TPTP Problem File: SLH0750^1.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 : Actuarial_Mathematics/0000_Preliminaries/prob_00038_001161__12818618_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1418 ( 583 unt; 147 typ; 0 def)
% Number of atoms : 3629 (1206 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 12363 ( 197 ~; 40 |; 278 &;10286 @)
% ( 0 <=>;1562 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 1203 (1203 >; 0 *; 0 +; 0 <<)
% Number of symbols : 135 ( 132 usr; 18 con; 0-3 aty)
% Number of variables : 4351 ( 414 ^;3763 !; 174 ?;4351 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:08:48.410
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J_J,type,
set_set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $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__Real__Oreal_J_J,type,
set_set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_Eo_J_J,type,
set_real_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (132)
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__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Real__Oreal_M_Eo_J,type,
comple3015195443809154064real_o: set_real_o > real > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
comple1385675409528146559p_real: set_real > real ).
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__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Real__Oreal_J,type,
comple3096694443085538997t_real: set_set_real > set_real ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001_Eo,type,
finite_Fpow_o: set_o > set_set_o ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Real__Oreal,type,
finite_Fpow_real: set_real > set_set_real ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_I_Eo_J,type,
finite_Fpow_set_o: set_set_o > set_set_set_o ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Nat__Onat_J,type,
finite_Fpow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Real__Oreal_J,type,
finite_Fpow_set_real: set_set_real > set_set_set_real ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Real__Oreal,type,
formal3683295897622742886n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
times_times_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
zero_z7760665558314615101s_real: formal3361831859752904756s_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
ord_le4374716579403074808_set_o: set_set_o > set_set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
ord_le3558479182127378552t_real: set_set_real > set_set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_I_Eo_J_J_J,type,
ord_le3178852226150452184_set_o: set_set_set_o > set_set_set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J_J,type,
ord_le561408886441773742t_real: set_set_set_real > set_set_set_real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
top_top_o_o: $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
top_top_real_o: real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
top_top_set_set_o: set_set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
top_top_set_set_real: set_set_real ).
thf(sy_c_Set_OBex_001_Eo,type,
bex_o: set_o > ( $o > $o ) > $o ).
thf(sy_c_Set_OBex_001t__Nat__Onat,type,
bex_nat: set_nat > ( nat > $o ) > $o ).
thf(sy_c_Set_OBex_001t__Real__Oreal,type,
bex_real: set_real > ( real > $o ) > $o ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_Eo_J,type,
collect_set_o: ( set_o > $o ) > set_set_o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Real__Oreal_J,type,
collect_set_real: ( set_real > $o ) > set_set_real ).
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__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Real__Oreal_M_Eo_J_001t__Set__Oset_It__Real__Oreal_J,type,
image_2734271470692514752t_real: ( ( real > $o ) > set_real ) > set_real_o > set_set_real ).
thf(sy_c_Set_Oimage_001_Eo_001_062_I_Eo_M_Eo_J,type,
image_o_o_o: ( $o > $o > $o ) > set_o > set_o_o ).
thf(sy_c_Set_Oimage_001_Eo_001_062_It__Nat__Onat_M_Eo_J,type,
image_o_nat_o: ( $o > nat > $o ) > set_o > set_nat_o ).
thf(sy_c_Set_Oimage_001_Eo_001_062_It__Real__Oreal_M_Eo_J,type,
image_o_real_o: ( $o > real > $o ) > set_o > set_real_o ).
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__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Real__Oreal,type,
image_o_real: ( $o > real ) > set_o > set_real ).
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__Nat__Onat_J,type,
image_o_set_nat: ( $o > set_nat ) > set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Real__Oreal_J,type,
image_o_set_real: ( $o > set_real ) > set_o > set_set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J,type,
image_nat_nat_o: ( nat > nat > $o ) > set_nat > set_nat_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Real__Oreal_J,type,
image_nat_set_real: ( nat > set_real ) > set_nat > set_set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001_Eo,type,
image_real_o: ( real > $o ) > set_real > set_o ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
image_real_nat: ( real > nat ) > set_real > set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_I_Eo_J,type,
image_real_set_o: ( real > set_o ) > set_real > set_set_o ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Nat__Onat_J,type,
image_real_set_nat: ( real > set_nat ) > set_real > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Set__Oset_It__Real__Oreal_J,type,
image_real_set_real: ( real > set_real ) > set_real > set_set_real ).
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__Nat__Onat_J,type,
image_set_o_set_nat: ( set_o > set_nat ) > set_set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Real__Oreal_J,type,
image_set_o_set_real: ( set_o > set_real ) > set_set_o > set_set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
image_5023573440332574309_set_o: ( set_o > set_set_o ) > set_set_o > set_set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_7698617416147310703et_nat: ( set_o > set_set_nat ) > set_set_o > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
image_6191879853830326987t_real: ( set_o > set_set_real ) > set_set_o > set_set_set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J,type,
image_set_nat_nat_o: ( set_nat > nat > $o ) > set_set_nat > set_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_Eo_J,type,
image_set_nat_set_o: ( set_nat > set_o ) > set_set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Real__Oreal_J,type,
image_6333053925516494319t_real: ( set_nat > set_real ) > set_set_nat > set_set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001_062_It__Real__Oreal_M_Eo_J,type,
image_5650221686686655994real_o: ( set_real > real > $o ) > set_set_real > set_real_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001_Eo,type,
image_set_real_o: ( set_real > $o ) > set_set_real > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_I_Eo_J,type,
image_set_real_set_o: ( set_real > set_o ) > set_set_real > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7270232309134952815et_nat: ( set_real > set_nat ) > set_set_real > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
image_2436557299294012491t_real: ( set_real > set_real ) > set_set_real > set_set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5842784325960735177et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_7884819252390400639et_nat: ( set_set_nat > set_set_nat ) > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
set_or66887138388493659n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_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__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_v_A,type,
a: nat > set_nat ).
thf(sy_v_m,type,
m: nat ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1265)
thf(fact_0_assms_I1_J,axiom,
m != zero_zero_nat ).
% assms(1)
thf(fact_1_lessThan__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_2_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_3_UN__I,axiom,
! [A: set_nat,A2: set_set_nat,B: nat,B2: set_nat > set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_4_UN__I,axiom,
! [A: nat,A2: set_nat,B: real,B2: nat > set_real] :
( ( member_nat @ A @ A2 )
=> ( ( member_real @ B @ ( B2 @ A ) )
=> ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_5_UN__I,axiom,
! [A: nat,A2: set_nat,B: $o,B2: nat > set_o] :
( ( member_nat @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_6_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_7_UN__I,axiom,
! [A: $o,A2: set_o,B: real,B2: $o > set_real] :
( ( member_o @ A @ A2 )
=> ( ( member_real @ B @ ( B2 @ A ) )
=> ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_8_UN__I,axiom,
! [A: real,A2: set_real,B: $o,B2: real > set_o] :
( ( member_real @ A @ A2 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_real_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_9_UN__I,axiom,
! [A: real,A2: set_real,B: real,B2: real > set_real] :
( ( member_real @ A @ A2 )
=> ( ( member_real @ B @ ( B2 @ A ) )
=> ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_real_set_real @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_10_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat,B2: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_11_UN__I,axiom,
! [A: $o,A2: set_o,B: nat,B2: $o > set_nat] :
( ( member_o @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_12_UN__I,axiom,
! [A: real,A2: set_real,B: nat,B2: real > set_nat] :
( ( member_real @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_13_UN__iff,axiom,
! [B: nat,B2: $o > set_nat,A2: set_o] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( member_nat @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_14_UN__iff,axiom,
! [B: nat,B2: set_nat > set_nat,A2: set_set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( member_nat @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_15_UN__iff,axiom,
! [B: real,B2: $o > set_real,A2: set_o] :
( ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( member_real @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_16_UN__iff,axiom,
! [B: $o,B2: $o > set_o,A2: set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( member_o @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_17_UN__iff,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( member_nat @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_18_SUP__identity__eq,axiom,
! [A2: set_set_real] :
( ( comple3096694443085538997t_real
@ ( image_2436557299294012491t_real
@ ^ [X2: set_real] : X2
@ A2 ) )
= ( comple3096694443085538997t_real @ A2 ) ) ).
% SUP_identity_eq
thf(fact_19_SUP__identity__eq,axiom,
! [A2: set_set_o] :
( ( comple90263536869209701_set_o
@ ( image_set_o_set_o
@ ^ [X2: set_o] : X2
@ A2 ) )
= ( comple90263536869209701_set_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_20_SUP__identity__eq,axiom,
! [A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X2: set_nat] : X2
@ A2 ) )
= ( comple7399068483239264473et_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_21_SUP__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A2 ) )
= ( complete_Sup_Sup_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_22_SUP__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_23_cSup__lessThan,axiom,
! [X: real] :
( ( comple1385675409528146559p_real @ ( set_or5984915006950818249n_real @ X ) )
= X ) ).
% cSup_lessThan
thf(fact_24_UN__ball__bex__simps_I4_J,axiom,
! [B2: $o > set_nat,A2: set_o,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_25_UN__ball__bex__simps_I4_J,axiom,
! [B2: set_nat > set_nat,A2: set_set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_26_UN__ball__bex__simps_I4_J,axiom,
! [B2: $o > set_real,A2: set_o,P: real > $o] :
( ( ? [X2: real] :
( ( member_real @ X2 @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ? [Y2: real] :
( ( member_real @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_27_UN__ball__bex__simps_I4_J,axiom,
! [B2: $o > set_o,A2: set_o,P: $o > $o] :
( ( ? [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ? [Y2: $o] :
( ( member_o @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_28_UN__ball__bex__simps_I4_J,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_29_UN__ball__bex__simps_I2_J,axiom,
! [B2: $o > set_nat,A2: set_o,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_30_UN__ball__bex__simps_I2_J,axiom,
! [B2: set_nat > set_nat,A2: set_set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_31_UN__ball__bex__simps_I2_J,axiom,
! [B2: $o > set_real,A2: set_o,P: real > $o] :
( ( ! [X2: real] :
( ( member_real @ X2 @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ! [Y2: real] :
( ( member_real @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_32_UN__ball__bex__simps_I2_J,axiom,
! [B2: $o > set_o,A2: set_o,P: $o > $o] :
( ( ! [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_33_UN__ball__bex__simps_I2_J,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_34_bex__UN,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_35_bex__UN,axiom,
! [B2: $o > set_nat,A2: set_o,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_36_bex__UN,axiom,
! [B2: set_nat > set_nat,A2: set_set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_37_bex__UN,axiom,
! [B2: $o > set_real,A2: set_o,P: real > $o] :
( ( ? [X2: real] :
( ( member_real @ X2 @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ? [Y2: real] :
( ( member_real @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_38_bex__UN,axiom,
! [B2: $o > set_o,A2: set_o,P: $o > $o] :
( ( ? [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ? [Y2: $o] :
( ( member_o @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_39_ball__UN,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_40_ball__UN,axiom,
! [B2: $o > set_nat,A2: set_o,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_41_ball__UN,axiom,
! [B2: set_nat > set_nat,A2: set_set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_42_ball__UN,axiom,
! [B2: $o > set_real,A2: set_o,P: real > $o] :
( ( ! [X2: real] :
( ( member_real @ X2 @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ! [Y2: real] :
( ( member_real @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_43_ball__UN,axiom,
! [B2: $o > set_o,A2: set_o,P: $o > $o] :
( ( ! [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_44_SUP__UNION,axiom,
! [F: nat > $o,G: nat > set_nat,A2: set_nat] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y2: nat] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_45_SUP__UNION,axiom,
! [F: nat > $o,G: $o > set_nat,A2: set_o] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y2: $o] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_46_SUP__UNION,axiom,
! [F: real > $o,G: $o > set_real,A2: set_o] :
( ( complete_Sup_Sup_o @ ( image_real_o @ F @ ( comple3096694443085538997t_real @ ( image_o_set_real @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y2: $o] : ( complete_Sup_Sup_o @ ( image_real_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_47_SUP__UNION,axiom,
! [F: $o > $o,G: $o > set_o,A2: set_o] :
( ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [Y2: $o] : ( complete_Sup_Sup_o @ ( image_o_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_48_SUP__UNION,axiom,
! [F: nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_49_SUP__UNION,axiom,
! [F: nat > set_nat,G: $o > set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y2: $o] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_50_SUP__UNION,axiom,
! [F: real > set_nat,G: nat > set_real,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_51_SUP__UNION,axiom,
! [F: real > set_nat,G: $o > set_real,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ ( comple3096694443085538997t_real @ ( image_o_set_real @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y2: $o] : ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_52_SUP__UNION,axiom,
! [F: $o > set_nat,G: nat > set_o,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_53_SUP__UNION,axiom,
! [F: $o > set_nat,G: $o > set_o,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y2: $o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_54_Union__iff,axiom,
! [A2: nat,C: set_set_nat] :
( ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ C )
& ( member_nat @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_55_Union__iff,axiom,
! [A2: real,C: set_set_real] :
( ( member_real @ A2 @ ( comple3096694443085538997t_real @ C ) )
= ( ? [X2: set_real] :
( ( member_set_real @ X2 @ C )
& ( member_real @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_56_Union__iff,axiom,
! [A2: $o,C: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) )
= ( ? [X2: set_o] :
( ( member_set_o @ X2 @ C )
& ( member_o @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_57_UnionI,axiom,
! [X3: set_nat,C: set_set_nat,A2: nat] :
( ( member_set_nat @ X3 @ C )
=> ( ( member_nat @ A2 @ X3 )
=> ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C ) ) ) ) ).
% UnionI
thf(fact_58_UnionI,axiom,
! [X3: set_real,C: set_set_real,A2: real] :
( ( member_set_real @ X3 @ C )
=> ( ( member_real @ A2 @ X3 )
=> ( member_real @ A2 @ ( comple3096694443085538997t_real @ C ) ) ) ) ).
% UnionI
thf(fact_59_UnionI,axiom,
! [X3: set_o,C: set_set_o,A2: $o] :
( ( member_set_o @ X3 @ C )
=> ( ( member_o @ A2 @ X3 )
=> ( member_o @ A2 @ ( comple90263536869209701_set_o @ C ) ) ) ) ).
% UnionI
thf(fact_60_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ X2 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_61_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_real,P: real > $o] :
( ( ! [X2: real] :
( ( member_real @ X2 @ ( comple3096694443085538997t_real @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ A2 )
=> ! [Y2: real] :
( ( member_real @ Y2 @ X2 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_62_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_o,P: $o > $o] :
( ( ! [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A2 )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ X2 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_63_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ X2 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_64_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_real,P: real > $o] :
( ( ? [X2: real] :
( ( member_real @ X2 @ ( comple3096694443085538997t_real @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_real] :
( ( member_set_real @ X2 @ A2 )
& ? [Y2: real] :
( ( member_real @ Y2 @ X2 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_65_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_o,P: $o > $o] :
( ( ? [X2: $o] :
( ( member_o @ X2 @ ( comple90263536869209701_set_o @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_o] :
( ( member_set_o @ X2 @ A2 )
& ? [Y2: $o] :
( ( member_o @ Y2 @ X2 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_66_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_67_lessThan__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( set_or5984915006950818249n_real @ X )
= ( set_or5984915006950818249n_real @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_68_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_69_Sup__set__def,axiom,
( comple3096694443085538997t_real
= ( ^ [A3: set_set_real] :
( collect_real
@ ^ [X2: real] : ( complete_Sup_Sup_o @ ( image_set_real_o @ ( member_real @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_70_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X2: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_71_Sup_OSUP__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat > set_nat,D: set_nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_7916887816326733075et_nat @ C @ A2 ) )
= ( Sup @ ( image_7916887816326733075et_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_72_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_real,D: $o > set_real,Sup: set_set_real > set_real] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_o_set_real @ C @ A2 ) )
= ( Sup @ ( image_o_set_real @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_73_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_o,D: $o > set_o,Sup: set_set_o > set_o] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_o_set_o @ C @ A2 ) )
= ( Sup @ ( image_o_set_o @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_74_Sup_OSUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_nat,D: $o > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_o_set_nat @ C @ A2 ) )
= ( Sup @ ( image_o_set_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_75_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_nat,D: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C @ A2 ) )
= ( Sup @ ( image_nat_set_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_76_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A2 ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_77_Inf_OINF__cong,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_nat > set_nat,D: set_nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_7916887816326733075et_nat @ C @ A2 ) )
= ( Inf @ ( image_7916887816326733075et_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_78_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_real,D: $o > set_real,Inf: set_set_real > set_real] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_o_set_real @ C @ A2 ) )
= ( Inf @ ( image_o_set_real @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_79_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_o,D: $o > set_o,Inf: set_set_o > set_o] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_o_set_o @ C @ A2 ) )
= ( Inf @ ( image_o_set_o @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_80_Inf_OINF__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_nat,D: $o > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_o_set_nat @ C @ A2 ) )
= ( Inf @ ( image_o_set_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_81_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_nat,D: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C @ A2 ) )
= ( Inf @ ( image_nat_set_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_82_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A2 ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_83_UnionE,axiom,
! [A2: nat,C: set_set_nat] :
( ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C ) )
=> ~ ! [X5: set_nat] :
( ( member_nat @ A2 @ X5 )
=> ~ ( member_set_nat @ X5 @ C ) ) ) ).
% UnionE
thf(fact_84_UnionE,axiom,
! [A2: real,C: set_set_real] :
( ( member_real @ A2 @ ( comple3096694443085538997t_real @ C ) )
=> ~ ! [X5: set_real] :
( ( member_real @ A2 @ X5 )
=> ~ ( member_set_real @ X5 @ C ) ) ) ).
