat] : ( sup_sup_set_rat @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_1122_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F: rat > set_rat,A2: set_rat,G: rat > set_rat] :
( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ G @ A2 ) ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [A4: rat] : ( sup_sup_set_rat @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_1123_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F: set_rat > set_set_rat,A2: set_set_rat,G: set_rat > set_set_rat] :
( ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ F @ A2 ) ) @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ G @ A2 ) ) )
= ( comple3392050375588816791et_rat
@ ( image_9010471772881432025et_rat
@ ^ [A4: set_rat] : ( sup_sup_set_set_rat @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_1124_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F: rat > set_set_rat,A2: set_rat,G: rat > set_set_rat] :
( ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ F @ A2 ) ) @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ G @ A2 ) ) )
= ( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [A4: rat] : ( sup_sup_set_set_rat @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_1125_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F: rat > set_o,A2: set_rat,G: rat > set_o] :
( ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ G @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [A4: rat] : ( sup_sup_set_o @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_1126_SUP__absorb,axiom,
! [K: rat,I3: set_rat,A2: rat > $o] :
( ( member_rat @ K @ I3 )
=> ( ( sup_sup_o @ ( A2 @ K ) @ ( complete_Sup_Sup_o @ ( image_rat_o @ A2 @ I3 ) ) )
= ( complete_Sup_Sup_o @ ( image_rat_o @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_1127_SUP__absorb,axiom,
! [K: $o,I3: set_o,A2: $o > $o] :
( ( member_o @ K @ I3 )
=> ( ( sup_sup_o @ ( A2 @ K ) @ ( complete_Sup_Sup_o @ ( image_o_o @ A2 @ I3 ) ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_1128_SUP__absorb,axiom,
! [K: dedekind_preal,I3: set_Dedekind_preal,A2: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ K @ I3 )
=> ( ( sup_sup_o @ ( A2 @ K ) @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ A2 @ I3 ) ) )
= ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_1129_SUP__absorb,axiom,
! [K: rat,I3: set_rat,A2: rat > set_rat] :
( ( member_rat @ K @ I3 )
=> ( ( sup_sup_set_rat @ ( A2 @ K ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ A2 @ I3 ) ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_1130_SUP__absorb,axiom,
! [K: $o,I3: set_o,A2: $o > set_rat] :
( ( member_o @ K @ I3 )
=> ( ( sup_sup_set_rat @ ( A2 @ K ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ A2 @ I3 ) ) )
= ( comple3890839924845867745et_rat @ ( image_o_set_rat @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_1131_SUP__absorb,axiom,
! [K: dedekind_preal,I3: set_Dedekind_preal,A2: dedekind_preal > set_rat] :
( ( member6871284927547481791_preal @ K @ I3 )
=> ( ( sup_sup_set_rat @ ( A2 @ K ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ A2 @ I3 ) ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_1132_SUP__absorb,axiom,
! [K: set_rat,I3: set_set_rat,A2: set_rat > $o] :
( ( member_set_rat @ K @ I3 )
=> ( ( sup_sup_o @ ( A2 @ K ) @ ( complete_Sup_Sup_o @ ( image_set_rat_o @ A2 @ I3 ) ) )
= ( complete_Sup_Sup_o @ ( image_set_rat_o @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_1133_SUP__absorb,axiom,
! [K: rat,I3: set_rat,A2: rat > set_o] :
( ( member_rat @ K @ I3 )
=> ( ( sup_sup_set_o @ ( A2 @ K ) @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ A2 @ I3 ) ) )
= ( comple90263536869209701_set_o @ ( image_rat_set_o @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_1134_SUP__absorb,axiom,
! [K: $o,I3: set_o,A2: $o > set_o] :
( ( member_o @ K @ I3 )
=> ( ( sup_sup_set_o @ ( A2 @ K ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ I3 ) ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_1135_SUP__absorb,axiom,
! [K: dedekind_preal,I3: set_Dedekind_preal,A2: dedekind_preal > set_o] :
( ( member6871284927547481791_preal @ K @ I3 )
=> ( ( sup_sup_set_o @ ( A2 @ K ) @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ A2 @ I3 ) ) )
= ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ A2 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_1136_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ~ ! [S5: rat] :
( ( ord_less_rat @ zero_zero_rat @ S5 )
=> ! [T2: rat] :
( ( ord_less_rat @ zero_zero_rat @ T2 )
=> ( R2
!= ( plus_plus_rat @ S5 @ T2 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_1137_Un__Union__image,axiom,
! [A2: dedekind_preal > set_rat,B2: dedekind_preal > set_rat,C: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ A2 @ C ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ C ) ) ) ) ).
% Un_Union_image
thf(fact_1138_Un__Union__image,axiom,
! [A2: set_rat > set_rat,B2: set_rat > set_rat,C: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ A2 @ C ) ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ C ) ) ) ) ).
% Un_Union_image
thf(fact_1139_Un__Union__image,axiom,
! [A2: rat > set_rat,B2: rat > set_rat,C: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ A2 @ C ) ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ C ) ) ) ) ).
% Un_Union_image
thf(fact_1140_Un__Union__image,axiom,
! [A2: set_rat > set_set_rat,B2: set_rat > set_set_rat,C: set_set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_9010471772881432025et_rat
@ ^ [X4: set_rat] : ( sup_sup_set_set_rat @ ( A2 @ X4 ) @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ A2 @ C ) ) @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ C ) ) ) ) ).
% Un_Union_image
thf(fact_1141_Un__Union__image,axiom,
! [A2: rat > set_set_rat,B2: rat > set_set_rat,C: set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [X4: rat] : ( sup_sup_set_set_rat @ ( A2 @ X4 ) @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ A2 @ C ) ) @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ C ) ) ) ) ).
% Un_Union_image
thf(fact_1142_Un__Union__image,axiom,
! [A2: rat > set_o,B2: rat > set_o,C: set_rat] :
( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( sup_sup_set_o @ ( A2 @ X4 ) @ ( B2 @ X4 ) )
@ C ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ A2 @ C ) ) @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ C ) ) ) ) ).
% Un_Union_image
thf(fact_1143_UN__Un__distrib,axiom,
! [A2: dedekind_preal > set_rat,B2: dedekind_preal > set_rat,I3: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [I2: dedekind_preal] : ( sup_sup_set_rat @ ( A2 @ I2 ) @ ( B2 @ I2 ) )
@ I3 ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ A2 @ I3 ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ I3 ) ) ) ) ).