% UnionE
thf(fact_85_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_86_Sup_OSUP__identity__eq,axiom,
! [Sup: set_set_nat > set_nat,A2: set_set_nat] :
( ( Sup
@ ( image_7916887816326733075et_nat
@ ^ [X2: set_nat] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_87_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_88_Inf_OINF__identity__eq,axiom,
! [Inf: set_set_nat > set_nat,A2: set_set_nat] :
( ( Inf
@ ( image_7916887816326733075et_nat
@ ^ [X2: set_nat] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_89_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_90_cSup__eq,axiom,
! [X3: set_real,A: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X3 )
=> ( ord_less_eq_real @ X4 @ A ) )
=> ( ! [Y3: real] :
( ! [X6: real] :
( ( member_real @ X6 @ X3 )
=> ( ord_less_eq_real @ X6 @ Y3 ) )
=> ( ord_less_eq_real @ A @ Y3 ) )
=> ( ( comple1385675409528146559p_real @ X3 )
= A ) ) ) ).
% cSup_eq
thf(fact_91_cSup__eq__maximum,axiom,
! [Z: real,X3: set_real] :
( ( member_real @ Z @ X3 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ X3 )
=> ( ord_less_eq_real @ X4 @ Z ) )
=> ( ( comple1385675409528146559p_real @ X3 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_92_cSup__eq__maximum,axiom,
! [Z: set_nat,X3: set_set_nat] :
( ( member_set_nat @ Z @ X3 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X3 )
=> ( ord_less_eq_set_nat @ X4 @ Z ) )
=> ( ( comple7399068483239264473et_nat @ X3 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_93_cSup__eq__maximum,axiom,
! [Z: $o,X3: set_o] :
( ( member_o @ Z @ X3 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X3 )
=> ( ord_less_eq_o @ X4 @ Z ) )
=> ( ( complete_Sup_Sup_o @ X3 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_94_cSup__eq__maximum,axiom,
! [Z: nat,X3: set_nat] :
( ( member_nat @ Z @ X3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X3 )
=> ( ord_less_eq_nat @ X4 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X3 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_95_cSup__eq__maximum,axiom,
! [Z: set_real,X3: set_set_real] :
( ( member_set_real @ Z @ X3 )
=> ( ! [X4: set_real] :
( ( member_set_real @ X4 @ X3 )
=> ( ord_less_eq_set_real @ X4 @ Z ) )
=> ( ( comple3096694443085538997t_real @ X3 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_96_cSup__eq__maximum,axiom,
! [Z: set_o,X3: set_set_o] :
( ( member_set_o @ Z @ X3 )
=> ( ! [X4: set_o] :
( ( member_set_o @ X4 @ X3 )
=> ( ord_less_eq_set_o @ X4 @ Z ) )
=> ( ( comple90263536869209701_set_o @ X3 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_97_Sup__upper2,axiom,
! [U: set_nat,A2: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A2 )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_98_Sup__upper2,axiom,
! [U: $o,A2: set_o,V: $o] :
( ( member_o @ U @ A2 )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_99_Sup__upper2,axiom,
! [U: set_real,A2: set_set_real,V: set_real] :
( ( member_set_real @ U @ A2 )
=> ( ( ord_less_eq_set_real @ V @ U )
=> ( ord_less_eq_set_real @ V @ ( comple3096694443085538997t_real @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_100_Sup__upper2,axiom,
! [U: set_o,A2: set_set_o,V: set_o] :
( ( member_set_o @ U @ A2 )
=> ( ( ord_less_eq_set_o @ V @ U )
=> ( ord_less_eq_set_o @ V @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_101_Sup__le__iff,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ B )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_102_Sup__le__iff,axiom,
! [A2: set_o,B: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ B )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_o @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_103_Sup__le__iff,axiom,
! [A2: set_set_real,B: set_real] :
( ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A2 ) @ B )
= ( ! [X2: set_real] :
( ( member_set_real @ X2 @ A2 )
=> ( ord_less_eq_set_real @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_104_Sup__le__iff,axiom,
! [A2: set_set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ B )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A2 )
=> ( ord_less_eq_set_o @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_105_Sup__upper,axiom,
! [X: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_106_Sup__upper,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ X @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_107_Sup__upper,axiom,
! [X: set_real,A2: set_set_real] :
( ( member_set_real @ X @ A2 )
=> ( ord_less_eq_set_real @ X @ ( comple3096694443085538997t_real @ A2 ) ) ) ).
% Sup_upper
thf(fact_108_Sup__upper,axiom,
! [X: set_o,A2: set_set_o] :
( ( member_set_o @ X @ A2 )
=> ( ord_less_eq_set_o @ X @ ( comple90263536869209701_set_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_109_Sup__least,axiom,
! [A2: set_set_nat,Z: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ X4 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_110_Sup__least,axiom,
! [A2: set_o,Z: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_o @ X4 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_111_Sup__least,axiom,
! [A2: set_set_real,Z: set_real] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ( ord_less_eq_set_real @ X4 @ Z ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_112_Sup__least,axiom,
! [A2: set_set_o,Z: set_o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ A2 )
=> ( ord_less_eq_set_o @ X4 @ Z ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_113_Sup__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ A2 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ B2 )
& ( ord_less_eq_set_nat @ A4 @ X6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_114_Sup__mono,axiom,
! [A2: set_o,B2: set_o] :
( ! [A4: $o] :
( ( member_o @ A4 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ A4 @ X6 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_115_Sup__mono,axiom,
! [A2: set_set_real,B2: set_set_real] :
( ! [A4: set_real] :
( ( member_set_real @ A4 @ A2 )
=> ? [X6: set_real] :
( ( member_set_real @ X6 @ B2 )
& ( ord_less_eq_set_real @ A4 @ X6 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A2 ) @ ( comple3096694443085538997t_real @ B2 ) ) ) ).
% Sup_mono
thf(fact_116_Sup__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ! [A4: set_o] :
( ( member_set_o @ A4 @ A2 )
=> ? [X6: set_o] :
( ( member_set_o @ X6 @ B2 )
& ( ord_less_eq_set_o @ A4 @ X6 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_117_Sup__eqI,axiom,
! [A2: set_set_nat,X: set_nat] :
( ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ A2 )
=> ( ord_less_eq_set_nat @ Y3 @ X ) )
=> ( ! [Y3: set_nat] :
( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ A2 )
=> ( ord_less_eq_set_nat @ Z2 @ Y3 ) )
=> ( ord_less_eq_set_nat @ X @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_118_Sup__eqI,axiom,
! [A2: set_o,X: $o] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ A2 )
=> ( ord_less_eq_o @ Y3 @ X ) )
=> ( ! [Y3: $o] :
( ! [Z2: $o] :
( ( member_o @ Z2 @ A2 )
=> ( ord_less_eq_o @ Z2 @ Y3 ) )
=> ( ord_less_eq_o @ X @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_119_Sup__eqI,axiom,
! [A2: set_set_real,X: set_real] :
( ! [Y3: set_real] :
( ( member_set_real @ Y3 @ A2 )
=> ( ord_less_eq_set_real @ Y3 @ X ) )
=> ( ! [Y3: set_real] :
( ! [Z2: set_real] :
( ( member_set_real @ Z2 @ A2 )
=> ( ord_less_eq_set_real @ Z2 @ Y3 ) )
=> ( ord_less_eq_set_real @ X @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_120_Sup__eqI,axiom,
! [A2: set_set_o,X: set_o] :
( ! [Y3: set_o] :
( ( member_set_o @ Y3 @ A2 )
=> ( ord_less_eq_set_o @ Y3 @ X ) )
=> ( ! [Y3: set_o] :
( ! [Z2: set_o] :
( ( member_set_o @ Z2 @ A2 )
=> ( ord_less_eq_set_o @ Z2 @ Y3 ) )
=> ( ord_less_eq_set_o @ X @ Y3 ) )
=> ( ( comple90263536869209701_set_o @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_121_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > $o,D: $o > $o] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_122_SUP__cong,axiom,
! [A2: set_real,B2: set_real,C: real > $o,D: real > $o] :
( ( A2 = B2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_123_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > $o,D: nat > $o] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_124_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > nat,D: $o > nat] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_125_SUP__cong,axiom,
! [A2: set_real,B2: set_real,C: real > nat,D: real > nat] :
( ( A2 = B2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_real_nat @ C @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_real_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_126_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > nat,D: nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_127_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_nat,D: $o > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_128_SUP__cong,axiom,
! [A2: set_real,B2: set_real,C: real > set_nat,D: real > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ C @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_real_set_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_129_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C: nat > set_nat,D: nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_130_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C: $o > set_real,D: $o > set_real] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( C @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple3096694443085538997t_real @ ( image_o_set_real @ C @ A2 ) )
= ( comple3096694443085538997t_real @ ( image_o_set_real @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_131_Union__subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ? [Y4: set_nat] :
( ( member_set_nat @ Y4 @ B2 )
& ( ord_less_eq_set_nat @ X4 @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_132_Union__subsetI,axiom,
! [A2: set_set_real,B2: set_set_real] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ A2 )
=> ? [Y4: set_real] :
( ( member_set_real @ Y4 @ B2 )
& ( ord_less_eq_set_real @ X4 @ Y4 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A2 ) @ ( comple3096694443085538997t_real @ B2 ) ) ) ).
% Union_subsetI
thf(fact_133_Union__subsetI,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ A2 )
=> ? [Y4: set_o] :
( ( member_set_o @ Y4 @ B2 )
& ( ord_less_eq_set_o @ X4 @ Y4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Union_subsetI
thf(fact_134_Union__upper,axiom,
! [B2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_135_Union__upper,axiom,
! [B2: set_real,A2: set_set_real] :
( ( member_set_real @ B2 @ A2 )
=> ( ord_less_eq_set_real @ B2 @ ( comple3096694443085538997t_real @ A2 ) ) ) ).
% Union_upper
thf(fact_136_Union__upper,axiom,
! [B2: set_o,A2: set_set_o] :
( ( member_set_o @ B2 @ A2 )
=> ( ord_less_eq_set_o @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ).
% Union_upper
thf(fact_137_Union__least,axiom,
! [A2: set_set_nat,C: set_nat] :
( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
=> ( ord_less_eq_set_nat @ X5 @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ C ) ) ).
% Union_least
thf(fact_138_Union__least,axiom,
! [A2: set_set_real,C: set_real] :
( ! [X5: set_real] :
( ( member_set_real @ X5 @ A2 )
=> ( ord_less_eq_set_real @ X5 @ C ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A2 ) @ C ) ) ).
% Union_least
thf(fact_139_Union__least,axiom,
! [A2: set_set_o,C: set_o] :
( ! [X5: set_o] :
( ( member_set_o @ X5 @ A2 )
=> ( ord_less_eq_set_o @ X5 @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ C ) ) ).
% Union_least
thf(fact_140_Union__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_141_Union__mono,axiom,
! [A2: set_set_real,B2: set_set_real] :
( ( ord_le3558479182127378552t_real @ A2 @ B2 )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A2 ) @ ( comple3096694443085538997t_real @ B2 ) ) ) ).
% Union_mono
thf(fact_142_Union__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Union_mono
thf(fact_143_SUP__commute,axiom,
! [F: nat > nat > set_nat,B2: set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_144_SUP__commute,axiom,
! [F: nat > $o > set_nat,B2: set_o,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [J: $o] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_145_SUP__commute,axiom,
! [F: $o > nat > set_nat,B2: set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_146_SUP__commute,axiom,
! [F: $o > $o > set_nat,B2: set_o,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [J: $o] :
( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_147_SUP__commute,axiom,
! [F: $o > $o > set_real,B2: set_o,A2: set_o] :
( ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [I: $o] : ( comple3096694443085538997t_real @ ( image_o_set_real @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [J: $o] :
( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_148_SUP__commute,axiom,
! [F: $o > $o > set_o,B2: set_o,A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [I: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [J: $o] :
( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_149_SUP__commute,axiom,
! [F: nat > set_nat > set_nat,B2: set_set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [J: set_nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_150_SUP__commute,axiom,
! [F: $o > set_nat > set_nat,B2: set_set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [J: set_nat] :
( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_151_SUP__commute,axiom,
! [F: set_nat > nat > set_nat,B2: set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [I: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [I: set_nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_152_SUP__commute,axiom,
! [F: set_nat > $o > set_nat,B2: set_o,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [I: set_nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [J: $o] :
( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [I: set_nat] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_153_image__Union,axiom,
! [F: nat > nat,S: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_154_image__Union,axiom,
! [F: nat > real,S: set_set_nat] :
( ( image_nat_real @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple3096694443085538997t_real @ ( image_6333053925516494319t_real @ ( image_nat_real @ F ) @ S ) ) ) ).
% image_Union
thf(fact_155_image__Union,axiom,
! [F: nat > $o,S: set_set_nat] :
( ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ ( image_nat_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_156_image__Union,axiom,
! [F: real > nat,S: set_set_real] :
( ( image_real_nat @ F @ ( comple3096694443085538997t_real @ S ) )
= ( comple7399068483239264473et_nat @ ( image_7270232309134952815et_nat @ ( image_real_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_157_image__Union,axiom,
! [F: real > real,S: set_set_real] :
( ( image_real_real @ F @ ( comple3096694443085538997t_real @ S ) )
= ( comple3096694443085538997t_real @ ( image_2436557299294012491t_real @ ( image_real_real @ F ) @ S ) ) ) ).
% image_Union
thf(fact_158_image__Union,axiom,
! [F: real > $o,S: set_set_real] :
( ( image_real_o @ F @ ( comple3096694443085538997t_real @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_real_set_o @ ( image_real_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_159_image__Union,axiom,
! [F: $o > nat,S: set_set_o] :
( ( image_o_nat @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple7399068483239264473et_nat @ ( image_set_o_set_nat @ ( image_o_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_160_image__Union,axiom,
! [F: $o > real,S: set_set_o] :
( ( image_o_real @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple3096694443085538997t_real @ ( image_set_o_set_real @ ( image_o_real @ F ) @ S ) ) ) ).
% image_Union
thf(fact_161_image__Union,axiom,
! [F: $o > $o,S: set_set_o] :
( ( image_o_o @ F @ ( comple90263536869209701_set_o @ S ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o @ ( image_o_o @ F ) @ S ) ) ) ).
% image_Union
thf(fact_162_image__Union,axiom,
! [F: nat > set_nat,S: set_set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_163_UN__UN__flatten,axiom,
! [C: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_164_UN__UN__flatten,axiom,
! [C: nat > set_nat,B2: $o > set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y2: $o] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_165_UN__UN__flatten,axiom,
! [C: real > set_nat,B2: nat > set_real,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ C @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : ( comple7399068483239264473et_nat @ ( image_real_set_nat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_166_UN__UN__flatten,axiom,
! [C: real > set_nat,B2: $o > set_real,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ C @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y2: $o] : ( comple7399068483239264473et_nat @ ( image_real_set_nat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_167_UN__UN__flatten,axiom,
! [C: $o > set_nat,B2: nat > set_o,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_168_UN__UN__flatten,axiom,
! [C: $o > set_nat,B2: $o > set_o,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y2: $o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_169_UN__UN__flatten,axiom,
! [C: nat > set_real,B2: nat > set_nat,A2: set_nat] :
( ( comple3096694443085538997t_real @ ( image_nat_set_real @ C @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) )
= ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [Y2: nat] : ( comple3096694443085538997t_real @ ( image_nat_set_real @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_170_UN__UN__flatten,axiom,
! [C: nat > set_real,B2: $o > set_nat,A2: set_o] :
( ( comple3096694443085538997t_real @ ( image_nat_set_real @ C @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) )
= ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [Y2: $o] : ( comple3096694443085538997t_real @ ( image_nat_set_real @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_171_UN__UN__flatten,axiom,
! [C: real > set_real,B2: $o > set_real,A2: set_o] :
( ( comple3096694443085538997t_real @ ( image_real_set_real @ C @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) ) )
= ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [Y2: $o] : ( comple3096694443085538997t_real @ ( image_real_set_real @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_172_UN__UN__flatten,axiom,
! [C: $o > set_real,B2: $o > set_o,A2: set_o] :
( ( comple3096694443085538997t_real @ ( image_o_set_real @ C @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) )
= ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [Y2: $o] : ( comple3096694443085538997t_real @ ( image_o_set_real @ C @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_173_UN__E,axiom,
! [B: nat,B2: set_nat > set_nat,A2: set_set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) )
=> ~ ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_174_UN__E,axiom,
! [B: nat,B2: $o > set_nat,A2: set_o] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_175_UN__E,axiom,
! [B: nat,B2: real > set_nat,A2: set_real] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B2 @ A2 ) ) )
=> ~ ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_176_UN__E,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ~ ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_177_UN__E,axiom,
! [B: real,B2: $o > set_real,A2: set_o] :
( ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ~ ( member_real @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_178_UN__E,axiom,
! [B: real,B2: real > set_real,A2: set_real] :
( ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_real_set_real @ B2 @ A2 ) ) )
=> ~ ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ~ ( member_real @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_179_UN__E,axiom,
! [B: real,B2: nat > set_real,A2: set_nat] :
( ( member_real @ B @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ A2 ) ) )
=> ~ ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ( member_real @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_180_UN__E,axiom,
! [B: $o,B2: $o > set_o,A2: set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_181_UN__E,axiom,
! [B: $o,B2: real > set_o,A2: set_real] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_real_set_o @ B2 @ A2 ) ) )
=> ~ ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_182_UN__E,axiom,
! [B: $o,B2: nat > set_o,A2: set_nat] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) )
=> ~ ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ( member_o @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_183_UN__extend__simps_I8_J,axiom,
! [B2: set_nat > set_nat,A2: set_set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_5842784325960735177et_nat
@ ^ [Y2: set_set_nat] : ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ ( comple548664676211718543et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_184_UN__extend__simps_I8_J,axiom,
! [B2: nat > set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y2: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_185_UN__extend__simps_I8_J,axiom,
! [B2: real > set_nat,A2: set_set_real] :
( ( comple7399068483239264473et_nat
@ ( image_7270232309134952815et_nat
@ ^ [Y2: set_real] : ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B2 @ ( comple3096694443085538997t_real @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_186_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_nat,A2: set_set_o] :
( ( comple7399068483239264473et_nat
@ ( image_set_o_set_nat
@ ^ [Y2: set_o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_187_UN__extend__simps_I8_J,axiom,
! [B2: nat > set_real,A2: set_set_nat] :
( ( comple3096694443085538997t_real
@ ( image_6333053925516494319t_real
@ ^ [Y2: set_nat] : ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ Y2 ) )
@ A2 ) )
= ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_188_UN__extend__simps_I8_J,axiom,
! [B2: real > set_real,A2: set_set_real] :
( ( comple3096694443085538997t_real
@ ( image_2436557299294012491t_real
@ ^ [Y2: set_real] : ( comple3096694443085538997t_real @ ( image_real_set_real @ B2 @ Y2 ) )
@ A2 ) )
= ( comple3096694443085538997t_real @ ( image_real_set_real @ B2 @ ( comple3096694443085538997t_real @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_189_UN__extend__simps_I8_J,axiom,
! [B2: $o > set_real,A2: set_set_o] :
( ( comple3096694443085538997t_real
@ ( image_set_o_set_real
@ ^ [Y2: set_o] : ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ Y2 ) )
@ A2 ) )
= ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_190_UN__extend__simps_I8_J,axiom,
! [B2: nat > set_o,A2: set_set_nat] :
( ( comple90263536869209701_set_o
@ ( image_set_nat_set_o
@ ^ [Y2: set_nat] : ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ Y2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_191_UN__extend__simps_I8_J,axiom,
! [B2: real > set_o,A2: set_set_real] :
( ( comple90263536869209701_set_o
@ ( image_set_real_set_o
@ ^ [Y2: set_real] : ( comple90263536869209701_set_o @ ( image_real_set_o @ B2 @ Y2 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_real_set_o @ B2 @ ( comple3096694443085538997t_real @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_192_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_193_UN__extend__simps_I9_J,axiom,
! [C: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_194_UN__extend__simps_I9_J,axiom,
! [C: nat > set_nat,B2: $o > set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X2: $o] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_195_UN__extend__simps_I9_J,axiom,
! [C: real > set_nat,B2: nat > set_real,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( comple7399068483239264473et_nat @ ( image_real_set_nat @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_real_set_nat @ C @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_196_UN__extend__simps_I9_J,axiom,
! [C: real > set_nat,B2: $o > set_real,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X2: $o] : ( comple7399068483239264473et_nat @ ( image_real_set_nat @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_real_set_nat @ C @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_197_UN__extend__simps_I9_J,axiom,
! [C: $o > set_nat,B2: nat > set_o,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_198_UN__extend__simps_I9_J,axiom,
! [C: $o > set_nat,B2: $o > set_o,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X2: $o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_199_UN__extend__simps_I9_J,axiom,
! [C: nat > set_real,B2: nat > set_nat,A2: set_nat] :
( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [X2: nat] : ( comple3096694443085538997t_real @ ( image_nat_set_real @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple3096694443085538997t_real @ ( image_nat_set_real @ C @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_200_UN__extend__simps_I9_J,axiom,
! [C: nat > set_real,B2: $o > set_nat,A2: set_o] :
( ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [X2: $o] : ( comple3096694443085538997t_real @ ( image_nat_set_real @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple3096694443085538997t_real @ ( image_nat_set_real @ C @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_201_UN__extend__simps_I9_J,axiom,
! [C: real > set_real,B2: $o > set_real,A2: set_o] :
( ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [X2: $o] : ( comple3096694443085538997t_real @ ( image_real_set_real @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple3096694443085538997t_real @ ( image_real_set_real @ C @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_202_UN__extend__simps_I9_J,axiom,
! [C: $o > set_real,B2: $o > set_o,A2: set_o] :
( ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [X2: $o] : ( comple3096694443085538997t_real @ ( image_o_set_real @ C @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple3096694443085538997t_real @ ( image_o_set_real @ C @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_203_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_204_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_205_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_206_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_207_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_208_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_209_SUP__eq,axiom,
! [A2: set_o,B2: set_o,F: $o > $o,G: $o > $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_210_SUP__eq,axiom,
! [A2: set_o,B2: set_real,F: $o > $o,G: real > $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ? [X6: real] :
( ( member_real @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: real] :
( ( member_real @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_211_SUP__eq,axiom,
! [A2: set_o,B2: set_nat,F: $o > $o,G: nat > $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_212_SUP__eq,axiom,
! [A2: set_real,B2: set_o,F: real > $o,G: $o > $o] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: real] :
( ( member_real @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_213_SUP__eq,axiom,
! [A2: set_real,B2: set_real,F: real > $o,G: real > $o] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ? [X6: real] :
( ( member_real @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: real] :
( ( member_real @ J2 @ B2 )
=> ? [X6: real] :
( ( member_real @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_214_SUP__eq,axiom,
! [A2: set_real,B2: set_nat,F: real > $o,G: nat > $o] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X6: real] :
( ( member_real @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_215_SUP__eq,axiom,
! [A2: set_nat,B2: set_o,F: nat > $o,G: $o > $o] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_216_SUP__eq,axiom,
! [A2: set_nat,B2: set_real,F: nat > $o,G: real > $o] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ? [X6: real] :
( ( member_real @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: real] :
( ( member_real @ J2 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_real_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_217_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > $o,G: nat > $o] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_218_SUP__eq,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat,G: $o > set_nat] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_219_Sup__subset__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_220_Sup__subset__mono,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_221_Sup__subset__mono,axiom,
! [A2: set_set_real,B2: set_set_real] :
( ( ord_le3558479182127378552t_real @ A2 @ B2 )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ A2 ) @ ( comple3096694443085538997t_real @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_222_Sup__subset__mono,axiom,
! [A2: set_set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_223_SUP__upper2,axiom,
! [I3: $o,A2: set_o,U: $o,F: $o > $o] :
( ( member_o @ I3 @ A2 )
=> ( ( ord_less_eq_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_224_SUP__upper2,axiom,
! [I3: real,A2: set_real,U: $o,F: real > $o] :
( ( member_real @ I3 @ A2 )
=> ( ( ord_less_eq_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_225_SUP__upper2,axiom,
! [I3: nat,A2: set_nat,U: $o,F: nat > $o] :
( ( member_nat @ I3 @ A2 )
=> ( ( ord_less_eq_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_226_SUP__upper2,axiom,
! [I3: $o,A2: set_o,U: set_nat,F: $o > set_nat] :
( ( member_o @ I3 @ A2 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_227_SUP__upper2,axiom,
! [I3: real,A2: set_real,U: set_nat,F: real > set_nat] :
( ( member_real @ I3 @ A2 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_228_SUP__upper2,axiom,
! [I3: nat,A2: set_nat,U: set_nat,F: nat > set_nat] :
( ( member_nat @ I3 @ A2 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_229_SUP__upper2,axiom,
! [I3: $o,A2: set_o,U: set_real,F: $o > set_real] :
( ( member_o @ I3 @ A2 )
=> ( ( ord_less_eq_set_real @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_real @ U @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_230_SUP__upper2,axiom,
! [I3: real,A2: set_real,U: set_real,F: real > set_real] :
( ( member_real @ I3 @ A2 )
=> ( ( ord_less_eq_set_real @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_real @ U @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_231_SUP__upper2,axiom,
! [I3: nat,A2: set_nat,U: set_real,F: nat > set_real] :
( ( member_nat @ I3 @ A2 )
=> ( ( ord_less_eq_set_real @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_real @ U @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_232_SUP__upper2,axiom,
! [I3: $o,A2: set_o,U: set_o,F: $o > set_o] :
( ( member_o @ I3 @ A2 )
=> ( ( ord_less_eq_set_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_o @ U @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_233_SUP__le__iff,axiom,
! [F: nat > set_nat,A2: set_nat,U: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ U )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_234_SUP__le__iff,axiom,
! [F: $o > set_nat,A2: set_o,U: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ U )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_235_SUP__le__iff,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,U: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) @ U )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_236_SUP__le__iff,axiom,
! [F: $o > set_real,A2: set_o,U: set_real] :
( ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) @ U )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_237_SUP__le__iff,axiom,
! [F: $o > set_o,A2: set_o,U: set_o] :
( ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ U )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_238_SUP__upper,axiom,
! [I3: $o,A2: set_o,F: $o > $o] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_239_SUP__upper,axiom,
! [I3: real,A2: set_real,F: real > $o] :
( ( member_real @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_240_SUP__upper,axiom,
! [I3: nat,A2: set_nat,F: nat > $o] :
( ( member_nat @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_241_SUP__upper,axiom,
! [I3: $o,A2: set_o,F: $o > set_nat] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_242_SUP__upper,axiom,
! [I3: real,A2: set_real,F: real > set_nat] :
( ( member_real @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_243_SUP__upper,axiom,
! [I3: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_244_SUP__upper,axiom,
! [I3: $o,A2: set_o,F: $o > set_real] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_245_SUP__upper,axiom,
! [I3: real,A2: set_real,F: real > set_real] :
( ( member_real @ I3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_246_SUP__upper,axiom,
! [I3: nat,A2: set_nat,F: nat > set_real] :
( ( member_nat @ I3 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I3 ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_247_SUP__upper,axiom,
! [I3: $o,A2: set_o,F: $o > set_o] :
( ( member_o @ I3 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I3 ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_248_SUP__mono_H,axiom,
! [F: nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ! [X4: nat] : ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_249_SUP__mono_H,axiom,
! [F: $o > set_nat,G: $o > set_nat,A2: set_o] :
( ! [X4: $o] : ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_250_SUP__mono_H,axiom,
! [F: set_nat > set_nat,G: set_nat > set_nat,A2: set_set_nat] :
( ! [X4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_251_SUP__mono_H,axiom,
! [F: $o > set_real,G: $o > set_real,A2: set_o] :
( ! [X4: $o] : ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_o_set_real @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_252_SUP__mono_H,axiom,
! [F: $o > set_o,G: $o > set_o,A2: set_o] :
( ! [X4: $o] : ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_253_SUP__least,axiom,
! [A2: set_o,F: $o > $o,U: $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_254_SUP__least,axiom,
! [A2: set_real,F: real > $o,U: $o] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_255_SUP__least,axiom,
! [A2: set_nat,F: nat > $o,U: $o] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_256_SUP__least,axiom,
! [A2: set_o,F: $o > set_nat,U: set_nat] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_257_SUP__least,axiom,
! [A2: set_real,F: real > set_nat,U: set_nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_258_SUP__least,axiom,
! [A2: set_nat,F: nat > set_nat,U: set_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_259_SUP__least,axiom,
! [A2: set_o,F: $o > set_real,U: set_real] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_260_SUP__least,axiom,
! [A2: set_real,F: real > set_real,U: set_real] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_261_SUP__least,axiom,
! [A2: set_nat,F: nat > set_real,U: set_real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_262_SUP__least,axiom,
! [A2: set_o,F: $o > set_o,U: set_o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_263_SUP__mono,axiom,
! [A2: set_o,B2: set_nat,F: $o > set_nat,G: nat > set_nat] :
( ! [N: $o] :
( ( member_o @ N @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_264_SUP__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat,G: $o > set_nat] :
( ! [N: $o] :
( ( member_o @ N @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_265_SUP__mono,axiom,
! [A2: set_real,B2: set_nat,F: real > set_nat,G: nat > set_nat] :
( ! [N: real] :
( ( member_real @ N @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_266_SUP__mono,axiom,
! [A2: set_real,B2: set_o,F: real > set_nat,G: $o > set_nat] :
( ! [N: real] :
( ( member_real @ N @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_267_SUP__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [N: nat] :
( ( member_nat @ N @ A2 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_268_SUP__mono,axiom,
! [A2: set_nat,B2: set_o,F: nat > set_nat,G: $o > set_nat] :
( ! [N: nat] :
( ( member_nat @ N @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_269_SUP__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_real,G: $o > set_real] :
( ! [N: $o] :
( ( member_o @ N @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_real @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_o_set_real @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_270_SUP__mono,axiom,
! [A2: set_real,B2: set_o,F: real > set_real,G: $o > set_real] :
( ! [N: real] :
( ( member_real @ N @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_real @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_o_set_real @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_271_SUP__mono,axiom,
! [A2: set_nat,B2: set_o,F: nat > set_real,G: $o > set_real] :
( ! [N: nat] :
( ( member_nat @ N @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_real @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_o_set_real @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_272_SUP__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o,G: $o > set_o] :
( ! [N: $o] :
( ( member_o @ N @ A2 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B2 )
& ( ord_less_eq_set_o @ ( F @ N ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_273_SUP__eqI,axiom,
! [A2: set_o,F: $o > $o,X: $o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_o @ X @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_274_SUP__eqI,axiom,
! [A2: set_real,F: real > $o,X: $o] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: $o] :
( ! [I4: real] :
( ( member_real @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_o @ X @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_275_SUP__eqI,axiom,
! [A2: set_nat,F: nat > $o,X: $o] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_o @ X @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_276_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_nat,X: set_nat] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_277_SUP__eqI,axiom,
! [A2: set_real,F: real > set_nat,X: set_nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_nat] :
( ! [I4: real] :
( ( member_real @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_278_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_nat,X: set_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_279_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_real,X: set_real] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_real] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_real @ X @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_280_SUP__eqI,axiom,
! [A2: set_real,F: real > set_real,X: set_real] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_real] :
( ! [I4: real] :
( ( member_real @ I4 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_real @ X @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_281_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_real,X: set_real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_real] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_real @ X @ Y3 ) )
=> ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_282_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_o,X: set_o] :
( ! [I2: $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I2 ) @ X ) )
=> ( ! [Y3: set_o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_o @ X @ Y3 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_283_image__UN,axiom,
! [F: nat > nat,B2: nat > set_nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_nat_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_284_image__UN,axiom,
! [F: nat > nat,B2: $o > set_nat,A2: set_o] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X2: $o] : ( image_nat_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_285_image__UN,axiom,
! [F: nat > real,B2: nat > set_nat,A2: set_nat] :
( ( image_nat_real @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [X2: nat] : ( image_nat_real @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_286_image__UN,axiom,
! [F: nat > real,B2: $o > set_nat,A2: set_o] :
( ( image_nat_real @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
= ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [X2: $o] : ( image_nat_real @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_287_image__UN,axiom,
! [F: nat > $o,B2: nat > set_nat,A2: set_nat] :
( ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X2: nat] : ( image_nat_o @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_288_image__UN,axiom,
! [F: nat > $o,B2: $o > set_nat,A2: set_o] :
( ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X2: $o] : ( image_nat_o @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_289_image__UN,axiom,
! [F: real > nat,B2: nat > set_real,A2: set_nat] :
( ( image_real_nat @ F @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_real_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_290_image__UN,axiom,
! [F: real > nat,B2: $o > set_real,A2: set_o] :
( ( image_real_nat @ F @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X2: $o] : ( image_real_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_291_image__UN,axiom,
! [F: real > real,B2: $o > set_real,A2: set_o] :
( ( image_real_real @ F @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) )
= ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [X2: $o] : ( image_real_real @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_292_image__UN,axiom,
! [F: real > $o,B2: $o > set_real,A2: set_o] :
( ( image_real_o @ F @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X2: $o] : ( image_real_o @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_293_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_nat,F: nat > nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_294_UN__extend__simps_I10_J,axiom,
! [B2: $o > set_nat,F: nat > $o,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_295_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_nat,F: $o > nat,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [A5: $o] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_296_UN__extend__simps_I10_J,axiom,
! [B2: $o > set_nat,F: $o > $o,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [A5: $o] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_297_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_real,F: nat > nat,A2: set_nat] :
( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [A5: nat] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_298_UN__extend__simps_I10_J,axiom,
! [B2: $o > set_real,F: $o > $o,A2: set_o] :
( ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [A5: $o] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_299_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_o,F: nat > nat,A2: set_nat] :
( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [A5: nat] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_300_UN__extend__simps_I10_J,axiom,
! [B2: $o > set_o,F: $o > $o,A2: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [A5: $o] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_301_UN__extend__simps_I10_J,axiom,
! [B2: set_nat > set_nat,F: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_302_UN__extend__simps_I10_J,axiom,
! [B2: set_real > set_nat,F: $o > set_real,A2: set_o] :
( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [A5: $o] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7270232309134952815et_nat @ B2 @ ( image_o_set_real @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_303_UN__subset__iff,axiom,
! [A2: nat > set_nat,I5: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I5 )
=> ( ord_less_eq_set_nat @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_304_UN__subset__iff,axiom,
! [A2: $o > set_nat,I5: set_o,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I5 )
=> ( ord_less_eq_set_nat @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_305_UN__subset__iff,axiom,
! [A2: set_nat > set_nat,I5: set_set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ I5 )
=> ( ord_less_eq_set_nat @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_306_UN__subset__iff,axiom,
! [A2: $o > set_real,I5: set_o,B2: set_real] :
( ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I5 )
=> ( ord_less_eq_set_real @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_307_UN__subset__iff,axiom,
! [A2: $o > set_o,I5: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ I5 ) ) @ B2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I5 )
=> ( ord_less_eq_set_o @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_308_UN__upper,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_nat > set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_309_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_nat] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_310_UN__upper,axiom,
! [A: real,A2: set_real,B2: real > set_nat] :
( ( member_real @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_311_UN__upper,axiom,
! [A: nat,A2: set_nat,B2: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_312_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_real] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_real @ ( B2 @ A ) @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_313_UN__upper,axiom,
! [A: real,A2: set_real,B2: real > set_real] :
( ( member_real @ A @ A2 )
=> ( ord_less_eq_set_real @ ( B2 @ A ) @ ( comple3096694443085538997t_real @ ( image_real_set_real @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_314_UN__upper,axiom,
! [A: nat,A2: set_nat,B2: nat > set_real] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_real @ ( B2 @ A ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_315_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_o] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ A ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_316_UN__upper,axiom,
! [A: real,A2: set_real,B2: real > set_o] :
( ( member_real @ A @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ A ) @ ( comple90263536869209701_set_o @ ( image_real_set_o @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_317_UN__upper,axiom,
! [A: nat,A2: set_nat,B2: nat > set_o] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ A ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_318_UN__least,axiom,
! [A2: set_set_nat,B2: set_nat > set_nat,C: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_319_UN__least,axiom,
! [A2: set_o,B2: $o > set_nat,C: set_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_320_UN__least,axiom,
! [A2: set_real,B2: real > set_nat,C: set_nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_321_UN__least,axiom,
! [A2: set_nat,B2: nat > set_nat,C: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_322_UN__least,axiom,
! [A2: set_o,B2: $o > set_real,C: set_real] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_real @ ( B2 @ X4 ) @ C ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_323_UN__least,axiom,
! [A2: set_real,B2: real > set_real,C: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_set_real @ ( B2 @ X4 ) @ C ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_324_UN__least,axiom,
! [A2: set_nat,B2: nat > set_real,C: set_real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_real @ ( B2 @ X4 ) @ C ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_325_UN__least,axiom,
! [A2: set_o,B2: $o > set_o,C: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X4 ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_326_UN__least,axiom,
! [A2: set_real,B2: real > set_o,C: set_o] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X4 ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_real_set_o @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_327_UN__least,axiom,
! [A2: set_nat,B2: nat > set_o,C: set_o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_o @ ( B2 @ X4 ) @ C ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) @ C ) ) ).