% UN_Un_distrib
thf(fact_1144_UN__Un__distrib,axiom,
! [A2: set_rat > set_rat,B2: set_rat > set_rat,I3: set_set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [I2: set_rat] : ( sup_sup_set_rat @ ( A2 @ I2 ) @ ( B2 @ I2 ) )
@ I3 ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ A2 @ I3 ) ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ I3 ) ) ) ) ).
% UN_Un_distrib
thf(fact_1145_UN__Un__distrib,axiom,
! [A2: rat > set_rat,B2: rat > set_rat,I3: set_rat] :
( ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [I2: rat] : ( sup_sup_set_rat @ ( A2 @ I2 ) @ ( B2 @ I2 ) )
@ I3 ) )
= ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ A2 @ I3 ) ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ I3 ) ) ) ) ).
% UN_Un_distrib
thf(fact_1146_UN__Un__distrib,axiom,
! [A2: set_rat > set_set_rat,B2: set_rat > set_set_rat,I3: set_set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_9010471772881432025et_rat
@ ^ [I2: set_rat] : ( sup_sup_set_set_rat @ ( A2 @ I2 ) @ ( B2 @ I2 ) )
@ I3 ) )
= ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ A2 @ I3 ) ) @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ B2 @ I3 ) ) ) ) ).
% UN_Un_distrib
thf(fact_1147_UN__Un__distrib,axiom,
! [A2: rat > set_set_rat,B2: rat > set_set_rat,I3: set_rat] :
( ( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [I2: rat] : ( sup_sup_set_set_rat @ ( A2 @ I2 ) @ ( B2 @ I2 ) )
@ I3 ) )
= ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ A2 @ I3 ) ) @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ I3 ) ) ) ) ).
% UN_Un_distrib
thf(fact_1148_UN__Un__distrib,axiom,
! [A2: rat > set_o,B2: rat > set_o,I3: set_rat] :
( ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [I2: rat] : ( sup_sup_set_o @ ( A2 @ I2 ) @ ( B2 @ I2 ) )
@ I3 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ A2 @ I3 ) ) @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ I3 ) ) ) ) ).
% UN_Un_distrib
thf(fact_1149_UN__absorb,axiom,
! [K: rat,I3: set_rat,A2: rat > set_rat] :
( ( member_rat @ K @ I3 )
=> ( ( sup_sup_set_rat @ ( A2 @ K ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ A2 @ I3 ) ) )
= ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_1150_UN__absorb,axiom,
! [K: $o,I3: set_o,A2: $o > set_rat] :
( ( member_o @ K @ I3 )
=> ( ( sup_sup_set_rat @ ( A2 @ K ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ A2 @ I3 ) ) )
= ( comple3890839924845867745et_rat @ ( image_o_set_rat @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_1151_UN__absorb,axiom,
! [K: dedekind_preal,I3: set_Dedekind_preal,A2: dedekind_preal > set_rat] :
( ( member6871284927547481791_preal @ K @ I3 )
=> ( ( sup_sup_set_rat @ ( A2 @ K ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ A2 @ I3 ) ) )
= ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_1152_UN__absorb,axiom,
! [K: rat,I3: set_rat,A2: rat > set_o] :
( ( member_rat @ K @ I3 )
=> ( ( sup_sup_set_o @ ( A2 @ K ) @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ A2 @ I3 ) ) )
= ( comple90263536869209701_set_o @ ( image_rat_set_o @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_1153_UN__absorb,axiom,
! [K: $o,I3: set_o,A2: $o > set_o] :
( ( member_o @ K @ I3 )
=> ( ( sup_sup_set_o @ ( A2 @ K ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ I3 ) ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_1154_UN__absorb,axiom,
! [K: dedekind_preal,I3: set_Dedekind_preal,A2: dedekind_preal > set_o] :
( ( member6871284927547481791_preal @ K @ I3 )
=> ( ( sup_sup_set_o @ ( A2 @ K ) @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ A2 @ I3 ) ) )
= ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_1155_UN__absorb,axiom,
! [K: set_rat,I3: set_set_rat,A2: set_rat > set_rat] :
( ( member_set_rat @ K @ I3 )
=> ( ( sup_sup_set_rat @ ( A2 @ K ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ A2 @ I3 ) ) )
= ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_1156_UN__absorb,axiom,
! [K: rat,I3: set_rat,A2: rat > set_set_rat] :
( ( member_rat @ K @ I3 )
=> ( ( sup_sup_set_set_rat @ ( A2 @ K ) @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ A2 @ I3 ) ) )
= ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_1157_UN__absorb,axiom,
! [K: $o,I3: set_o,A2: $o > set_set_rat] :
( ( member_o @ K @ I3 )
=> ( ( sup_sup_set_set_rat @ ( A2 @ K ) @ ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ A2 @ I3 ) ) )
= ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_1158_UN__absorb,axiom,
! [K: dedekind_preal,I3: set_Dedekind_preal,A2: dedekind_preal > set_set_rat] :
( ( member6871284927547481791_preal @ K @ I3 )
=> ( ( sup_sup_set_set_rat @ ( A2 @ K ) @ ( comple3392050375588816791et_rat @ ( image_8268678932589273365et_rat @ A2 @ I3 ) ) )
= ( comple3392050375588816791et_rat @ ( image_8268678932589273365et_rat @ A2 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_1159_is__singletonI_H,axiom,
! [A2: set_set_rat] :
( ( A2 != bot_bot_set_set_rat )
=> ( ! [X3: set_rat,Y4: set_rat] :
( ( member_set_rat @ X3 @ A2 )
=> ( ( member_set_rat @ Y4 @ A2 )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_set_rat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1160_is__singletonI_H,axiom,
! [A2: set_Dedekind_preal] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ! [X3: dedekind_preal,Y4: dedekind_preal] :
( ( member6871284927547481791_preal @ X3 @ A2 )
=> ( ( member6871284927547481791_preal @ Y4 @ A2 )
=> ( X3 = Y4 ) ) )
=> ( is_sin4170067188624449034_preal @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1161_is__singletonI_H,axiom,
! [A2: set_rat] :
( ( A2 != bot_bot_set_rat )
=> ( ! [X3: rat,Y4: rat] :
( ( member_rat @ X3 @ A2 )
=> ( ( member_rat @ Y4 @ A2 )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_rat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1162_is__singletonI_H,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o,Y4: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( member_o @ Y4 @ A2 )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_o @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1163_bind__UNION,axiom,
( bind_D1371630106014942811al_rat
= ( ^ [A3: set_Dedekind_preal,F2: dedekind_preal > set_rat] : ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F2 @ A3 ) ) ) ) ).