% UN_least
thf(fact_328_UN__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_329_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_330_UN__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_331_UN__mono,axiom,
! [A2: set_real,B2: set_real,F: real > set_nat,G: real > set_nat] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_332_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_real,G: $o > set_real] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_o_set_real @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_333_UN__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_real,G: nat > set_real] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_334_UN__mono,axiom,
! [A2: set_real,B2: set_real,F: real > set_real,G: real > set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_real_set_real @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_335_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o,G: $o > set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_336_UN__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_o,G: nat > set_o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_337_UN__mono,axiom,
! [A2: set_real,B2: set_real,F: real > set_o,G: real > set_o] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_real_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_real_set_o @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_338_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_339_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > $o,G: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_340_SUP__subset__mono,axiom,
! [A2: set_real,B2: set_real,F: real > $o,G: real > $o] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_real_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_real_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_341_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_342_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_343_SUP__subset__mono,axiom,
! [A2: set_real,B2: set_real,F: real > set_nat,G: real > set_nat] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_real_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_344_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_real,G: $o > set_real] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_o_set_real @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_345_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_real,G: nat > set_real] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_346_SUP__subset__mono,axiom,
! [A2: set_real,B2: set_real,F: real > set_real,G: real > set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) ) @ ( comple3096694443085538997t_real @ ( image_real_set_real @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_347_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o,G: $o > set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_348_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_349_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_350_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_351_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_352_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_353_mult__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_354_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_355_mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_356_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X: real,B: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_357_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X: real,Y: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_358_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_359_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_360_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( times_times_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_361_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_362_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_363_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_364_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_365_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_366_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_367_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_368_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_369_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_370_vector__space__over__itself_Oscale__scale,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_371_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_372_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_373_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_374_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_375_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_376_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_377_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_378_mult__le__mono,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J3 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_379_mult__le__mono1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J3 @ K ) ) ) ).
% mult_le_mono1
thf(fact_380_mult__le__mono2,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J3 ) ) ) ).
% mult_le_mono2
thf(fact_381_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_382_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_383_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_384_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_385_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_386_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_387_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_388_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A: real,B: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_389_mult__left__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_390_mult__left__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_391_mult__right__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_392_mult__right__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_393_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_394_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_395_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_396_mult__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_397_mult__mono,axiom,
! [A: real,B: real,C2: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_398_mult__mono_H,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_399_mult__mono_H,axiom,
! [A: real,B: real,C2: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_400_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_401_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_402_mult__left__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_403_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_404_mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_405_mult__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_406_mult__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_right_mono_neg
thf(fact_407_mult__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_408_mult__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_409_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_410_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_411_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_412_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_413_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_414_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_415_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_416_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_417_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_418_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_419_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_420_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_421_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_422_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_423_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_424_image__ident,axiom,
! [Y5: set_set_nat] :
( ( image_7916887816326733075et_nat
@ ^ [X2: set_nat] : X2
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_425_image__ident,axiom,
! [Y5: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_426_set__times__mono2,axiom,
! [C: set_nat,D: set_nat,E: set_nat,F2: set_nat] :
( ( ord_less_eq_set_nat @ C @ D )
=> ( ( ord_less_eq_set_nat @ E @ F2 )
=> ( ord_less_eq_set_nat @ ( times_times_set_nat @ C @ E ) @ ( times_times_set_nat @ D @ F2 ) ) ) ) ).
% set_times_mono2
thf(fact_427_set__times__mono2,axiom,
! [C: set_real,D: set_real,E: set_real,F2: set_real] :
( ( ord_less_eq_set_real @ C @ D )
=> ( ( ord_less_eq_set_real @ E @ F2 )
=> ( ord_less_eq_set_real @ ( times_times_set_real @ C @ E ) @ ( times_times_set_real @ D @ F2 ) ) ) ) ).
% set_times_mono2
thf(fact_428_subsetI,axiom,
! [A2: set_o,B2: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_o @ X4 @ B2 ) )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% subsetI
thf(fact_429_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_430_subsetI,axiom,
! [A2: set_real,B2: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_real @ X4 @ B2 ) )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% subsetI
thf(fact_431_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_432_subset__antisym,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_433_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_434_image__eqI,axiom,
! [B: real,F: $o > real,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_real @ B @ ( image_o_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_435_image__eqI,axiom,
! [B: nat,F: $o > nat,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_nat @ B @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_436_image__eqI,axiom,
! [B: $o,F: real > $o,X: real,A2: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_o @ B @ ( image_real_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_437_image__eqI,axiom,
! [B: real,F: real > real,X: real,A2: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ B @ ( image_real_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_438_image__eqI,axiom,
! [B: nat,F: real > nat,X: real,A2: set_real] :
( ( B
= ( F @ X ) )
=> ( ( member_real @ X @ A2 )
=> ( member_nat @ B @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_439_image__eqI,axiom,
! [B: $o,F: nat > $o,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_o @ B @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_440_image__eqI,axiom,
! [B: real,F: nat > real,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_real @ B @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_441_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_442_image__eqI,axiom,
! [B: set_real,F: $o > set_real,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_set_real @ B @ ( image_o_set_real @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_443_set__times__intro,axiom,
! [A: nat,C: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_444_set__times__intro,axiom,
! [A: real,C: set_real,B: real,D: set_real] :
( ( member_real @ A @ C )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C @ D ) ) ) ) ).
% set_times_intro
thf(fact_445_set__times__UNION__distrib_I2_J,axiom,
! [M2: nat > set_nat,I5: set_nat,A2: set_nat] :
( ( times_times_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ I5 ) ) @ A2 )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( times_times_set_nat @ ( M2 @ I ) @ A2 )
@ I5 ) ) ) ).
% set_times_UNION_distrib(2)
thf(fact_446_set__times__UNION__distrib_I2_J,axiom,
! [M2: $o > set_nat,I5: set_o,A2: set_nat] :
( ( times_times_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ M2 @ I5 ) ) @ A2 )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( times_times_set_nat @ ( M2 @ I ) @ A2 )
@ I5 ) ) ) ).
% set_times_UNION_distrib(2)
thf(fact_447_set__times__UNION__distrib_I2_J,axiom,
! [M2: set_nat > set_nat,I5: set_set_nat,A2: set_nat] :
( ( times_times_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ M2 @ I5 ) ) @ A2 )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [I: set_nat] : ( times_times_set_nat @ ( M2 @ I ) @ A2 )
@ I5 ) ) ) ).
% set_times_UNION_distrib(2)
thf(fact_448_set__times__UNION__distrib_I2_J,axiom,
! [M2: $o > set_real,I5: set_o,A2: set_real] :
( ( times_times_set_real @ ( comple3096694443085538997t_real @ ( image_o_set_real @ M2 @ I5 ) ) @ A2 )
= ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [I: $o] : ( times_times_set_real @ ( M2 @ I ) @ A2 )
@ I5 ) ) ) ).
% set_times_UNION_distrib(2)
thf(fact_449_set__times__UNION__distrib_I1_J,axiom,
! [A2: set_nat,M2: nat > set_nat,I5: set_nat] :
( ( times_times_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ I5 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( times_times_set_nat @ A2 @ ( M2 @ I ) )
@ I5 ) ) ) ).
% set_times_UNION_distrib(1)
thf(fact_450_set__times__UNION__distrib_I1_J,axiom,
! [A2: set_nat,M2: $o > set_nat,I5: set_o] :
( ( times_times_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ M2 @ I5 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( times_times_set_nat @ A2 @ ( M2 @ I ) )
@ I5 ) ) ) ).
% set_times_UNION_distrib(1)
thf(fact_451_set__times__UNION__distrib_I1_J,axiom,
! [A2: set_nat,M2: set_nat > set_nat,I5: set_set_nat] :
( ( times_times_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ M2 @ I5 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [I: set_nat] : ( times_times_set_nat @ A2 @ ( M2 @ I ) )
@ I5 ) ) ) ).
% set_times_UNION_distrib(1)
thf(fact_452_set__times__UNION__distrib_I1_J,axiom,
! [A2: set_real,M2: $o > set_real,I5: set_o] :
( ( times_times_set_real @ A2 @ ( comple3096694443085538997t_real @ ( image_o_set_real @ M2 @ I5 ) ) )
= ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [I: $o] : ( times_times_set_real @ A2 @ ( M2 @ I ) )
@ I5 ) ) ) ).
% set_times_UNION_distrib(1)
thf(fact_453_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_454_le__trans,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_455_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_456_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_457_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_458_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_459_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_460_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_461_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_462_set__times__elim,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X @ ( times_times_set_nat @ A2 @ B2 ) )
=> ~ ! [A4: nat,B3: nat] :
( ( X
= ( times_times_nat @ A4 @ B3 ) )
=> ( ( member_nat @ A4 @ A2 )
=> ~ ( member_nat @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_463_set__times__elim,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ( member_real @ X @ ( times_times_set_real @ A2 @ B2 ) )
=> ~ ! [A4: real,B3: real] :
( ( X
= ( times_times_real @ A4 @ B3 ) )
=> ( ( member_real @ A4 @ A2 )
=> ~ ( member_real @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_464_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_465_mult_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C2 ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_466_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_467_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A5: real,B4: real] : ( times_times_real @ B4 @ A5 ) ) ) ).
% mult.commute
thf(fact_468_mult_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_469_mult_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_470_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_471_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_472_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_473_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: real,F: $o > real] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_o_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_474_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: nat,F: $o > nat] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_475_rev__image__eqI,axiom,
! [X: real,A2: set_real,B: $o,F: real > $o] :
( ( member_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_real_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_476_rev__image__eqI,axiom,
! [X: real,A2: set_real,B: real,F: real > real] :
( ( member_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_real_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_477_rev__image__eqI,axiom,
! [X: real,A2: set_real,B: nat,F: real > nat] :
( ( member_real @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_real_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_478_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: $o,F: nat > $o] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_479_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: real,F: nat > real] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_nat_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_480_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_481_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: set_real,F: $o > set_real] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_real @ B @ ( image_o_set_real @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_482_ball__imageD,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_483_ball__imageD,axiom,
! [F: $o > set_real,A2: set_o,P: set_real > $o] :
( ! [X4: set_real] :
( ( member_set_real @ X4 @ ( image_o_set_real @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X6: $o] :
( ( member_o @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_484_ball__imageD,axiom,
! [F: $o > set_o,A2: set_o,P: set_o > $o] :
( ! [X4: set_o] :
( ( member_set_o @ X4 @ ( image_o_set_o @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X6: $o] :
( ( member_o @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_485_ball__imageD,axiom,
! [F: $o > set_nat,A2: set_o,P: set_nat > $o] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_o_set_nat @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X6: $o] :
( ( member_o @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_486_ball__imageD,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_nat > $o] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_487_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_488_image__cong,axiom,
! [M2: set_set_nat,N3: set_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ( M2 = N3 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ N3 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_7916887816326733075et_nat @ F @ M2 )
= ( image_7916887816326733075et_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_489_image__cong,axiom,
! [M2: set_o,N3: set_o,F: $o > set_real,G: $o > set_real] :
( ( M2 = N3 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ N3 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_o_set_real @ F @ M2 )
= ( image_o_set_real @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_490_image__cong,axiom,
! [M2: set_o,N3: set_o,F: $o > set_o,G: $o > set_o] :
( ( M2 = N3 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ N3 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_o_set_o @ F @ M2 )
= ( image_o_set_o @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_491_image__cong,axiom,
! [M2: set_o,N3: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( M2 = N3 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ N3 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_o_set_nat @ F @ M2 )
= ( image_o_set_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_492_image__cong,axiom,
! [M2: set_nat,N3: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M2 = N3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N3 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_set_nat @ F @ M2 )
= ( image_nat_set_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_493_image__cong,axiom,
! [M2: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
( ( M2 = N3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N3 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_nat @ F @ M2 )
= ( image_nat_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_494_bex__imageD,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_495_bex__imageD,axiom,
! [F: $o > set_real,A2: set_o,P: set_real > $o] :
( ? [X6: set_real] :
( ( member_set_real @ X6 @ ( image_o_set_real @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_496_bex__imageD,axiom,
! [F: $o > set_o,A2: set_o,P: set_o > $o] :
( ? [X6: set_o] :
( ( member_set_o @ X6 @ ( image_o_set_o @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_497_bex__imageD,axiom,
! [F: $o > set_nat,A2: set_o,P: set_nat > $o] :
( ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( image_o_set_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_498_bex__imageD,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_nat > $o] :
( ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_499_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_500_image__iff,axiom,
! [Z: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_501_image__iff,axiom,
! [Z: set_real,F: $o > set_real,A2: set_o] :
( ( member_set_real @ Z @ ( image_o_set_real @ F @ A2 ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_502_image__iff,axiom,
! [Z: set_o,F: $o > set_o,A2: set_o] :
( ( member_set_o @ Z @ ( image_o_set_o @ F @ A2 ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_503_image__iff,axiom,
! [Z: set_nat,F: $o > set_nat,A2: set_o] :
( ( member_set_nat @ Z @ ( image_o_set_nat @ F @ A2 ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_504_image__iff,axiom,
! [Z: set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( member_set_nat @ Z @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_505_image__iff,axiom,
! [Z: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_506_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_507_imageI,axiom,
! [X: $o,A2: set_o,F: $o > real] :
( ( member_o @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_o_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_508_imageI,axiom,
! [X: $o,A2: set_o,F: $o > nat] :
( ( member_o @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_o_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_509_imageI,axiom,
! [X: real,A2: set_real,F: real > $o] :
( ( member_real @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_real_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_510_imageI,axiom,
! [X: real,A2: set_real,F: real > real] :
( ( member_real @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_real_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_511_imageI,axiom,
! [X: real,A2: set_real,F: real > nat] :
( ( member_real @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_real_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_512_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > $o] :
( ( member_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_nat_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_513_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > real] :
( ( member_nat @ X @ A2 )
=> ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_514_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_515_imageI,axiom,
! [X: $o,A2: set_o,F: $o > set_real] :
( ( member_o @ X @ A2 )
=> ( member_set_real @ ( F @ X ) @ ( image_o_set_real @ F @ A2 ) ) ) ).
% imageI
thf(fact_516_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_517_Collect__mono__iff,axiom,
! [P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
= ( ! [X2: real] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_518_set__eq__subset,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [A3: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_519_set__eq__subset,axiom,
( ( ^ [Y6: set_real,Z3: set_real] : ( Y6 = Z3 ) )
= ( ^ [A3: set_real,B5: set_real] :
( ( ord_less_eq_set_real @ A3 @ B5 )
& ( ord_less_eq_set_real @ B5 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_520_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_521_subset__trans,axiom,
! [A2: set_real,B2: set_real,C: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( ord_less_eq_set_real @ B2 @ C )
=> ( ord_less_eq_set_real @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_522_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_523_Collect__mono,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X4: real] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% Collect_mono
thf(fact_524_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_525_subset__refl,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ A2 ) ).
% subset_refl
thf(fact_526_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B5: set_o] :
! [T: $o] :
( ( member_o @ T @ A3 )
=> ( member_o @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_527_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_528_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B5: set_real] :
! [T: real] :
( ( member_real @ T @ A3 )
=> ( member_real @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_529_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_530_equalityD2,axiom,
! [A2: set_real,B2: set_real] :
( ( A2 = B2 )
=> ( ord_less_eq_set_real @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_531_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_532_equalityD1,axiom,
! [A2: set_real,B2: set_real] :
( ( A2 = B2 )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_533_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B5: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ( member_o @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_534_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_535_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B5: set_real] :
! [X2: real] :
( ( member_real @ X2 @ A3 )
=> ( member_real @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_536_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_537_equalityE,axiom,
! [A2: set_real,B2: set_real] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ~ ( ord_less_eq_set_real @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_538_subsetD,axiom,
! [A2: set_o,B2: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_539_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_540_subsetD,axiom,
! [A2: set_real,B2: set_real,C2: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ C2 @ A2 )
=> ( member_real @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_541_in__mono,axiom,
! [A2: set_o,B2: set_o,X: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ X @ A2 )
=> ( member_o @ X @ B2 ) ) ) ).
% in_mono
thf(fact_542_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_543_in__mono,axiom,
! [A2: set_real,B2: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B2 ) ) ) ).
% in_mono
thf(fact_544_Compr__image__eq,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_o_o @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_545_Compr__image__eq,axiom,
! [F: real > $o,A2: set_real,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_real_o @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_real_o @ F
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_546_Compr__image__eq,axiom,
! [F: nat > $o,A2: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_nat_o @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_547_Compr__image__eq,axiom,
! [F: $o > real,A2: set_o,P: real > $o] :
( ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ ( image_o_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_o_real @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_548_Compr__image__eq,axiom,
! [F: real > real,A2: set_real,P: real > $o] :
( ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ ( image_real_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_real_real @ F
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_549_Compr__image__eq,axiom,
! [F: nat > real,A2: set_nat,P: real > $o] :
( ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ ( image_nat_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_real @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_550_Compr__image__eq,axiom,
! [F: $o > nat,A2: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_o_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_551_Compr__image__eq,axiom,
! [F: real > nat,A2: set_real,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_real_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_real_nat @ F
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_552_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_553_Compr__image__eq,axiom,
! [F: $o > set_real,A2: set_o,P: set_real > $o] :
( ( collect_set_real
@ ^ [X2: set_real] :
( ( member_set_real @ X2 @ ( image_o_set_real @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_o_set_real @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_554_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_555_image__image,axiom,
! [F: set_nat > nat,G: nat > set_nat,A2: set_nat] :
( ( image_set_nat_nat @ F @ ( image_nat_set_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_556_image__image,axiom,
! [F: nat > set_nat,G: $o > nat,A2: set_o] :
( ( image_nat_set_nat @ F @ ( image_o_nat @ G @ A2 ) )
= ( image_o_set_nat
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_557_image__image,axiom,
! [F: nat > set_nat,G: nat > nat,A2: set_nat] :
( ( image_nat_set_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_558_image__image,axiom,
! [F: $o > set_real,G: $o > $o,A2: set_o] :
( ( image_o_set_real @ F @ ( image_o_o @ G @ A2 ) )
= ( image_o_set_real
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_559_image__image,axiom,
! [F: $o > set_o,G: $o > $o,A2: set_o] :
( ( image_o_set_o @ F @ ( image_o_o @ G @ A2 ) )
= ( image_o_set_o
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_560_image__image,axiom,
! [F: $o > set_nat,G: nat > $o,A2: set_nat] :
( ( image_o_set_nat @ F @ ( image_nat_o @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_561_image__image,axiom,
! [F: $o > set_nat,G: $o > $o,A2: set_o] :
( ( image_o_set_nat @ F @ ( image_o_o @ G @ A2 ) )
= ( image_o_set_nat
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_562_image__image,axiom,
! [F: set_real > set_real,G: $o > set_real,A2: set_o] :
( ( image_2436557299294012491t_real @ F @ ( image_o_set_real @ G @ A2 ) )
= ( image_o_set_real
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_563_image__image,axiom,
! [F: set_real > set_o,G: $o > set_real,A2: set_o] :
( ( image_set_real_set_o @ F @ ( image_o_set_real @ G @ A2 ) )
= ( image_o_set_o
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_564_imageE,axiom,
! [B: $o,F: $o > $o,A2: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A2 ) )
=> ~ ! [X4: $o] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_565_imageE,axiom,
! [B: $o,F: real > $o,A2: set_real] :
( ( member_o @ B @ ( image_real_o @ F @ A2 ) )
=> ~ ! [X4: real] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_real @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_566_imageE,axiom,
! [B: $o,F: nat > $o,A2: set_nat] :
( ( member_o @ B @ ( image_nat_o @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_567_imageE,axiom,
! [B: real,F: $o > real,A2: set_o] :
( ( member_real @ B @ ( image_o_real @ F @ A2 ) )
=> ~ ! [X4: $o] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_568_imageE,axiom,
! [B: real,F: real > real,A2: set_real] :
( ( member_real @ B @ ( image_real_real @ F @ A2 ) )
=> ~ ! [X4: real] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_real @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_569_imageE,axiom,
! [B: real,F: nat > real,A2: set_nat] :
( ( member_real @ B @ ( image_nat_real @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_570_imageE,axiom,
! [B: nat,F: $o > nat,A2: set_o] :
( ( member_nat @ B @ ( image_o_nat @ F @ A2 ) )
=> ~ ! [X4: $o] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_571_imageE,axiom,
! [B: nat,F: real > nat,A2: set_real] :
( ( member_nat @ B @ ( image_real_nat @ F @ A2 ) )
=> ~ ! [X4: real] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_real @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_572_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_573_imageE,axiom,
! [B: set_real,F: $o > set_real,A2: set_o] :
( ( member_set_real @ B @ ( image_o_set_real @ F @ A2 ) )
=> ~ ! [X4: $o] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_574_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B5: set_o] :
( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A3 )
@ ^ [X2: $o] : ( member_o @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_575_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_576_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B5: set_real] :
( ord_less_eq_real_o
@ ^ [X2: real] : ( member_real @ X2 @ A3 )
@ ^ [X2: real] : ( member_real @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_577_conj__subset__def,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_578_conj__subset__def,axiom,
! [A2: set_real,P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ A2
@ ( collect_real
@ ^ [X2: real] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq_set_real @ A2 @ ( collect_real @ P ) )
& ( ord_less_eq_set_real @ A2 @ ( collect_real @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_579_Collect__subset,axiom,
! [A2: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_580_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_581_Collect__subset,axiom,
! [A2: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_582_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_583_subset__image__iff,axiom,
! [B2: set_set_real,F: $o > set_real,A2: set_o] :
( ( ord_le3558479182127378552t_real @ B2 @ ( image_o_set_real @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_set_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_584_subset__image__iff,axiom,
! [B2: set_set_o,F: $o > set_o,A2: set_o] :
( ( ord_le4374716579403074808_set_o @ B2 @ ( image_o_set_o @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_set_o @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_585_subset__image__iff,axiom,
! [B2: set_set_nat,F: $o > set_nat,A2: set_o] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_o_set_nat @ F @ A2 ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A2 )
& ( B2
= ( image_o_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_586_subset__image__iff,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B2
= ( image_7916887816326733075et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_587_subset__image__iff,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_588_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_589_subset__image__iff,axiom,
! [B2: set_nat,F: real > nat,A2: set_real] :
( ( ord_less_eq_set_nat @ B2 @ ( image_real_nat @ F @ A2 ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A2 )
& ( B2
= ( image_real_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_590_subset__image__iff,axiom,
! [B2: set_real,F: nat > real,A2: set_nat] :
( ( ord_less_eq_set_real @ B2 @ ( image_nat_real @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_591_subset__image__iff,axiom,
! [B2: set_real,F: real > real,A2: set_real] :
( ( ord_less_eq_set_real @ B2 @ ( image_real_real @ F @ A2 ) )
= ( ? [AA: set_real] :
( ( ord_less_eq_set_real @ AA @ A2 )
& ( B2
= ( image_real_real @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_592_image__subset__iff,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_593_image__subset__iff,axiom,
! [F: $o > set_real,A2: set_o,B2: set_set_real] :
( ( ord_le3558479182127378552t_real @ ( image_o_set_real @ F @ A2 ) @ B2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( member_set_real @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_594_image__subset__iff,axiom,
! [F: $o > set_o,A2: set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ A2 ) @ B2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( member_set_o @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_595_image__subset__iff,axiom,
! [F: $o > set_nat,A2: set_o,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_o_set_nat @ F @ A2 ) @ B2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_596_image__subset__iff,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_597_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_598_subset__imageE,axiom,
! [B2: set_set_real,F: $o > set_real,A2: set_o] :
( ( ord_le3558479182127378552t_real @ B2 @ ( image_o_set_real @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_o_set_real @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_599_subset__imageE,axiom,
! [B2: set_set_o,F: $o > set_o,A2: set_o] :
( ( ord_le4374716579403074808_set_o @ B2 @ ( image_o_set_o @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_o_set_o @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_600_subset__imageE,axiom,
! [B2: set_set_nat,F: $o > set_nat,A2: set_o] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_o_set_nat @ F @ A2 ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A2 )
=> ( B2
!= ( image_o_set_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_601_subset__imageE,axiom,
! [B2: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
=> ( B2
!= ( image_7916887816326733075et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_602_subset__imageE,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_set_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_603_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_604_subset__imageE,axiom,
! [B2: set_nat,F: real > nat,A2: set_real] :
( ( ord_less_eq_set_nat @ B2 @ ( image_real_nat @ F @ A2 ) )
=> ~ ! [C3: set_real] :
( ( ord_less_eq_set_real @ C3 @ A2 )
=> ( B2
!= ( image_real_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_605_subset__imageE,axiom,
! [B2: set_real,F: nat > real,A2: set_nat] :
( ( ord_less_eq_set_real @ B2 @ ( image_nat_real @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B2
!= ( image_nat_real @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_606_subset__imageE,axiom,
! [B2: set_real,F: real > real,A2: set_real] :
( ( ord_less_eq_set_real @ B2 @ ( image_real_real @ F @ A2 ) )
=> ~ ! [C3: set_real] :
( ( ord_less_eq_set_real @ C3 @ A2 )
=> ( B2
!= ( image_real_real @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_607_image__subsetI,axiom,
! [A2: set_o,F: $o > $o,B2: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_o @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_608_image__subsetI,axiom,
! [A2: set_real,F: real > $o,B2: set_o] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_o @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_real_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_609_image__subsetI,axiom,
! [A2: set_nat,F: nat > $o,B2: set_o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_o @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_610_image__subsetI,axiom,
! [A2: set_o,F: $o > nat,B2: set_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_611_image__subsetI,axiom,
! [A2: set_real,F: real > nat,B2: set_nat] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_612_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_613_image__subsetI,axiom,
! [A2: set_o,F: $o > real,B2: set_real] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_real @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_real @ ( image_o_real @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_614_image__subsetI,axiom,
! [A2: set_real,F: real > real,B2: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_real @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_615_image__subsetI,axiom,
! [A2: set_nat,F: nat > real,B2: set_real] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_real @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_616_image__subsetI,axiom,
! [A2: set_o,F: $o > set_real,B2: set_set_real] :
( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( member_set_real @ ( F @ X4 ) @ B2 ) )
=> ( ord_le3558479182127378552t_real @ ( image_o_set_real @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_617_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_real] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_le3558479182127378552t_real @ ( image_o_set_real @ F @ A2 ) @ ( image_o_set_real @ F @ B2 ) ) ) ).
% image_mono
thf(fact_618_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ A2 ) @ ( image_o_set_o @ F @ B2 ) ) ) ).
% image_mono
thf(fact_619_image__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_o_set_nat @ F @ A2 ) @ ( image_o_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_620_image__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_621_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_622_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_623_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > real] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A2 ) @ ( image_nat_real @ F @ B2 ) ) ) ).
% image_mono
thf(fact_624_image__mono,axiom,
! [A2: set_real,B2: set_real,F: real > nat] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_real_nat @ F @ A2 ) @ ( image_real_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_625_image__mono,axiom,
! [A2: set_real,B2: set_real,F: real > real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ ( image_real_real @ F @ B2 ) ) ) ).
% image_mono
thf(fact_626_SUP__UN__eq,axiom,
! [R: nat > set_nat,S: set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_nat_nat_o
@ ^ [I: nat,X2: nat] : ( member_nat @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_627_SUP__UN__eq,axiom,
! [R: $o > set_nat,S: set_o] :
( ( comple8317665133742190828_nat_o
@ ( image_o_nat_o
@ ^ [I: $o,X2: nat] : ( member_nat @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_628_SUP__UN__eq,axiom,
! [R: set_nat > set_nat,S: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o
@ ^ [I: set_nat,X2: nat] : ( member_nat @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_629_SUP__UN__eq,axiom,
! [R: $o > set_real,S: set_o] :
( ( comple3015195443809154064real_o
@ ( image_o_real_o
@ ^ [I: $o,X2: real] : ( member_real @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: real] : ( member_real @ X2 @ ( comple3096694443085538997t_real @ ( image_o_set_real @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_630_SUP__UN__eq,axiom,
! [R: $o > set_o,S: set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_o_o_o
@ ^ [I: $o,X2: $o] : ( member_o @ X2 @ ( R @ I ) )
@ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_631_UN__constant__eq,axiom,
! [A: set_nat,A2: set_set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_632_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_nat,C2: set_nat] :
( ( member_o @ A @ A2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_633_UN__constant__eq,axiom,
! [A: real,A2: set_real,F: real > set_nat,C2: set_nat] :
( ( member_real @ A @ A2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_real_set_nat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_634_UN__constant__eq,axiom,
! [A: nat,A2: set_nat,F: nat > set_nat,C2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_635_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_real,C2: set_real] :
( ( member_o @ A @ A2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple3096694443085538997t_real @ ( image_o_set_real @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_636_UN__constant__eq,axiom,
! [A: real,A2: set_real,F: real > set_real,C2: set_real] :
( ( member_real @ A @ A2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple3096694443085538997t_real @ ( image_real_set_real @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_637_UN__constant__eq,axiom,
! [A: nat,A2: set_nat,F: nat > set_real,C2: set_real] :
( ( member_nat @ A @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_638_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_o,C2: set_o] :
( ( member_o @ A @ A2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_639_UN__constant__eq,axiom,
! [A: real,A2: set_real,F: real > set_o,C2: set_o] :
( ( member_real @ A @ A2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_real_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_640_UN__constant__eq,axiom,
! [A: nat,A2: set_nat,F: nat > set_o,C2: set_o] :
( ( member_nat @ A @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_641_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_642_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_643_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_644_dual__order_Orefl,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ A ) ).
% dual_order.refl
thf(fact_645_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_646_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_647_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_648_order__refl,axiom,
! [X: set_real] : ( ord_less_eq_set_real @ X @ X ) ).
% order_refl
thf(fact_649_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > set_nat,B2: set_set_nat] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( member_set_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_650_image__Collect__subsetI,axiom,
! [P: $o > $o,F: $o > set_real,B2: set_set_real] :
( ! [X4: $o] :
( ( P @ X4 )
=> ( member_set_real @ ( F @ X4 ) @ B2 ) )
=> ( ord_le3558479182127378552t_real @ ( image_o_set_real @ F @ ( collect_o @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_651_image__Collect__subsetI,axiom,
! [P: $o > $o,F: $o > set_o,B2: set_set_o] :
( ! [X4: $o] :
( ( P @ X4 )
=> ( member_set_o @ ( F @ X4 ) @ B2 ) )
=> ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ ( collect_o @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_652_image__Collect__subsetI,axiom,
! [P: $o > $o,F: $o > set_nat,B2: set_set_nat] :
( ! [X4: $o] :
( ( P @ X4 )
=> ( member_set_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_o_set_nat @ F @ ( collect_o @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_653_image__Collect__subsetI,axiom,
! [P: set_nat > $o,F: set_nat > set_nat,B2: set_set_nat] :
( ! [X4: set_nat] :
( ( P @ X4 )
=> ( member_set_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ ( collect_set_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_654_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B2: set_nat] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_655_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_656_all__subset__image,axiom,
! [F: $o > set_real,A2: set_o,P: set_set_real > $o] :
( ( ! [B5: set_set_real] :
( ( ord_le3558479182127378552t_real @ B5 @ ( image_o_set_real @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_o] :
( ( ord_less_eq_set_o @ B5 @ A2 )
=> ( P @ ( image_o_set_real @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_657_all__subset__image,axiom,
! [F: $o > set_o,A2: set_o,P: set_set_o > $o] :
( ( ! [B5: set_set_o] :
( ( ord_le4374716579403074808_set_o @ B5 @ ( image_o_set_o @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_o] :
( ( ord_less_eq_set_o @ B5 @ A2 )
=> ( P @ ( image_o_set_o @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_658_all__subset__image,axiom,
! [F: $o > set_nat,A2: set_o,P: set_set_nat > $o] :
( ( ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ ( image_o_set_nat @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_o] :
( ( ord_less_eq_set_o @ B5 @ A2 )
=> ( P @ ( image_o_set_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_659_all__subset__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ A2 )
=> ( P @ ( image_7916887816326733075et_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_660_all__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( P @ ( image_nat_set_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_661_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_662_all__subset__image,axiom,
! [F: real > nat,A2: set_real,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_real_nat @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_real] :
( ( ord_less_eq_set_real @ B5 @ A2 )
=> ( P @ ( image_real_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_663_all__subset__image,axiom,
! [F: nat > real,A2: set_nat,P: set_real > $o] :
( ( ! [B5: set_real] :
( ( ord_less_eq_set_real @ B5 @ ( image_nat_real @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( P @ ( image_nat_real @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_664_all__subset__image,axiom,
! [F: real > real,A2: set_real,P: set_real > $o] :
( ( ! [B5: set_real] :
( ( ord_less_eq_set_real @ B5 @ ( image_real_real @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_real] :
( ( ord_less_eq_set_real @ B5 @ A2 )
=> ( P @ ( image_real_real @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_665_mult__delta__right,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_666_mult__delta__right,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_667_mult__delta__left,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_668_mult__delta__left,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_669_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_670_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_671_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_672_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_673_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
& ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_674_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_675_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_676_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_real,Z3: set_real] : ( Y6 = Z3 ) )
= ( ^ [X2: set_real,Y2: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y2 )
& ( ord_less_eq_set_real @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_677_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_678_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_679_ord__eq__le__trans,axiom,
! [A: real,B: real,C2: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_680_ord__eq__le__trans,axiom,
! [A: set_real,B: set_real,C2: set_real] :
( ( A = B )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ord_less_eq_set_real @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_681_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_682_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_683_ord__le__eq__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_684_ord__le__eq__trans,axiom,
! [A: set_real,B: set_real,C2: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_real @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_685_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_686_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_687_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_688_order__antisym,axiom,
! [X: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( ord_less_eq_set_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_689_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_690_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_691_order_Otrans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% order.trans
thf(fact_692_order_Otrans,axiom,
! [A: set_real,B: set_real,C2: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ord_less_eq_set_real @ A @ C2 ) ) ) ).