% bind_UNION
thf(fact_1164_bind__UNION,axiom,
( bind_set_rat_rat
= ( ^ [A3: set_set_rat,F2: set_rat > set_rat] : ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F2 @ A3 ) ) ) ) ).
% bind_UNION
thf(fact_1165_bind__UNION,axiom,
( bind_rat_rat
= ( ^ [A3: set_rat,F2: rat > set_rat] : ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F2 @ A3 ) ) ) ) ).
% bind_UNION
thf(fact_1166_bind__UNION,axiom,
( bind_set_rat_set_rat
= ( ^ [A3: set_set_rat,F2: set_rat > set_set_rat] : ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ F2 @ A3 ) ) ) ) ).
% bind_UNION
thf(fact_1167_bind__UNION,axiom,
( bind_rat_set_rat
= ( ^ [A3: set_rat,F2: rat > set_set_rat] : ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ F2 @ A3 ) ) ) ) ).
% bind_UNION
thf(fact_1168_bind__UNION,axiom,
( bind_rat_o
= ( ^ [A3: set_rat,F2: rat > set_o] : ( comple90263536869209701_set_o @ ( image_rat_set_o @ F2 @ A3 ) ) ) ) ).
% bind_UNION
thf(fact_1169_SUP__insert,axiom,
! [F: dedekind_preal > set_rat,A: dedekind_preal,A2: set_Dedekind_preal] :
( ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ ( insert4264952892349296358_preal @ A @ A2 ) ) )
= ( sup_sup_set_rat @ ( F @ A ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_1170_SUP__insert,axiom,
! [F: set_rat > set_rat,A: set_rat,A2: set_set_rat] :
( ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ ( insert_set_rat @ A @ A2 ) ) )
= ( sup_sup_set_rat @ ( F @ A ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_1171_SUP__insert,axiom,
! [F: rat > set_rat,A: rat,A2: set_rat] :
( ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ ( insert_rat @ A @ A2 ) ) )
= ( sup_sup_set_rat @ ( F @ A ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_1172_SUP__insert,axiom,
! [F: $o > set_rat,A: $o,A2: set_o] :
( ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ ( insert_o @ A @ A2 ) ) )
= ( sup_sup_set_rat @ ( F @ A ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_1173_SUP__insert,axiom,
! [F: $o > $o,A: $o,A2: set_o] :
( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( insert_o @ A @ A2 ) ) )
= ( sup_sup_o @ ( F @ A ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_1174_SUP__insert,axiom,
! [F: set_rat > set_set_rat,A: set_rat,A2: set_set_rat] :
( ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ F @ ( insert_set_rat @ A @ A2 ) ) )
= ( sup_sup_set_set_rat @ ( F @ A ) @ ( comple3392050375588816791et_rat @ ( image_9010471772881432025et_rat @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_1175_SUP__insert,axiom,
! [F: rat > set_set_rat,A: rat,A2: set_rat] :
( ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ F @ ( insert_rat @ A @ A2 ) ) )
= ( sup_sup_set_set_rat @ ( F @ A ) @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_1176_SUP__insert,axiom,
! [F: $o > set_set_rat,A: $o,A2: set_o] :
( ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ F @ ( insert_o @ A @ A2 ) ) )
= ( sup_sup_set_set_rat @ ( F @ A ) @ ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_1177_SUP__insert,axiom,
! [F: rat > set_o,A: rat,A2: set_rat] :
( ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ ( insert_rat @ A @ A2 ) ) )
= ( sup_sup_set_o @ ( F @ A ) @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_1178_SUP__insert,axiom,
! [F: $o > set_o,A: $o,A2: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ ( insert_o @ A @ A2 ) ) )
= ( sup_sup_set_o @ ( F @ A ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% SUP_insert
thf(fact_1179_UN__extend__simps_I2_J,axiom,
! [C: set_Dedekind_preal,A2: dedekind_preal > set_rat,B2: set_rat] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ A2 @ C ) ) @ B2 )
= B2 ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ A2 @ C ) ) @ B2 )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1180_UN__extend__simps_I2_J,axiom,
! [C: set_rat,A2: rat > set_rat,B2: set_rat] :
( ( ( C = bot_bot_set_rat )
=> ( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ A2 @ C ) ) @ B2 )
= B2 ) )
& ( ( C != bot_bot_set_rat )
=> ( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ A2 @ C ) ) @ B2 )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1181_UN__extend__simps_I2_J,axiom,
! [C: set_o,A2: $o > set_rat,B2: set_rat] :
( ( ( C = bot_bot_set_o )
=> ( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ A2 @ C ) ) @ B2 )
= B2 ) )
& ( ( C != bot_bot_set_o )
=> ( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ A2 @ C ) ) @ B2 )
= ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1182_UN__extend__simps_I2_J,axiom,
! [C: set_Dedekind_preal,A2: dedekind_preal > set_o,B2: set_o] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ A2 @ C ) ) @ B2 )
= B2 ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ A2 @ C ) ) @ B2 )
= ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( sup_sup_set_o @ ( A2 @ X4 ) @ B2 )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1183_UN__extend__simps_I2_J,axiom,
! [C: set_rat,A2: rat > set_o,B2: set_o] :
( ( ( C = bot_bot_set_rat )
=> ( ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ A2 @ C ) ) @ B2 )
= B2 ) )
& ( ( C != bot_bot_set_rat )
=> ( ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ A2 @ C ) ) @ B2 )
= ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( sup_sup_set_o @ ( A2 @ X4 ) @ B2 )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1184_UN__extend__simps_I2_J,axiom,
! [C: set_o,A2: $o > set_o,B2: set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ C ) ) @ B2 )
= B2 ) )
& ( ( C != bot_bot_set_o )
=> ( ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ C ) ) @ B2 )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ ( A2 @ X4 ) @ B2 )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1185_UN__extend__simps_I2_J,axiom,
! [C: set_set_rat,A2: set_rat > set_rat,B2: set_rat] :
( ( ( C = bot_bot_set_set_rat )
=> ( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ A2 @ C ) ) @ B2 )
= B2 ) )
& ( ( C != bot_bot_set_set_rat )
=> ( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ A2 @ C ) ) @ B2 )
= ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( sup_sup_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1186_UN__extend__simps_I2_J,axiom,
! [C: set_Dedekind_preal,A2: dedekind_preal > set_set_rat,B2: set_set_rat] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_8268678932589273365et_rat @ A2 @ C ) ) @ B2 )
= B2 ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_8268678932589273365et_rat @ A2 @ C ) ) @ B2 )
= ( comple3392050375588816791et_rat
@ ( image_8268678932589273365et_rat
@ ^ [X4: dedekind_preal] : ( sup_sup_set_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1187_UN__extend__simps_I2_J,axiom,
! [C: set_rat,A2: rat > set_set_rat,B2: set_set_rat] :
( ( ( C = bot_bot_set_rat )
=> ( ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ A2 @ C ) ) @ B2 )
= B2 ) )
& ( ( C != bot_bot_set_rat )
=> ( ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ A2 @ C ) ) @ B2 )
= ( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [X4: rat] : ( sup_sup_set_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1188_UN__extend__simps_I2_J,axiom,
! [C: set_o,A2: $o > set_set_rat,B2: set_set_rat] :
( ( ( C = bot_bot_set_o )
=> ( ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ A2 @ C ) ) @ B2 )
= B2 ) )
& ( ( C != bot_bot_set_o )
=> ( ( sup_sup_set_set_rat @ ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ A2 @ C ) ) @ B2 )
= ( comple3392050375588816791et_rat
@ ( image_o_set_set_rat
@ ^ [X4: $o] : ( sup_sup_set_set_rat @ ( A2 @ X4 ) @ B2 )
@ C ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1189_UN__extend__simps_I3_J,axiom,
! [C: set_Dedekind_preal,A2: set_rat,B2: dedekind_preal > set_rat] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( sup_sup_set_rat @ A2 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ C ) ) )
= A2 ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( sup_sup_set_rat @ A2 @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ B2 @ C ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [X4: dedekind_preal] : ( sup_sup_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1190_UN__extend__simps_I3_J,axiom,
! [C: set_rat,A2: set_rat,B2: rat > set_rat] :
( ( ( C = bot_bot_set_rat )
=> ( ( sup_sup_set_rat @ A2 @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_rat )
=> ( ( sup_sup_set_rat @ A2 @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ B2 @ C ) ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [X4: rat] : ( sup_sup_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1191_UN__extend__simps_I3_J,axiom,
! [C: set_o,A2: set_rat,B2: $o > set_rat] :
( ( ( C = bot_bot_set_o )
=> ( ( sup_sup_set_rat @ A2 @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_o )
=> ( ( sup_sup_set_rat @ A2 @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B2 @ C ) ) )
= ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( sup_sup_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1192_UN__extend__simps_I3_J,axiom,
! [C: set_Dedekind_preal,A2: set_o,B2: dedekind_preal > set_o] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( sup_sup_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ C ) ) )
= A2 ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( sup_sup_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ B2 @ C ) ) )
= ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [X4: dedekind_preal] : ( sup_sup_set_o @ A2 @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1193_UN__extend__simps_I3_J,axiom,
! [C: set_rat,A2: set_o,B2: rat > set_o] :
( ( ( C = bot_bot_set_rat )
=> ( ( sup_sup_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_rat )
=> ( ( sup_sup_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ B2 @ C ) ) )
= ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [X4: rat] : ( sup_sup_set_o @ A2 @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1194_UN__extend__simps_I3_J,axiom,
! [C: set_o,A2: set_o,B2: $o > set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( sup_sup_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_o )
=> ( ( sup_sup_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ C ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ A2 @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1195_UN__extend__simps_I3_J,axiom,
! [C: set_set_rat,A2: set_rat,B2: set_rat > set_rat] :
( ( ( C = bot_bot_set_set_rat )
=> ( ( sup_sup_set_rat @ A2 @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_set_rat )
=> ( ( sup_sup_set_rat @ A2 @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ B2 @ C ) ) )
= ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [X4: set_rat] : ( sup_sup_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1196_UN__extend__simps_I3_J,axiom,
! [C: set_Dedekind_preal,A2: set_set_rat,B2: dedekind_preal > set_set_rat] :
( ( ( C = bot_bo4848840305443107682_preal )
=> ( ( sup_sup_set_set_rat @ A2 @ ( comple3392050375588816791et_rat @ ( image_8268678932589273365et_rat @ B2 @ C ) ) )
= A2 ) )
& ( ( C != bot_bo4848840305443107682_preal )
=> ( ( sup_sup_set_set_rat @ A2 @ ( comple3392050375588816791et_rat @ ( image_8268678932589273365et_rat @ B2 @ C ) ) )
= ( comple3392050375588816791et_rat
@ ( image_8268678932589273365et_rat
@ ^ [X4: dedekind_preal] : ( sup_sup_set_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1197_UN__extend__simps_I3_J,axiom,
! [C: set_rat,A2: set_set_rat,B2: rat > set_set_rat] :
( ( ( C = bot_bot_set_rat )
=> ( ( sup_sup_set_set_rat @ A2 @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_rat )
=> ( ( sup_sup_set_set_rat @ A2 @ ( comple3392050375588816791et_rat @ ( image_5169559123115255331et_rat @ B2 @ C ) ) )
= ( comple3392050375588816791et_rat
@ ( image_5169559123115255331et_rat
@ ^ [X4: rat] : ( sup_sup_set_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1198_UN__extend__simps_I3_J,axiom,
! [C: set_o,A2: set_set_rat,B2: $o > set_set_rat] :
( ( ( C = bot_bot_set_o )
=> ( ( sup_sup_set_set_rat @ A2 @ ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ B2 @ C ) ) )
= A2 ) )
& ( ( C != bot_bot_set_o )
=> ( ( sup_sup_set_set_rat @ A2 @ ( comple3392050375588816791et_rat @ ( image_o_set_set_rat @ B2 @ C ) ) )
= ( comple3392050375588816791et_rat
@ ( image_o_set_set_rat
@ ^ [X4: $o] : ( sup_sup_set_set_rat @ A2 @ ( B2 @ X4 ) )
@ C ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1199_Gleason9__34__contra,axiom,
! [A2: set_rat,U2: rat,Y: rat] :
( ( dedekind_cut @ A2 )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ A2 )
=> ( member_rat @ ( plus_plus_rat @ X3 @ U2 ) @ A2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ U2 )
=> ( ( ord_less_rat @ zero_zero_rat @ Y )
=> ( member_rat @ Y @ A2 ) ) ) ) ) ).