% order.trans
thf(fact_693_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_694_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_695_order__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_696_order__trans,axiom,
! [X: set_real,Y: set_real,Z: set_real] :
( ( ord_less_eq_set_real @ X @ Y )
=> ( ( ord_less_eq_set_real @ Y @ Z )
=> ( ord_less_eq_set_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_697_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_698_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_699_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_700_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_701_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_702_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_real,Z3: set_real] : ( Y6 = Z3 ) )
= ( ^ [A5: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ B4 @ A5 )
& ( ord_less_eq_set_real @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_703_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_704_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_705_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_706_dual__order_Oantisym,axiom,
! [B: set_real,A: set_real] :
( ( ord_less_eq_set_real @ B @ A )
=> ( ( ord_less_eq_set_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_707_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_708_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_709_dual__order_Otrans,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_710_dual__order_Otrans,axiom,
! [B: set_real,A: set_real,C2: set_real] :
( ( ord_less_eq_set_real @ B @ A )
=> ( ( ord_less_eq_set_real @ C2 @ B )
=> ( ord_less_eq_set_real @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_711_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_712_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_713_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_714_antisym,axiom,
! [A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( ord_less_eq_set_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_715_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_716_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_717_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_718_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_real,Z3: set_real] : ( Y6 = Z3 ) )
= ( ^ [A5: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ A5 @ B4 )
& ( ord_less_eq_set_real @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_719_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_720_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_721_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_722_order__subst1,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_723_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_724_order__subst1,axiom,
! [A: set_nat,F: real > set_nat,B: real,C2: real] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_725_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_726_order__subst1,axiom,
! [A: nat,F: set_real > nat,B: set_real,C2: set_real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ! [X4: set_real,Y3: set_real] :
( ( ord_less_eq_set_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_727_order__subst1,axiom,
! [A: real,F: set_nat > real,B: set_nat,C2: set_nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_728_order__subst1,axiom,
! [A: real,F: set_real > real,B: set_real,C2: set_real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_real @ B @ C2 )
=> ( ! [X4: set_real,Y3: set_real] :
( ( ord_less_eq_set_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_729_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_730_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_731_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_732_order__subst2,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_733_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_734_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > real,C2: real] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C2 )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_735_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_736_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_real,C2: set_real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_real @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_737_order__subst2,axiom,
! [A: real,B: real,F: real > set_nat,C2: set_nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_738_order__subst2,axiom,
! [A: real,B: real,F: real > set_real,C2: set_real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_set_real @ ( F @ B ) @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_739_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_740_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_741_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_742_order__eq__refl,axiom,
! [X: set_real,Y: set_real] :
( ( X = Y )
=> ( ord_less_eq_set_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_743_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_744_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_745_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_746_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_747_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_748_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_749_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C2: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_750_ord__eq__le__subst,axiom,
! [A: real,F: set_nat > real,B: set_nat,C2: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_751_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_752_ord__eq__le__subst,axiom,
! [A: set_real,F: nat > set_real,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_753_ord__eq__le__subst,axiom,
! [A: set_nat,F: real > set_nat,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_754_ord__eq__le__subst,axiom,
! [A: set_real,F: real > set_real,B: real,C2: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_755_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_756_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_757_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_758_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_759_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_760_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > real,C2: real] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_761_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_762_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_real,C2: set_real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_763_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > set_nat,C2: set_nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_764_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > set_real,C2: set_real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: real,Y3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ord_less_eq_set_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_765_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_766_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_767_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_768_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_769_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_770_order__antisym__conv,axiom,
! [Y: set_real,X: set_real] :
( ( ord_less_eq_set_real @ Y @ X )
=> ( ( ord_less_eq_set_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_771_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S2: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_772_Sup__SUP__eq,axiom,
( comple3015195443809154064real_o
= ( ^ [S2: set_real_o,X2: real] : ( member_real @ X2 @ ( comple3096694443085538997t_real @ ( image_2734271470692514752t_real @ collect_real @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_773_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S2: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_774_prop__restrict,axiom,
! [X: $o,Z4: set_o,X3: set_o,P: $o > $o] :
( ( member_o @ X @ Z4 )
=> ( ( ord_less_eq_set_o @ Z4
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X3 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_775_prop__restrict,axiom,
! [X: nat,Z4: set_nat,X3: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X3 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_776_prop__restrict,axiom,
! [X: real,Z4: set_real,X3: set_real,P: real > $o] :
( ( member_real @ X @ Z4 )
=> ( ( ord_less_eq_set_real @ Z4
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ X3 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_777_Collect__restrict,axiom,
! [X3: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ X3 )
& ( P @ X2 ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_778_Collect__restrict,axiom,
! [X3: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X3 )
& ( P @ X2 ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_779_Collect__restrict,axiom,
! [X3: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ X3 )
& ( P @ X2 ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_780_pred__subset__eq,axiom,
! [R2: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ R2 )
@ ^ [X2: $o] : ( member_o @ X2 @ S ) )
= ( ord_less_eq_set_o @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_781_pred__subset__eq,axiom,
! [R2: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R2 )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_782_pred__subset__eq,axiom,
! [R2: set_real,S: set_real] :
( ( ord_less_eq_real_o
@ ^ [X2: real] : ( member_real @ X2 @ R2 )
@ ^ [X2: real] : ( member_real @ X2 @ S ) )
= ( ord_less_eq_set_real @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_783_SUP__Sup__eq,axiom,
! [S: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o
@ ^ [I: set_nat,X2: nat] : ( member_nat @ X2 @ I )
@ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_784_SUP__Sup__eq,axiom,
! [S: set_set_real] :
( ( comple3015195443809154064real_o
@ ( image_5650221686686655994real_o
@ ^ [I: set_real,X2: real] : ( member_real @ X2 @ I )
@ S ) )
= ( ^ [X2: real] : ( member_real @ X2 @ ( comple3096694443085538997t_real @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_785_SUP__Sup__eq,axiom,
! [S: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I: set_o,X2: $o] : ( member_o @ X2 @ I )
@ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_786_image__Fpow__mono,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_set_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_787_image__Fpow__mono,axiom,
! [F: $o > set_real,A2: set_o,B2: set_set_real] :
( ( ord_le3558479182127378552t_real @ ( image_o_set_real @ F @ A2 ) @ B2 )
=> ( ord_le561408886441773742t_real @ ( image_6191879853830326987t_real @ ( image_o_set_real @ F ) @ ( finite_Fpow_o @ A2 ) ) @ ( finite_Fpow_set_real @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_788_image__Fpow__mono,axiom,
! [F: $o > set_o,A2: set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ A2 ) @ B2 )
=> ( ord_le3178852226150452184_set_o @ ( image_5023573440332574309_set_o @ ( image_o_set_o @ F ) @ ( finite_Fpow_o @ A2 ) ) @ ( finite_Fpow_set_o @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_789_image__Fpow__mono,axiom,
! [F: $o > set_nat,A2: set_o,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_o_set_nat @ F @ A2 ) @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( image_7698617416147310703et_nat @ ( image_o_set_nat @ F ) @ ( finite_Fpow_o @ A2 ) ) @ ( finite_Fpow_set_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_790_image__Fpow__mono,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ ( image_7916887816326733075et_nat @ F ) @ ( finite_Fpow_set_nat @ A2 ) ) @ ( finite_Fpow_set_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_791_image__Fpow__mono,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_792_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_793_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_794_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_795_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_796_A__def,axiom,
( a
= ( ^ [I: nat] : ( set_or4665077453230672383an_nat @ ( times_times_nat @ I @ m ) @ ( times_times_nat @ ( plus_plus_nat @ I @ one_one_nat ) @ m ) ) ) ) ).
% A_def
thf(fact_797_subset__CollectI,axiom,
! [B2: set_o,A2: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B2 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ B2 )
& ( Q @ X2 ) ) )
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_798_subset__CollectI,axiom,
! [B2: set_nat,A2: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ( Q @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_799_subset__CollectI,axiom,
! [B2: set_real,A2: set_real,Q: real > $o,P: real > $o] :
( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ B2 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ B2 )
& ( Q @ X2 ) ) )
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_800_subset__Collect__iff,axiom,
! [B2: set_o,A2: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ( ord_less_eq_set_o @ B2
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_801_subset__Collect__iff,axiom,
! [B2: set_nat,A2: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ B2
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_802_subset__Collect__iff,axiom,
! [B2: set_real,A2: set_real,P: real > $o] :
( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( ( ord_less_eq_set_real @ B2
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( ! [X2: real] :
( ( member_real @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_803_set__plus__intro,axiom,
! [A: nat,C: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C @ D ) ) ) ) ).
% set_plus_intro
thf(fact_804_set__plus__intro,axiom,
! [A: real,C: set_real,B: real,D: set_real] :
( ( member_real @ A @ C )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C @ D ) ) ) ) ).
% set_plus_intro
thf(fact_805_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_806_add__right__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_807_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_808_add__left__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_809_UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_I
thf(fact_810_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_811_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_812_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_813_add__le__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_814_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_815_add__le__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_816_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_817_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_818_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_819_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_820_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_821_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_822_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_823_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_824_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_825_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_826_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_827_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_828_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_829_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_830_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_831_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_832_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_833_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_834_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_835_vector__space__over__itself_Oscale__one,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% vector_space_over_itself.scale_one
thf(fact_836_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_837_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_838_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_839_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_840_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N2 ) )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_841_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_842_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_843_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_844_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_845_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_846_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_847_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_848_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_849_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_850_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_851_mult__cancel__left1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_852_mult__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ( times_times_real @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_853_mult__cancel__right1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_854_mult__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ( times_times_real @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_855_image__add__0,axiom,
! [S: set_set_nat] :
( ( image_7916887816326733075et_nat @ ( plus_plus_set_nat @ zero_zero_set_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_856_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_857_image__add__0,axiom,
! [S: set_real] :
( ( image_real_real @ ( plus_plus_real @ zero_zero_real ) @ S )
= S ) ).
% image_add_0
thf(fact_858_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_859_Sup__UNIV,axiom,
( ( complete_Sup_Sup_o @ top_top_set_o )
= top_top_o ) ).
% Sup_UNIV
thf(fact_860_Sup__UNIV,axiom,
( ( comple3096694443085538997t_real @ top_top_set_set_real )
= top_top_set_real ) ).
% Sup_UNIV
thf(fact_861_Sup__UNIV,axiom,
( ( comple90263536869209701_set_o @ top_top_set_set_o )
= top_top_set_o ) ).
% Sup_UNIV
thf(fact_862_ivl__subset,axiom,
! [I3: real,J3: real,M: real,N2: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I3 @ J3 ) @ ( set_or66887138388493659n_real @ M @ N2 ) )
= ( ( ord_less_eq_real @ J3 @ I3 )
| ( ( ord_less_eq_real @ M @ I3 )
& ( ord_less_eq_real @ J3 @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_863_ivl__subset,axiom,
! [I3: nat,J3: nat,M: nat,N2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I3 @ J3 ) @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
= ( ( ord_less_eq_nat @ J3 @ I3 )
| ( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_eq_nat @ J3 @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_864_image__add__atLeastLessThan,axiom,
! [K: real,I3: real,J3: real] :
( ( image_real_real @ ( plus_plus_real @ K ) @ ( set_or66887138388493659n_real @ I3 @ J3 ) )
= ( set_or66887138388493659n_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ K ) ) ) ).
% image_add_atLeastLessThan
thf(fact_865_image__add__atLeastLessThan,axiom,
! [K: nat,I3: nat,J3: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ K ) @ ( set_or4665077453230672383an_nat @ I3 @ J3 ) )
= ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% image_add_atLeastLessThan
thf(fact_866_image__add__atLeastLessThan_H,axiom,
! [K: real,I3: real,J3: real] :
( ( image_real_real
@ ^ [N4: real] : ( plus_plus_real @ N4 @ K )
@ ( set_or66887138388493659n_real @ I3 @ J3 ) )
= ( set_or66887138388493659n_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ K ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_867_image__add__atLeastLessThan_H,axiom,
! [K: nat,I3: nat,J3: nat] :
( ( image_nat_nat
@ ^ [N4: nat] : ( plus_plus_nat @ N4 @ K )
@ ( set_or4665077453230672383an_nat @ I3 @ J3 ) )
= ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_868_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_869_Union__UNIV,axiom,
( ( comple3096694443085538997t_real @ top_top_set_set_real )
= top_top_set_real ) ).
% Union_UNIV
thf(fact_870_Union__UNIV,axiom,
( ( comple90263536869209701_set_o @ top_top_set_set_o )
= top_top_set_o ) ).
% Union_UNIV
thf(fact_871_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_872_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_873_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X2: nat] : $true ) ) ).
% UNIV_def
thf(fact_874_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_875_set__plus__elim,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A2 @ B2 ) )
=> ~ ! [A4: nat,B3: nat] :
( ( X
= ( plus_plus_nat @ A4 @ B3 ) )
=> ( ( member_nat @ A4 @ A2 )
=> ~ ( member_nat @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_876_set__plus__elim,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ( member_real @ X @ ( plus_plus_set_real @ A2 @ B2 ) )
=> ~ ! [A4: real,B3: real] :
( ( X
= ( plus_plus_real @ A4 @ B3 ) )
=> ( ( member_real @ A4 @ A2 )
=> ~ ( member_real @ B3 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_877_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_878_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_879_UNIV__witness,axiom,
? [X4: $o] : ( member_o @ X4 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_880_UNIV__witness,axiom,
? [X4: real] : ( member_real @ X4 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_881_UNIV__witness,axiom,
? [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_882_UNIV__eq__I,axiom,
! [A2: set_o] :
( ! [X4: $o] : ( member_o @ X4 @ A2 )
=> ( top_top_set_o = A2 ) ) ).
% UNIV_eq_I
thf(fact_883_UNIV__eq__I,axiom,
! [A2: set_real] :
( ! [X4: real] : ( member_real @ X4 @ A2 )
=> ( top_top_set_real = A2 ) ) ).
% UNIV_eq_I
thf(fact_884_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X4: nat] : ( member_nat @ X4 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_885_Bex__def,axiom,
( bex_o
= ( ^ [A3: set_o,P2: $o > $o] :
? [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_886_Bex__def,axiom,
( bex_real
= ( ^ [A3: set_real,P2: real > $o] :
? [X2: real] :
( ( member_real @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_887_Bex__def,axiom,
( bex_nat
= ( ^ [A3: set_nat,P2: nat > $o] :
? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_888_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_889_add__right__imp__eq,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_890_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_891_add__left__imp__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_892_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_893_add_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C2 ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_894_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_895_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A5: real,B4: real] : ( plus_plus_real @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_896_add_Oright__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_897_add_Oleft__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_898_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_899_add_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_900_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_901_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_902_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_903_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_904_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( I3 = J3 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_905_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( I3 = J3 )
& ( K = L ) )
=> ( ( plus_plus_real @ I3 @ K )
= ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_906_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_907_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_908_set__plus__def,axiom,
( plus_plus_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ^ [C4: nat] :
? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ B5 )
& ( C4
= ( plus_plus_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% set_plus_def
thf(fact_909_set__plus__def,axiom,
( plus_plus_set_real
= ( ^ [A3: set_real,B5: set_real] :
( collect_real
@ ^ [C4: real] :
? [X2: real] :
( ( member_real @ X2 @ A3 )
& ? [Y2: real] :
( ( member_real @ Y2 @ B5 )
& ( C4
= ( plus_plus_real @ X2 @ Y2 ) ) ) ) ) ) ) ).
% set_plus_def
thf(fact_910_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_911_top__greatest,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% top_greatest
thf(fact_912_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_913_top_Oextremum__unique,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A )
= ( A = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_914_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_915_top_Oextremum__uniqueI,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A )
=> ( A = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_916_UN__finite2__eq,axiom,
! [A2: nat > set_nat,B2: nat > set_nat,K: nat] :
( ! [N: nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N @ K ) ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ top_top_set_nat ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ top_top_set_nat ) ) ) ) ).
% UN_finite2_eq
thf(fact_917_UN__finite2__eq,axiom,
! [A2: nat > set_real,B2: nat > set_real,K: nat] :
( ! [N: nat] :
( ( comple3096694443085538997t_real @ ( image_nat_set_real @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
= ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N @ K ) ) ) ) )
=> ( ( comple3096694443085538997t_real @ ( image_nat_set_real @ A2 @ top_top_set_nat ) )
= ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ top_top_set_nat ) ) ) ) ).
% UN_finite2_eq
thf(fact_918_UN__finite2__eq,axiom,
! [A2: nat > set_o,B2: nat > set_o,K: nat] :
( ! [N: nat] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N @ K ) ) ) ) )
=> ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ top_top_set_nat ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ top_top_set_nat ) ) ) ) ).
% UN_finite2_eq
thf(fact_919_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_920_add__le__imp__le__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_921_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_922_add__le__imp__le__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_923_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C4: nat] :
( B4
= ( plus_plus_nat @ A5 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_924_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_925_add__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_926_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C5: nat] :
( B
!= ( plus_plus_nat @ A @ C5 ) ) ) ).
% less_eqE
thf(fact_927_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_928_add__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_929_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_930_add__mono,axiom,
! [A: real,B: real,C2: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_931_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J3 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_932_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J3 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_933_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( I3 = J3 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_934_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( I3 = J3 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_935_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J3 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_936_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: real,J3: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J3 )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_937_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_938_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_939_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_940_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_941_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_942_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( times_times_real @ A @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ A @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_943_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ B @ X ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_944_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_945_combine__common__factor,axiom,
! [A: real,E2: real,B: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_946_distrib__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_947_distrib__right,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_948_distrib__left,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_949_distrib__left,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_950_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_951_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_952_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_953_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_954_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_955_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= M )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_956_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_957_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_958_add__leE,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_959_le__add1,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% le_add1
thf(fact_960_le__add2,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% le_add2
thf(fact_961_add__leD1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% add_leD1
thf(fact_962_add__leD2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_963_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N: nat] :
( L
= ( plus_plus_nat @ K @ N ) ) ) ).
% le_Suc_ex
thf(fact_964_add__le__mono,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_le_mono
thf(fact_965_add__le__mono1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% add_le_mono1
thf(fact_966_trans__le__add1,axiom,
! [I3: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_le_add1
thf(fact_967_trans__le__add2,axiom,
! [I3: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_le_add2
thf(fact_968_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_969_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_970_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_971_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_972_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_973_add__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_974_add__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_975_left__add__mult__distrib,axiom,
! [I3: nat,U: nat,J3: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I3 @ J3 ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_976_range__eqI,axiom,
! [B: set_real,F: $o > set_real,X: $o] :
( ( B
= ( F @ X ) )
=> ( member_set_real @ B @ ( image_o_set_real @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_977_range__eqI,axiom,
! [B: set_o,F: $o > set_o,X: $o] :
( ( B
= ( F @ X ) )
=> ( member_set_o @ B @ ( image_o_set_o @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_978_range__eqI,axiom,
! [B: set_nat,F: $o > set_nat,X: $o] :
( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_o_set_nat @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_979_range__eqI,axiom,
! [B: set_nat,F: set_nat > set_nat,X: set_nat] :
( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat ) ) ) ).
% range_eqI
thf(fact_980_range__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_981_range__eqI,axiom,
! [B: $o,F: nat > $o,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_982_range__eqI,axiom,
! [B: real,F: nat > real,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_real @ B @ ( image_nat_real @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_983_range__eqI,axiom,
! [B: nat,F: nat > nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_984_rangeI,axiom,
! [F: $o > set_real,X: $o] : ( member_set_real @ ( F @ X ) @ ( image_o_set_real @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_985_rangeI,axiom,
! [F: $o > set_o,X: $o] : ( member_set_o @ ( F @ X ) @ ( image_o_set_o @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_986_rangeI,axiom,
! [F: $o > set_nat,X: $o] : ( member_set_nat @ ( F @ X ) @ ( image_o_set_nat @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_987_rangeI,axiom,
! [F: set_nat > set_nat,X: set_nat] : ( member_set_nat @ ( F @ X ) @ ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat ) ) ).
% rangeI
thf(fact_988_rangeI,axiom,
! [F: nat > set_nat,X: nat] : ( member_set_nat @ ( F @ X ) @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_989_rangeI,axiom,
! [F: nat > $o,X: nat] : ( member_o @ ( F @ X ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_990_rangeI,axiom,
! [F: nat > real,X: nat] : ( member_real @ ( F @ X ) @ ( image_nat_real @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_991_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_992_subset__UNIV,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_993_subset__UNIV,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ top_top_set_real ) ).
% subset_UNIV
thf(fact_994_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_995_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_996_image__def,axiom,
( image_nat_set_nat
= ( ^ [F3: nat > set_nat,A3: set_nat] :
( collect_set_nat
@ ^ [Y2: set_nat] :
? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( Y2
= ( F3 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_997_image__def,axiom,
( image_o_set_real
= ( ^ [F3: $o > set_real,A3: set_o] :
( collect_set_real
@ ^ [Y2: set_real] :
? [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( Y2
= ( F3 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_998_image__def,axiom,
( image_o_set_o
= ( ^ [F3: $o > set_o,A3: set_o] :
( collect_set_o
@ ^ [Y2: set_o] :
? [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( Y2
= ( F3 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_999_image__def,axiom,
( image_o_set_nat
= ( ^ [F3: $o > set_nat,A3: set_o] :
( collect_set_nat
@ ^ [Y2: set_nat] :
? [X2: $o] :
( ( member_o @ X2 @ A3 )
& ( Y2
= ( F3 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_1000_image__def,axiom,
( image_7916887816326733075et_nat
= ( ^ [F3: set_nat > set_nat,A3: set_set_nat] :
( collect_set_nat
@ ^ [Y2: set_nat] :
? [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
& ( Y2
= ( F3 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_1001_image__def,axiom,
( image_nat_nat
= ( ^ [F3: nat > nat,A3: set_nat] :
( collect_nat
@ ^ [Y2: nat] :
? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( Y2
= ( F3 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_1002_UN__finite2__subset,axiom,
! [A2: nat > set_nat,B2: nat > set_nat,K: nat] :
( ! [N: nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N @ K ) ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ top_top_set_nat ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ top_top_set_nat ) ) ) ) ).
% UN_finite2_subset
thf(fact_1003_UN__finite2__subset,axiom,
! [A2: nat > set_real,B2: nat > set_real,K: nat] :
( ! [N: nat] : ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N @ K ) ) ) ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ A2 @ top_top_set_nat ) ) @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ B2 @ top_top_set_nat ) ) ) ) ).
% UN_finite2_subset
thf(fact_1004_UN__finite2__subset,axiom,
! [A2: nat > set_o,B2: nat > set_o,K: nat] :
( ! [N: nat] : ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N @ K ) ) ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ top_top_set_nat ) ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ top_top_set_nat ) ) ) ) ).
% UN_finite2_subset
thf(fact_1005_Union__eq,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X2: nat] :
? [Y2: set_nat] :
( ( member_set_nat @ Y2 @ A3 )
& ( member_nat @ X2 @ Y2 ) ) ) ) ) ).
% Union_eq
thf(fact_1006_Union__eq,axiom,
( comple3096694443085538997t_real
= ( ^ [A3: set_set_real] :
( collect_real
@ ^ [X2: real] :
? [Y2: set_real] :
( ( member_set_real @ Y2 @ A3 )
& ( member_real @ X2 @ Y2 ) ) ) ) ) ).
% Union_eq
thf(fact_1007_Union__eq,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X2: $o] :
? [Y2: set_o] :
( ( member_set_o @ Y2 @ A3 )
& ( member_o @ X2 @ Y2 ) ) ) ) ) ).
% Union_eq
thf(fact_1008_convex__bound__le,axiom,
! [X: real,A: real,Y: real,U: real,V: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U @ V )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1009_lambda__one,axiom,
( ( ^ [X2: nat] : X2 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_1010_lambda__one,axiom,
( ( ^ [X2: real] : X2 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_1011_range__composition,axiom,
! [F: nat > nat,G: nat > nat] :
( ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_nat_nat @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_1012_range__composition,axiom,
! [F: $o > set_real,G: $o > $o] :
( ( image_o_set_real
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ top_top_set_o )
= ( image_o_set_real @ F @ ( image_o_o @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_1013_range__composition,axiom,
! [F: $o > set_o,G: $o > $o] :
( ( image_o_set_o
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ top_top_set_o )
= ( image_o_set_o @ F @ ( image_o_o @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_1014_range__composition,axiom,
! [F: nat > set_nat,G: $o > nat] :
( ( image_o_set_nat
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ top_top_set_o )
= ( image_nat_set_nat @ F @ ( image_o_nat @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_1015_range__composition,axiom,
! [F: $o > set_nat,G: $o > $o] :
( ( image_o_set_nat
@ ^ [X2: $o] : ( F @ ( G @ X2 ) )
@ top_top_set_o )
= ( image_o_set_nat @ F @ ( image_o_o @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_1016_range__composition,axiom,
! [F: $o > set_real,G: nat > $o] :
( ( image_nat_set_real
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_o_set_real @ F @ ( image_nat_o @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_1017_range__composition,axiom,
! [F: $o > set_o,G: nat > $o] :
( ( image_nat_set_o
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_o_set_o @ F @ ( image_nat_o @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_1018_range__composition,axiom,
! [F: nat > set_nat,G: nat > nat] :
( ( image_nat_set_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_nat_set_nat @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_1019_range__composition,axiom,
! [F: $o > set_nat,G: nat > $o] :
( ( image_nat_set_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_o_set_nat @ F @ ( image_nat_o @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_1020_range__composition,axiom,
! [F: set_nat > nat,G: nat > set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_set_nat_nat @ F @ ( image_nat_set_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_1021_rangeE,axiom,
! [B: set_real,F: $o > set_real] :
( ( member_set_real @ B @ ( image_o_set_real @ F @ top_top_set_o ) )
=> ~ ! [X4: $o] :
( B
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_1022_rangeE,axiom,
! [B: set_o,F: $o > set_o] :
( ( member_set_o @ B @ ( image_o_set_o @ F @ top_top_set_o ) )
=> ~ ! [X4: $o] :
( B
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_1023_rangeE,axiom,
! [B: set_nat,F: $o > set_nat] :
( ( member_set_nat @ B @ ( image_o_set_nat @ F @ top_top_set_o ) )
=> ~ ! [X4: $o] :
( B
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_1024_rangeE,axiom,
! [B: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat ) )
=> ~ ! [X4: set_nat] :
( B
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_1025_rangeE,axiom,
! [B: set_nat,F: nat > set_nat] :
( ( member_set_nat @ B @ ( image_nat_set_nat @ F @ top_top_set_nat ) )
=> ~ ! [X4: nat] :
( B
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_1026_rangeE,axiom,
! [B: $o,F: nat > $o] :
( ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) )
=> ~ ! [X4: nat] :
( B
= ( ~ ( F @ X4 ) ) ) ) ).
% rangeE
thf(fact_1027_rangeE,axiom,
! [B: real,F: nat > real] :
( ( member_real @ B @ ( image_nat_real @ F @ top_top_set_nat ) )
=> ~ ! [X4: nat] :
( B
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_1028_rangeE,axiom,
! [B: nat,F: nat > nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ~ ! [X4: nat] :
( B
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_1029_UN__le__add__shift__strict,axiom,
! [M2: nat > set_nat,K: nat,N2: nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( M2 @ ( plus_plus_nat @ I @ K ) )
@ ( set_ord_lessThan_nat @ N2 ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ N2 @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_1030_UN__le__add__shift__strict,axiom,
! [M2: nat > set_real,K: nat,N2: nat] :
( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [I: nat] : ( M2 @ ( plus_plus_nat @ I @ K ) )
@ ( set_ord_lessThan_nat @ N2 ) ) )
= ( comple3096694443085538997t_real @ ( image_nat_set_real @ M2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ N2 @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_1031_UN__le__add__shift__strict,axiom,
! [M2: nat > set_o,K: nat,N2: nat] :
( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [I: nat] : ( M2 @ ( plus_plus_nat @ I @ K ) )
@ ( set_ord_lessThan_nat @ N2 ) ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ M2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ N2 @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_1032_UN__UN__finite__eq,axiom,
! [A2: nat > set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [N4: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
@ top_top_set_nat ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ top_top_set_nat ) ) ) ).
% UN_UN_finite_eq
thf(fact_1033_UN__UN__finite__eq,axiom,
! [A2: nat > set_real] :
( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [N4: nat] : ( comple3096694443085538997t_real @ ( image_nat_set_real @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
@ top_top_set_nat ) )
= ( comple3096694443085538997t_real @ ( image_nat_set_real @ A2 @ top_top_set_nat ) ) ) ).
% UN_UN_finite_eq
thf(fact_1034_UN__UN__finite__eq,axiom,
! [A2: nat > set_o] :
( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [N4: nat] : ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
@ top_top_set_nat ) )
= ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ top_top_set_nat ) ) ) ).
% UN_UN_finite_eq
thf(fact_1035_atLeastLessThan__subset__iff,axiom,
! [A: real,B: real,C2: real,D2: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or66887138388493659n_real @ C2 @ D2 ) )
=> ( ( ord_less_eq_real @ B @ A )
| ( ( ord_less_eq_real @ C2 @ A )
& ( ord_less_eq_real @ B @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1036_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C2 @ A )
& ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1037_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_1038_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1039_add__decreasing,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1040_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1041_add__increasing,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1042_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1043_add__decreasing2,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1044_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1045_add__increasing2,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1046_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1047_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1048_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1049_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_1050_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1051_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1052_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1053_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1054_UN__finite__subset,axiom,
! [A2: nat > set_nat,C: set_nat] :
( ! [N: nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) @ C )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ top_top_set_nat ) ) @ C ) ) ).
% UN_finite_subset
thf(fact_1055_UN__finite__subset,axiom,
! [A2: nat > set_real,C: set_real] :
( ! [N: nat] : ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) @ C )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ ( image_nat_set_real @ A2 @ top_top_set_nat ) ) @ C ) ) ).
% UN_finite_subset
thf(fact_1056_UN__finite__subset,axiom,
! [A2: nat > set_o,C: set_o] :
( ! [N: nat] : ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) @ C )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ A2 @ top_top_set_nat ) ) @ C ) ) ).
% UN_finite_subset
thf(fact_1057_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1058_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_1059_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1060_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1061_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1062_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_1063_range__subsetD,axiom,
! [F: $o > set_real,B2: set_set_real,I3: $o] :
( ( ord_le3558479182127378552t_real @ ( image_o_set_real @ F @ top_top_set_o ) @ B2 )
=> ( member_set_real @ ( F @ I3 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1064_range__subsetD,axiom,
! [F: $o > set_o,B2: set_set_o,I3: $o] :
( ( ord_le4374716579403074808_set_o @ ( image_o_set_o @ F @ top_top_set_o ) @ B2 )
=> ( member_set_o @ ( F @ I3 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1065_range__subsetD,axiom,
! [F: $o > set_nat,B2: set_set_nat,I3: $o] :
( ( ord_le6893508408891458716et_nat @ ( image_o_set_nat @ F @ top_top_set_o ) @ B2 )
=> ( member_set_nat @ ( F @ I3 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1066_range__subsetD,axiom,
! [F: set_nat > set_nat,B2: set_set_nat,I3: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat ) @ B2 )
=> ( member_set_nat @ ( F @ I3 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1067_range__subsetD,axiom,
! [F: nat > set_nat,B2: set_set_nat,I3: nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ top_top_set_nat ) @ B2 )
=> ( member_set_nat @ ( F @ I3 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1068_range__subsetD,axiom,
! [F: nat > $o,B2: set_o,I3: nat] :
( ( ord_less_eq_set_o @ ( image_nat_o @ F @ top_top_set_nat ) @ B2 )
=> ( member_o @ ( F @ I3 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1069_range__subsetD,axiom,
! [F: nat > nat,B2: set_nat,I3: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B2 )
=> ( member_nat @ ( F @ I3 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1070_range__subsetD,axiom,
! [F: nat > real,B2: set_real,I3: nat] :
( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ B2 )
=> ( member_real @ ( F @ I3 ) @ B2 ) ) ).
% range_subsetD
thf(fact_1071_Collect__bex__eq,axiom,
! [A2: set_nat,P: nat > nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
? [Y2: nat] :
( ( member_nat @ Y2 @ A2 )
& ( P @ X2 @ Y2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] :
( collect_nat
@ ^ [X2: nat] : ( P @ X2 @ Y2 ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_1072_Collect__bex__eq,axiom,
! [A2: set_o,P: nat > $o > $o] :
( ( collect_nat
@ ^ [X2: nat] :
? [Y2: $o] :
( ( member_o @ Y2 @ A2 )
& ( P @ X2 @ Y2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y2: $o] :
( collect_nat
@ ^ [X2: nat] : ( P @ X2 @ Y2 ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_1073_Collect__bex__eq,axiom,
! [A2: set_set_nat,P: nat > set_nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
? [Y2: set_nat] :
( ( member_set_nat @ Y2 @ A2 )
& ( P @ X2 @ Y2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y2: set_nat] :
( collect_nat
@ ^ [X2: nat] : ( P @ X2 @ Y2 ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_1074_Collect__bex__eq,axiom,
! [A2: set_o,P: real > $o > $o] :
( ( collect_real
@ ^ [X2: real] :
? [Y2: $o] :
( ( member_o @ Y2 @ A2 )
& ( P @ X2 @ Y2 ) ) )
= ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [Y2: $o] :
( collect_real
@ ^ [X2: real] : ( P @ X2 @ Y2 ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_1075_Collect__bex__eq,axiom,
! [A2: set_o,P: $o > $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
? [Y2: $o] :
( ( member_o @ Y2 @ A2 )
& ( P @ X2 @ Y2 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] :
( collect_o
@ ^ [X2: $o] : ( P @ X2 @ Y2 ) )
@ A2 ) ) ) ).
% Collect_bex_eq
thf(fact_1076_UNION__eq,axiom,
! [B2: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) )
= ( collect_nat
@ ^ [Y2: nat] :
? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( member_nat @ Y2 @ ( B2 @ X2 ) ) ) ) ) ).
% UNION_eq
thf(fact_1077_UNION__eq,axiom,
! [B2: $o > set_nat,A2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) )
= ( collect_nat
@ ^ [Y2: nat] :
? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( member_nat @ Y2 @ ( B2 @ X2 ) ) ) ) ) ).
% UNION_eq
thf(fact_1078_UNION__eq,axiom,
! [B2: set_nat > set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A2 ) )
= ( collect_nat
@ ^ [Y2: nat] :
? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( member_nat @ Y2 @ ( B2 @ X2 ) ) ) ) ) ).
% UNION_eq
thf(fact_1079_UNION__eq,axiom,
! [B2: $o > set_real,A2: set_o] :
( ( comple3096694443085538997t_real @ ( image_o_set_real @ B2 @ A2 ) )
= ( collect_real
@ ^ [Y2: real] :
? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( member_real @ Y2 @ ( B2 @ X2 ) ) ) ) ) ).
% UNION_eq
thf(fact_1080_UNION__eq,axiom,
! [B2: $o > set_o,A2: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) )
= ( collect_o
@ ^ [Y2: $o] :
? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ( member_o @ Y2 @ ( B2 @ X2 ) ) ) ) ) ).