% Gleason9_34_contra
thf(fact_1200_Gleason9__34,axiom,
! [A2: set_rat,U2: rat] :
( ( dedekind_cut @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ U2 )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ~ ( member_rat @ ( plus_plus_rat @ X3 @ U2 ) @ A2 ) ) ) ) ).
% Gleason9_34
thf(fact_1201_cSup__insert,axiom,
! [X2: set_set_rat,A: set_rat] :
( ( X2 != bot_bot_set_set_rat )
=> ( ( condit1969311730731577898et_rat @ X2 )
=> ( ( comple3890839924845867745et_rat @ ( insert_set_rat @ A @ X2 ) )
= ( sup_sup_set_rat @ A @ ( comple3890839924845867745et_rat @ X2 ) ) ) ) ) ).
% cSup_insert
thf(fact_1202_cSup__insert,axiom,
! [X2: set_o,A: $o] :
( ( X2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ X2 )
=> ( ( complete_Sup_Sup_o @ ( insert_o @ A @ X2 ) )
= ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ X2 ) ) ) ) ) ).
% cSup_insert
thf(fact_1203_cSup__insert,axiom,
! [X2: set_set_set_rat,A: set_set_rat] :
( ( X2 != bot_bo6619408370577057422et_rat )
=> ( ( condit6513867565548536544et_rat @ X2 )
=> ( ( comple3392050375588816791et_rat @ ( insert_set_set_rat @ A @ X2 ) )
= ( sup_sup_set_set_rat @ A @ ( comple3392050375588816791et_rat @ X2 ) ) ) ) ) ).
% cSup_insert
thf(fact_1204_cSup__insert,axiom,
! [X2: set_set_o,A: set_o] :
( ( X2 != bot_bot_set_set_o )
=> ( ( condit9194965722646069980_set_o @ X2 )
=> ( ( comple90263536869209701_set_o @ ( insert_set_o @ A @ X2 ) )
= ( sup_sup_set_o @ A @ ( comple90263536869209701_set_o @ X2 ) ) ) ) ) ).
% cSup_insert
thf(fact_1205_cSup__insert__If,axiom,
! [X2: set_set_rat,A: set_rat] :
( ( condit1969311730731577898et_rat @ X2 )
=> ( ( ( X2 = bot_bot_set_set_rat )
=> ( ( comple3890839924845867745et_rat @ ( insert_set_rat @ A @ X2 ) )
= A ) )
& ( ( X2 != bot_bot_set_set_rat )
=> ( ( comple3890839924845867745et_rat @ ( insert_set_rat @ A @ X2 ) )
= ( sup_sup_set_rat @ A @ ( comple3890839924845867745et_rat @ X2 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_1206_cSup__insert__If,axiom,
! [X2: set_o,A: $o] :
( ( condit5488710616941104124bove_o @ X2 )
=> ( ( complete_Sup_Sup_o @ ( insert_o @ A @ X2 ) )
= ( ( ( X2 = bot_bot_set_o )
=> A )
& ( ( X2 != bot_bot_set_o )
=> ( sup_sup_o @ A @ ( complete_Sup_Sup_o @ X2 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_1207_cSup__insert__If,axiom,
! [X2: set_set_set_rat,A: set_set_rat] :
( ( condit6513867565548536544et_rat @ X2 )
=> ( ( ( X2 = bot_bo6619408370577057422et_rat )
=> ( ( comple3392050375588816791et_rat @ ( insert_set_set_rat @ A @ X2 ) )
= A ) )
& ( ( X2 != bot_bo6619408370577057422et_rat )
=> ( ( comple3392050375588816791et_rat @ ( insert_set_set_rat @ A @ X2 ) )
= ( sup_sup_set_set_rat @ A @ ( comple3392050375588816791et_rat @ X2 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_1208_cSup__insert__If,axiom,
! [X2: set_set_o,A: set_o] :
( ( condit9194965722646069980_set_o @ X2 )
=> ( ( ( X2 = bot_bot_set_set_o )
=> ( ( comple90263536869209701_set_o @ ( insert_set_o @ A @ X2 ) )
= A ) )
& ( ( X2 != bot_bot_set_set_o )
=> ( ( comple90263536869209701_set_o @ ( insert_set_o @ A @ X2 ) )
= ( sup_sup_set_o @ A @ ( comple90263536869209701_set_o @ X2 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_1209_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_Dedekind_preal,F: dedekind_preal > $o,G: dedekind_preal > $o] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( condit5488710616941104124bove_o @ ( image_248103505173020295real_o @ F @ A2 ) )
=> ( ( condit5488710616941104124bove_o @ ( image_248103505173020295real_o @ G @ A2 ) )
=> ( ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_248103505173020295real_o @ G @ A2 ) ) )
= ( complete_Sup_Sup_o
@ ( image_248103505173020295real_o
@ ^ [A4: dedekind_preal] : ( sup_sup_o @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_1210_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_rat,F: rat > $o,G: rat > $o] :
( ( A2 != bot_bot_set_rat )
=> ( ( condit5488710616941104124bove_o @ ( image_rat_o @ F @ A2 ) )
=> ( ( condit5488710616941104124bove_o @ ( image_rat_o @ G @ A2 ) )
=> ( ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_rat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_rat_o @ G @ A2 ) ) )
= ( complete_Sup_Sup_o
@ ( image_rat_o
@ ^ [A4: rat] : ( sup_sup_o @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_1211_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_o,F: $o > $o,G: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ F @ A2 ) )
=> ( ( condit5488710616941104124bove_o @ ( image_o_o @ G @ A2 ) )
=> ( ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ A2 ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [A4: $o] : ( sup_sup_o @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_1212_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_Dedekind_preal,F: dedekind_preal > set_rat,G: dedekind_preal > set_rat] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) )
=> ( ( condit1969311730731577898et_rat @ ( image_2231269897757205599et_rat @ G @ A2 ) )
=> ( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ F @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ G @ A2 ) ) )
= ( comple3890839924845867745et_rat
@ ( image_2231269897757205599et_rat
@ ^ [A4: dedekind_preal] : ( sup_sup_set_rat @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_1213_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_rat,F: rat > set_rat,G: rat > set_rat] :
( ( A2 != bot_bot_set_rat )
=> ( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ F @ A2 ) )
=> ( ( condit1969311730731577898et_rat @ ( image_rat_set_rat @ G @ A2 ) )
=> ( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ F @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_rat_set_rat @ G @ A2 ) ) )
= ( comple3890839924845867745et_rat
@ ( image_rat_set_rat
@ ^ [A4: rat] : ( sup_sup_set_rat @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_1214_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_o,F: $o > set_rat,G: $o > set_rat] :
( ( A2 != bot_bot_set_o )
=> ( ( condit1969311730731577898et_rat @ ( image_o_set_rat @ F @ A2 ) )
=> ( ( condit1969311730731577898et_rat @ ( image_o_set_rat @ G @ A2 ) )
=> ( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ F @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ G @ A2 ) ) )
= ( comple3890839924845867745et_rat
@ ( image_o_set_rat
@ ^ [A4: $o] : ( sup_sup_set_rat @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_1215_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_Dedekind_preal,F: dedekind_preal > set_o,G: dedekind_preal > set_o] :
( ( A2 != bot_bo4848840305443107682_preal )
=> ( ( condit9194965722646069980_set_o @ ( image_8163909360452110695_set_o @ F @ A2 ) )
=> ( ( condit9194965722646069980_set_o @ ( image_8163909360452110695_set_o @ G @ A2 ) )
=> ( ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_8163909360452110695_set_o @ G @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_8163909360452110695_set_o
@ ^ [A4: dedekind_preal] : ( sup_sup_set_o @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_1216_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_rat,F: rat > set_o,G: rat > set_o] :
( ( A2 != bot_bot_set_rat )
=> ( ( condit9194965722646069980_set_o @ ( image_rat_set_o @ F @ A2 ) )
=> ( ( condit9194965722646069980_set_o @ ( image_rat_set_o @ G @ A2 ) )
=> ( ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_rat_set_o @ G @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_rat_set_o
@ ^ [A4: rat] : ( sup_sup_set_o @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_1217_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_o,F: $o > set_o,G: $o > set_o] :
( ( A2 != bot_bot_set_o )
=> ( ( condit9194965722646069980_set_o @ ( image_o_set_o @ F @ A2 ) )
=> ( ( condit9194965722646069980_set_o @ ( image_o_set_o @ G @ A2 ) )
=> ( ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [A4: $o] : ( sup_sup_set_o @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_1218_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A2: set_set_rat,F: set_rat > set_rat,G: set_rat > set_rat] :
( ( A2 != bot_bot_set_set_rat )
=> ( ( condit1969311730731577898et_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) )
=> ( ( condit1969311730731577898et_rat @ ( image_3939399684171694371et_rat @ G @ A2 ) )
=> ( ( sup_sup_set_rat @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ F @ A2 ) ) @ ( comple3890839924845867745et_rat @ ( image_3939399684171694371et_rat @ G @ A2 ) ) )
= ( comple3890839924845867745et_rat
@ ( image_3939399684171694371et_rat
@ ^ [A4: set_rat] : ( sup_sup_set_rat @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_1219_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_1220_sup__bot_Oright__neutral,axiom,
! [A: set_Dedekind_preal] :
( ( sup_su8384800326570918602_preal @ A @ bot_bo4848840305443107682_preal )
= A ) ).
% sup_bot.right_neutral
thf(fact_1221_sup__bot_Oright__neutral,axiom,
! [A: set_rat] :
( ( sup_sup_set_rat @ A @ bot_bot_set_rat )
= A ) ).
% sup_bot.right_neutral
thf(fact_1222_sup__bot_Oright__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_1223_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal] :
( ( bot_bo4848840305443107682_preal
= ( sup_su8384800326570918602_preal @ A @ B ) )
= ( ( A = bot_bo4848840305443107682_preal )
& ( B = bot_bo4848840305443107682_preal ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1224_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_rat,B: set_rat] :
( ( bot_bot_set_rat
= ( sup_sup_set_rat @ A @ B ) )
= ( ( A = bot_bot_set_rat )
& ( B = bot_bot_set_rat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1225_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_o,B: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ A @ B ) )
= ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1226_sup__bot_Oleft__neutral,axiom,
! [A: set_Dedekind_preal] :
( ( sup_su8384800326570918602_preal @ bot_bo4848840305443107682_preal @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1227_sup__bot_Oleft__neutral,axiom,
! [A: set_rat] :
( ( sup_sup_set_rat @ bot_bot_set_rat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1228_sup__bot_Oleft__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1229_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Dedekind_preal,B: set_Dedekind_preal] :
( ( ( sup_su8384800326570918602_preal @ A @ B )
= bot_bo4848840305443107682_preal )
= ( ( A = bot_bo4848840305443107682_preal )
& ( B = bot_bo4848840305443107682_preal ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1230_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_rat,B: set_rat] :
( ( ( sup_sup_set_rat @ A @ B )
= bot_bot_set_rat )
= ( ( A = bot_bot_set_rat )
& ( B = bot_bot_set_rat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1231_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A @ B )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1232_sup__eq__bot__iff,axiom,
! [X: set_Dedekind_preal,Y: set_Dedekind_preal] :
( ( ( sup_su8384800326570918602_preal @ X @ Y )
= bot_bo4848840305443107682_preal )
= ( ( X = bot_bo4848840305443107682_preal )
& ( Y = bot_bo4848840305443107682_preal ) ) ) ).
% sup_eq_bot_iff
thf(fact_1233_sup__eq__bot__iff,axiom,
! [X: set_rat,Y: set_rat] :
( ( ( sup_sup_set_rat @ X @ Y )
= bot_bot_set_rat )
= ( ( X = bot_bot_set_rat )
& ( Y = bot_bot_set_rat ) ) ) ).
% sup_eq_bot_iff
thf(fact_1234_sup__eq__bot__iff,axiom,
! [X: set_o,Y: set_o] :
( ( ( sup_sup_set_o @ X @ Y )
= bot_bot_set_o )
= ( ( X = bot_bot_set_o )
& ( Y = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_1235_bot__eq__sup__iff,axiom,
! [X: set_Dedekind_preal,Y: set_Dedekind_preal] :
( ( bot_bo4848840305443107682_preal
= ( sup_su8384800326570918602_preal @ X @ Y ) )
= ( ( X = bot_bo4848840305443107682_preal )
& ( Y = bot_bo4848840305443107682_preal ) ) ) ).
% bot_eq_sup_iff
thf(fact_1236_bot__eq__sup__iff,axiom,
! [X: set_rat,Y: set_rat] :
( ( bot_bot_set_rat
= ( sup_sup_set_rat @ X @ Y ) )
= ( ( X = bot_bot_set_rat )
& ( Y = bot_bot_set_rat ) ) ) ).
% bot_eq_sup_iff
thf(fact_1237_bot__eq__sup__iff,axiom,
! [X: set_o,Y: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ X @ Y ) )
= ( ( X = bot_bot_set_o )
& ( Y = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_1238_sup__bot__right,axiom,
! [X: set_Dedekind_preal] :
( ( sup_su8384800326570918602_preal @ X @ bot_bo4848840305443107682_preal )
= X ) ).
% sup_bot_right
thf(fact_1239_sup__bot__right,axiom,
! [X: set_rat] :
( ( sup_sup_set_rat @ X @ bot_bot_set_rat )
= X ) ).
% sup_bot_right
thf(fact_1240_sup__bot__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ bot_bot_set_o )
= X ) ).
% sup_bot_right
thf(fact_1241_Un__iff,axiom,
! [C2: rat,A2: set_rat,B2: set_rat] :
( ( member_rat @ C2 @ ( sup_sup_set_rat @ A2 @ B2 ) )
= ( ( member_rat @ C2 @ A2 )
| ( member_rat @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_1242_Un__iff,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B2 ) )
= ( ( member_o @ C2 @ A2 )
| ( member_o @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_1243_Un__iff,axiom,
! [C2: set_rat,A2: set_set_rat,B2: set_set_rat] :
( ( member_set_rat @ C2 @ ( sup_sup_set_set_rat @ A2 @ B2 ) )
= ( ( member_set_rat @ C2 @ A2 )
| ( member_set_rat @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_1244_Un__iff,axiom,
! [C2: dedekind_preal,A2: set_Dedekind_preal,B2: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ C2 @ ( sup_su8384800326570918602_preal @ A2 @ B2 ) )
= ( ( member6871284927547481791_preal @ C2 @ A2 )
| ( member6871284927547481791_preal @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_1245_UnCI,axiom,
! [C2: dedekind_preal,B2: set_Dedekind_preal,A2: set_Dedekind_preal] :
( ( ~ ( member6871284927547481791_preal @ C2 @ B2 )
=> ( member6871284927547481791_preal @ C2 @ A2 ) )
=> ( member6871284927547481791_preal @ C2 @ ( sup_su8384800326570918602_preal @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_1246_preal__add__assoc,axiom,
! [X: dedekind_preal,Y: dedekind_preal,Z2: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ X @ Y ) @ Z2 )
= ( plus_p3173629198307831117_preal @ X @ ( plus_p3173629198307831117_preal @ Y @ Z2 ) ) ) ).
% preal_add_assoc
thf(fact_1247_preal__add__commute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [X4: dedekind_preal,Y2: dedekind_preal] : ( plus_p3173629198307831117_preal @ Y2 @ X4 ) ) ) ).
% preal_add_commute
thf(fact_1248_preal__add__left__cancel,axiom,
! [C2: dedekind_preal,A: dedekind_preal,B: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ C2 @ A )
= ( plus_p3173629198307831117_preal @ C2 @ B ) )
=> ( A = B ) ) ).
% preal_add_left_cancel
thf(fact_1249_preal__add__right__cancel,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S3: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ R2 @ T )
= ( plus_p3173629198307831117_preal @ S3 @ T ) )
=> ( R2 = S3 ) ) ).
% preal_add_right_cancel
thf(fact_1250_mem__Rep__preal__diff__iff,axiom,
! [R2: dedekind_preal,S3: dedekind_preal,Z2: rat] :
( ( ord_le5708704896291381698_preal @ R2 @ S3 )
=> ( ( member_rat @ Z2 @ ( dedekind_Rep_preal @ ( minus_7336623429200594941_preal @ S3 @ R2 ) ) )
= ( ? [X4: rat] :
( ( ord_less_rat @ zero_zero_rat @ X4 )
& ( ord_less_rat @ zero_zero_rat @ Z2 )
& ~ ( member_rat @ X4 @ ( dedekind_Rep_preal @ R2 ) )
& ( member_rat @ ( plus_plus_rat @ X4 @ Z2 ) @ ( dedekind_Rep_preal @ S3 ) ) ) ) ) ) ).
% mem_Rep_preal_diff_iff
thf(fact_1251_diff__set__def,axiom,
( dedekind_diff_set
= ( ^ [A3: set_rat,B3: set_rat] :
( collect_rat
@ ^ [W: rat] :
? [X4: rat] :
( ( ord_less_rat @ zero_zero_rat @ W )
& ( ord_less_rat @ zero_zero_rat @ X4 )
& ~ ( member_rat @ X4 @ B3 )
& ( member_rat @ ( plus_plus_rat @ X4 @ W ) @ A3 ) ) ) ) ) ).
% diff_set_def
thf(fact_1252_less__add__left,axiom,
! [R2: dedekind_preal,S3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S3 )
=> ( ( plus_p3173629198307831117_preal @ R2 @ ( minus_7336623429200594941_preal @ S3 @ R2 ) )
= S3 ) ) ).
% less_add_left
thf(fact_1253_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_1254_inverse__set__def,axiom,
( dedekind_inverse_set
= ( ^ [A3: set_rat] :
( collect_rat
@ ^ [X4: rat] :
? [Y2: rat] :
( ( ord_less_rat @ zero_zero_rat @ X4 )
& ( ord_less_rat @ X4 @ Y2 )
& ~ ( member_rat @ ( inverse_inverse_rat @ Y2 ) @ A3 ) ) ) ) ) ).
% inverse_set_def
thf(fact_1255_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_1256_preal__mult__assoc,axiom,
! [X: dedekind_preal,Y: dedekind_preal,Z2: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ X @ Y ) @ Z2 )
= ( times_3000655703912201937_preal @ X @ ( times_3000655703912201937_preal @ Y @ Z2 ) ) ) ).
% preal_mult_assoc
thf(fact_1257_preal__mult__commute,axiom,
( times_3000655703912201937_preal
= ( ^ [X4: dedekind_preal,Y2: dedekind_preal] : ( times_3000655703912201937_preal @ Y2 @ X4 ) ) ) ).
% preal_mult_commute
thf(fact_1258_preal__add__mult__distrib2,axiom,
! [W2: dedekind_preal,X: dedekind_preal,Y: dedekind_preal] :
( ( times_3000655703912201937_preal @ W2 @ ( plus_p3173629198307831117_preal @ X @ Y ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W2 @ X ) @ ( times_3000655703912201937_preal @ W2 @ Y ) ) ) ).
% preal_add_mult_distrib2
thf(fact_1259_preal__add__mult__distrib,axiom,
! [X: dedekind_preal,Y: dedekind_preal,W2: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ X @ Y ) @ W2 )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X @ W2 ) @ ( times_3000655703912201937_preal @ Y @ W2 ) ) ) ).
% preal_add_mult_distrib
thf(fact_1260_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1261_preal__add__mult__distrib__mean,axiom,
! [A: rat,W2: dedekind_preal,B: rat,D3: rat,X: dedekind_preal,E2: rat,Y: dedekind_preal] :
( ( member_rat @ A @ ( dedekind_Rep_preal @ W2 ) )
=> ( ( member_rat @ B @ ( dedekind_Rep_preal @ W2 ) )
=> ( ( member_rat @ D3 @ ( dedekind_Rep_preal @ X ) )
=> ( ( member_rat @ E2 @ ( dedekind_Rep_preal @ Y ) )
=> ? [X3: rat] :
( ( member_rat @ X3 @ ( dedekind_Rep_preal @ W2 ) )
& ( ( plus_plus_rat @ ( times_times_rat @ A @ D3 ) @ ( times_times_rat @ B @ E2 ) )
= ( times_times_rat @ X3 @ ( plus_plus_rat @ D3 @ E2 ) ) ) ) ) ) ) ) ).
% preal_add_mult_distrib_mean
thf(fact_1262_mem__Rep__preal__mult__iff,axiom,
! [Z2: rat,R2: dedekind_preal,S3: dedekind_preal] :
( ( member_rat @ Z2 @ ( dedekind_Rep_preal @ ( times_3000655703912201937_preal @ R2 @ S3 ) ) )
= ( ? [X4: rat] :
( ( member_rat @ X4 @ ( dedekind_Rep_preal @ R2 ) )
& ? [Y2: rat] :
( ( member_rat @ Y2 @ ( dedekind_Rep_preal @ S3 ) )
& ( Z2
= ( times_times_rat @ X4 @ Y2 ) ) ) ) ) ) ).
% mem_Rep_preal_mult_iff
thf(fact_1263_preal__inverse__def,axiom,
( invers3090987106763476162_preal
= ( ^ [R: dedekind_preal] : ( dedekind_Abs_preal @ ( dedekind_inverse_set @ ( dedekind_Rep_preal @ R ) ) ) ) ) ).
% preal_inverse_def
thf(fact_1264_mem__Rep__preal__inverse__iff,axiom,
! [Z2: rat,R2: dedekind_preal] :
( ( member_rat @ Z2 @ ( dedekind_Rep_preal @ ( invers3090987106763476162_preal @ R2 ) ) )
= ( ( ord_less_rat @ zero_zero_rat @ Z2 )
& ? [Y2: rat] :
( ( ord_less_rat @ Z2 @ Y2 )
& ~ ( member_rat @ ( inverse_inverse_rat @ Y2 ) @ ( dedekind_Rep_preal @ R2 ) ) ) ) ) ).
% mem_Rep_preal_inverse_iff
thf(fact_1265_subset__inverse__mult__lemma,axiom,
! [X: rat,R3: dedekind_preal] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ X @ one_one_rat )
=> ? [V: rat,U3: rat,Y4: rat] :
( ( ord_less_rat @ zero_zero_rat @ V )
& ( ord_less_rat @ V @ Y4 )
& ~ ( member_rat @ ( inverse_inverse_rat @ Y4 ) @ ( dedekind_Rep_preal @ R3 ) )
& ( member_rat @ U3 @ ( dedekind_Rep_preal @ R3 ) )
& ( X
= ( times_times_rat @ V @ U3 ) ) ) ) ) ).
% subset_inverse_mult_lemma
thf(fact_1266_lemma__gleason9__36,axiom,
! [A2: set_rat,X: rat] :
( ( dedekind_cut @ A2 )
=> ( ( ord_less_rat @ one_one_rat @ X )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ~ ( member_rat @ ( times_times_rat @ X3 @ X ) @ A2 ) ) ) ) ).
% lemma_gleason9_36
thf(fact_1267_preal__mult__1,axiom,
! [Z2: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ Z2 )
= Z2 ) ).
% preal_mult_1
thf(fact_1268_preal__mult__inverse,axiom,
! [R2: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( invers3090987106763476162_preal @ R2 ) @ R2 )
= one_on9143529541772854033_preal ) ).
% preal_mult_inverse
thf(fact_1269_preal__mult__inverse__right,axiom,
! [R2: dedekind_preal] :
( ( times_3000655703912201937_preal @ R2 @ ( invers3090987106763476162_preal @ R2 ) )
= one_on9143529541772854033_preal ) ).
% preal_mult_inverse_right
thf(fact_1270_Rep__preal__one,axiom,
( ( dedekind_Rep_preal @ one_on9143529541772854033_preal )
= ( collect_rat
@ ^ [X4: rat] :
( ( ord_less_rat @ zero_zero_rat @ X4 )
& ( ord_less_rat @ X4 @ one_one_rat ) ) ) ) ).
% Rep_preal_one
thf(fact_1271_preal__one__def,axiom,
( one_on9143529541772854033_preal
= ( dedekind_Abs_preal
@ ( collect_rat
@ ^ [X4: rat] :
( ( ord_less_rat @ zero_zero_rat @ X4 )
& ( ord_less_rat @ X4 @ one_one_rat ) ) ) ) ) ).
% preal_one_def
thf(fact_1272_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A3: set_o] :
~ ( member_o @ $false @ A3 ) ) ) ).
% Inf_bool_def
thf(fact_1273_subset__inverse__mult,axiom,
! [R2: dedekind_preal] : ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ one_on9143529541772854033_preal ) @ ( dedekind_Rep_preal @ ( times_3000655703912201937_preal @ ( invers3090987106763476162_preal @ R2 ) @ R2 ) ) ) ).
% subset_inverse_mult
thf(fact_1274_inverse__mult__subset,axiom,
! [R2: dedekind_preal] : ( ord_less_eq_set_rat @ ( dedekind_Rep_preal @ ( times_3000655703912201937_preal @ ( invers3090987106763476162_preal @ R2 ) @ R2 ) ) @ ( dedekind_Rep_preal @ one_on9143529541772854033_preal ) ) ).
% inverse_mult_subset
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_set_rat @ ( comple3890839924845867745et_rat @ ( image_2231269897757205599et_rat @ dedekind_Rep_preal @ p ) ) @ ( set_or575021546402375026an_rat @ zero_zero_rat ) ).
%------------------------------------------------------------------------------