% UNION_eq
thf(fact_1081_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times_nat @ M @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1082_set__times__def,axiom,
( times_times_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ^ [C4: nat] :
? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ B5 )
& ( C4
= ( times_times_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% set_times_def
thf(fact_1083_set__times__def,axiom,
( times_times_set_real
= ( ^ [A3: set_real,B5: set_real] :
( collect_real
@ ^ [C4: real] :
? [X2: real] :
( ( member_real @ X2 @ A3 )
& ? [Y2: real] :
( ( member_real @ Y2 @ B5 )
& ( C4
= ( times_times_real @ X2 @ Y2 ) ) ) ) ) ) ) ).
% set_times_def
thf(fact_1084_sum__squares__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_1085_mult__left__le,axiom,
! [C2: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_1086_mult__left__le,axiom,
! [C2: real,A: real] :
( ( ord_less_eq_real @ C2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_1087_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1088_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_1089_mult__right__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1090_mult__left__le__one__le,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1091_Fpow__mono,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A2 ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% Fpow_mono
thf(fact_1092_Fpow__mono,axiom,
! [A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ord_le3558479182127378552t_real @ ( finite_Fpow_real @ A2 ) @ ( finite_Fpow_real @ B2 ) ) ) ).
% Fpow_mono
thf(fact_1093_surj__plus__right,axiom,
! [A: real] :
( ( image_real_real
@ ^ [B4: real] : ( plus_plus_real @ B4 @ A )
@ top_top_set_real )
= top_top_set_real ) ).
% surj_plus_right
thf(fact_1094_subset__translation__eq,axiom,
! [A: real,S3: set_real,T2: set_real] :
( ( ord_less_eq_set_real @ ( image_real_real @ ( plus_plus_real @ A ) @ S3 ) @ ( image_real_real @ ( plus_plus_real @ A ) @ T2 ) )
= ( ord_less_eq_set_real @ S3 @ T2 ) ) ).
% subset_translation_eq
thf(fact_1095_range__add,axiom,
! [A: real] :
( ( image_real_real @ ( plus_plus_real @ A ) @ top_top_set_real )
= top_top_set_real ) ).
% range_add
thf(fact_1096_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1097_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_1098_SUP__combine,axiom,
! [F: nat > nat > $o] :
( ! [A4: nat,B3: nat,C5: nat,D3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ C5 @ D3 )
=> ( ord_less_eq_o @ ( F @ A4 @ C5 ) @ ( F @ B3 @ D3 ) ) ) )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : ( complete_Sup_Sup_o @ ( image_nat_o @ ( F @ I ) @ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : ( F @ I @ I )
@ top_top_set_nat ) ) ) ) ).
% SUP_combine
thf(fact_1099_SUP__combine,axiom,
! [F: real > real > $o] :
( ! [A4: real,B3: real,C5: real,D3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ C5 @ D3 )
=> ( ord_less_eq_o @ ( F @ A4 @ C5 ) @ ( F @ B3 @ D3 ) ) ) )
=> ( ( complete_Sup_Sup_o
@ ( image_real_o
@ ^ [I: real] : ( complete_Sup_Sup_o @ ( image_real_o @ ( F @ I ) @ top_top_set_real ) )
@ top_top_set_real ) )
= ( complete_Sup_Sup_o
@ ( image_real_o
@ ^ [I: real] : ( F @ I @ I )
@ top_top_set_real ) ) ) ) ).
% SUP_combine
thf(fact_1100_SUP__combine,axiom,
! [F: $o > $o > set_nat] :
( ! [A4: $o,B3: $o,C5: $o,D3: $o] :
( ( ord_less_eq_o @ A4 @ B3 )
=> ( ( ord_less_eq_o @ C5 @ D3 )
=> ( ord_less_eq_set_nat @ ( F @ A4 @ C5 ) @ ( F @ B3 @ D3 ) ) ) )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( F @ I ) @ top_top_set_o ) )
@ top_top_set_o ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I: $o] : ( F @ I @ I )
@ top_top_set_o ) ) ) ) ).
% SUP_combine
thf(fact_1101_SUP__combine,axiom,
! [F: nat > nat > set_nat] :
( ! [A4: nat,B3: nat,C5: nat,D3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ C5 @ D3 )
=> ( ord_less_eq_set_nat @ ( F @ A4 @ C5 ) @ ( F @ B3 @ D3 ) ) ) )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I ) @ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( F @ I @ I )
@ top_top_set_nat ) ) ) ) ).
% SUP_combine
thf(fact_1102_SUP__combine,axiom,
! [F: real > real > set_nat] :
( ! [A4: real,B3: real,C5: real,D3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ C5 @ D3 )
=> ( ord_less_eq_set_nat @ ( F @ A4 @ C5 ) @ ( F @ B3 @ D3 ) ) ) )
=> ( ( comple7399068483239264473et_nat
@ ( image_real_set_nat
@ ^ [I: real] : ( comple7399068483239264473et_nat @ ( image_real_set_nat @ ( F @ I ) @ top_top_set_real ) )
@ top_top_set_real ) )
= ( comple7399068483239264473et_nat
@ ( image_real_set_nat
@ ^ [I: real] : ( F @ I @ I )
@ top_top_set_real ) ) ) ) ).
% SUP_combine
thf(fact_1103_SUP__combine,axiom,
! [F: set_nat > set_nat > $o] :
( ! [A4: set_nat,B3: set_nat,C5: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
=> ( ( ord_less_eq_set_nat @ C5 @ D3 )
=> ( ord_less_eq_o @ ( F @ A4 @ C5 ) @ ( F @ B3 @ D3 ) ) ) )
=> ( ( complete_Sup_Sup_o
@ ( image_set_nat_o
@ ^ [I: set_nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( F @ I ) @ top_top_set_set_nat ) )
@ top_top_set_set_nat ) )
= ( complete_Sup_Sup_o
@ ( image_set_nat_o
@ ^ [I: set_nat] : ( F @ I @ I )
@ top_top_set_set_nat ) ) ) ) ).
% SUP_combine
thf(fact_1104_SUP__combine,axiom,
! [F: set_real > set_real > $o] :
( ! [A4: set_real,B3: set_real,C5: set_real,D3: set_real] :
( ( ord_less_eq_set_real @ A4 @ B3 )
=> ( ( ord_less_eq_set_real @ C5 @ D3 )
=> ( ord_less_eq_o @ ( F @ A4 @ C5 ) @ ( F @ B3 @ D3 ) ) ) )
=> ( ( complete_Sup_Sup_o
@ ( image_set_real_o
@ ^ [I: set_real] : ( complete_Sup_Sup_o @ ( image_set_real_o @ ( F @ I ) @ top_top_set_set_real ) )
@ top_top_set_set_real ) )
= ( complete_Sup_Sup_o
@ ( image_set_real_o
@ ^ [I: set_real] : ( F @ I @ I )
@ top_top_set_set_real ) ) ) ) ).
% SUP_combine
thf(fact_1105_SUP__combine,axiom,
! [F: $o > $o > set_real] :
( ! [A4: $o,B3: $o,C5: $o,D3: $o] :
( ( ord_less_eq_o @ A4 @ B3 )
=> ( ( ord_less_eq_o @ C5 @ D3 )
=> ( ord_less_eq_set_real @ ( F @ A4 @ C5 ) @ ( F @ B3 @ D3 ) ) ) )
=> ( ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [I: $o] : ( comple3096694443085538997t_real @ ( image_o_set_real @ ( F @ I ) @ top_top_set_o ) )
@ top_top_set_o ) )
= ( comple3096694443085538997t_real
@ ( image_o_set_real
@ ^ [I: $o] : ( F @ I @ I )
@ top_top_set_o ) ) ) ) ).
% SUP_combine
thf(fact_1106_SUP__combine,axiom,
! [F: nat > nat > set_real] :
( ! [A4: nat,B3: nat,C5: nat,D3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( ( ord_less_eq_nat @ C5 @ D3 )
=> ( ord_less_eq_set_real @ ( F @ A4 @ C5 ) @ ( F @ B3 @ D3 ) ) ) )
=> ( ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [I: nat] : ( comple3096694443085538997t_real @ ( image_nat_set_real @ ( F @ I ) @ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( comple3096694443085538997t_real
@ ( image_nat_set_real
@ ^ [I: nat] : ( F @ I @ I )
@ top_top_set_nat ) ) ) ) ).
% SUP_combine
thf(fact_1107_SUP__combine,axiom,
! [F: real > real > set_real] :
( ! [A4: real,B3: real,C5: real,D3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ C5 @ D3 )
=> ( ord_less_eq_set_real @ ( F @ A4 @ C5 ) @ ( F @ B3 @ D3 ) ) ) )
=> ( ( comple3096694443085538997t_real
@ ( image_real_set_real
@ ^ [I: real] : ( comple3096694443085538997t_real @ ( image_real_set_real @ ( F @ I ) @ top_top_set_real ) )
@ top_top_set_real ) )
= ( comple3096694443085538997t_real
@ ( image_real_set_real
@ ^ [I: real] : ( F @ I @ I )
@ top_top_set_real ) ) ) ) ).
% SUP_combine
thf(fact_1108_affine__ineq,axiom,
! [X: real,V: real,U: real] :
( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( ord_less_eq_real @ V @ U )
=> ( ord_less_eq_real @ ( plus_plus_real @ V @ ( times_times_real @ X @ U ) ) @ ( plus_plus_real @ U @ ( times_times_real @ X @ V ) ) ) ) ) ).
% affine_ineq
thf(fact_1109_set__plus__mono2,axiom,
! [C: set_nat,D: set_nat,E: set_nat,F2: set_nat] :
( ( ord_less_eq_set_nat @ C @ D )
=> ( ( ord_less_eq_set_nat @ E @ F2 )
=> ( ord_less_eq_set_nat @ ( plus_plus_set_nat @ C @ E ) @ ( plus_plus_set_nat @ D @ F2 ) ) ) ) ).
% set_plus_mono2
thf(fact_1110_set__plus__mono2,axiom,
! [C: set_real,D: set_real,E: set_real,F2: set_real] :
( ( ord_less_eq_set_real @ C @ D )
=> ( ( ord_less_eq_set_real @ E @ F2 )
=> ( ord_less_eq_set_real @ ( plus_plus_set_real @ C @ E ) @ ( plus_plus_set_real @ D @ F2 ) ) ) ) ).
% set_plus_mono2
thf(fact_1111_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_1112_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_1113_top__empty__eq,axiom,
( top_top_real_o
= ( ^ [X2: real] : ( member_real @ X2 @ top_top_set_real ) ) ) ).
% top_empty_eq
thf(fact_1114_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_1115_set__zero__plus2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( member_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ ( plus_plus_set_nat @ A2 @ B2 ) ) ) ).
% set_zero_plus2
thf(fact_1116_set__zero__plus2,axiom,
! [A2: set_real,B2: set_real] :
( ( member_real @ zero_zero_real @ A2 )
=> ( ord_less_eq_set_real @ B2 @ ( plus_plus_set_real @ A2 @ B2 ) ) ) ).
% set_zero_plus2
thf(fact_1117_eq__add__iff,axiom,
! [X: real,Y: real] :
( ( X
= ( plus_plus_real @ X @ Y ) )
= ( Y = zero_zero_real ) ) ).
% eq_add_iff
thf(fact_1118_surj__def,axiom,
! [F: $o > set_real] :
( ( ( image_o_set_real @ F @ top_top_set_o )
= top_top_set_set_real )
= ( ! [Y2: set_real] :
? [X2: $o] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1119_surj__def,axiom,
! [F: $o > set_o] :
( ( ( image_o_set_o @ F @ top_top_set_o )
= top_top_set_set_o )
= ( ! [Y2: set_o] :
? [X2: $o] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1120_surj__def,axiom,
! [F: $o > set_nat] :
( ( ( image_o_set_nat @ F @ top_top_set_o )
= top_top_set_set_nat )
= ( ! [Y2: set_nat] :
? [X2: $o] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1121_surj__def,axiom,
! [F: set_nat > set_nat] :
( ( ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat )
= top_top_set_set_nat )
= ( ! [Y2: set_nat] :
? [X2: set_nat] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1122_surj__def,axiom,
! [F: nat > set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
= ( ! [Y2: set_nat] :
? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1123_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X2: nat] :
( Y2
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1124_surjI,axiom,
! [G: $o > set_real,F: set_real > $o] :
( ! [X4: set_real] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_o_set_real @ G @ top_top_set_o )
= top_top_set_set_real ) ) ).
% surjI
thf(fact_1125_surjI,axiom,
! [G: $o > set_o,F: set_o > $o] :
( ! [X4: set_o] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_o_set_o @ G @ top_top_set_o )
= top_top_set_set_o ) ) ).
% surjI
thf(fact_1126_surjI,axiom,
! [G: $o > set_nat,F: set_nat > $o] :
( ! [X4: set_nat] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_o_set_nat @ G @ top_top_set_o )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_1127_surjI,axiom,
! [G: set_nat > set_nat,F: set_nat > set_nat] :
( ! [X4: set_nat] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_7916887816326733075et_nat @ G @ top_top_set_set_nat )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_1128_surjI,axiom,
! [G: nat > set_nat,F: set_nat > nat] :
( ! [X4: set_nat] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_nat_set_nat @ G @ top_top_set_nat )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_1129_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X4: nat] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1130_surjE,axiom,
! [F: $o > set_real,Y: set_real] :
( ( ( image_o_set_real @ F @ top_top_set_o )
= top_top_set_set_real )
=> ~ ! [X4: $o] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_1131_surjE,axiom,
! [F: $o > set_o,Y: set_o] :
( ( ( image_o_set_o @ F @ top_top_set_o )
= top_top_set_set_o )
=> ~ ! [X4: $o] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_1132_surjE,axiom,
! [F: $o > set_nat,Y: set_nat] :
( ( ( image_o_set_nat @ F @ top_top_set_o )
= top_top_set_set_nat )
=> ~ ! [X4: $o] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_1133_surjE,axiom,
! [F: set_nat > set_nat,Y: set_nat] :
( ( ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat )
= top_top_set_set_nat )
=> ~ ! [X4: set_nat] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_1134_surjE,axiom,
! [F: nat > set_nat,Y: set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
=> ~ ! [X4: nat] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_1135_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X4: nat] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_1136_surjD,axiom,
! [F: $o > set_real,Y: set_real] :
( ( ( image_o_set_real @ F @ top_top_set_o )
= top_top_set_set_real )
=> ? [X4: $o] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_1137_surjD,axiom,
! [F: $o > set_o,Y: set_o] :
( ( ( image_o_set_o @ F @ top_top_set_o )
= top_top_set_set_o )
=> ? [X4: $o] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_1138_surjD,axiom,
! [F: $o > set_nat,Y: set_nat] :
( ( ( image_o_set_nat @ F @ top_top_set_o )
= top_top_set_set_nat )
=> ? [X4: $o] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_1139_surjD,axiom,
! [F: set_nat > set_nat,Y: set_nat] :
( ( ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat )
= top_top_set_set_nat )
=> ? [X4: set_nat] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_1140_surjD,axiom,
! [F: nat > set_nat,Y: set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
=> ? [X4: nat] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_1141_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X4: nat] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_1142_translation__assoc,axiom,
! [B: real,A: real,S: set_real] :
( ( image_real_real @ ( plus_plus_real @ B ) @ ( image_real_real @ ( plus_plus_real @ A ) @ S ) )
= ( image_real_real @ ( plus_plus_real @ ( plus_plus_real @ A @ B ) ) @ S ) ) ).
% translation_assoc
thf(fact_1143_translation__invert,axiom,
! [A: real,A2: set_real,B2: set_real] :
( ( ( image_real_real @ ( plus_plus_real @ A ) @ A2 )
= ( image_real_real @ ( plus_plus_real @ A ) @ B2 ) )
=> ( A2 = B2 ) ) ).
% translation_invert
thf(fact_1144_sum__squares__le__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1145_mult__eq__1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= one_one_nat )
= ( ( A = one_one_nat )
& ( B = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_1146_mult__eq__1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ( ( times_times_real @ A @ B )
= one_one_real )
= ( ( A = one_one_real )
& ( B = one_one_real ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_1147_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I2: nat] :
( ( Q @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I2 ) )
& ( ord_less_eq_real @ ( X4 @ I2 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X6: nat > real,I4: nat] : ( ord_less_eq_nat @ ( L2 @ X6 @ I4 ) @ one_one_nat )
& ! [X6: nat > real,I4: nat] :
( ( ( P @ X6 )
& ( Q @ I4 )
& ( ( X6 @ I4 )
= zero_zero_real ) )
=> ( ( L2 @ X6 @ I4 )
= zero_zero_nat ) )
& ! [X6: nat > real,I4: nat] :
( ( ( P @ X6 )
& ( Q @ I4 )
& ( ( X6 @ I4 )
= one_one_real ) )
=> ( ( L2 @ X6 @ I4 )
= one_one_nat ) )
& ! [X6: nat > real,I4: nat] :
( ( ( P @ X6 )
& ( Q @ I4 )
& ( ( L2 @ X6 @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X6 @ I4 ) @ ( F @ X6 @ I4 ) ) )
& ! [X6: nat > real,I4: nat] :
( ( ( P @ X6 )
& ( Q @ I4 )
& ( ( L2 @ X6 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X6 @ I4 ) @ ( X6 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1148_sum__le__prod1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% sum_le_prod1
thf(fact_1149_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C2: nat,D2: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C2 != D2 ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C2 ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1150_add__scale__eq__noteq,axiom,
! [R: real,A: real,B: real,C2: real,D2: real] :
( ( R != zero_zero_real )
=> ( ( ( A = B )
& ( C2 != D2 ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R @ C2 ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1151_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
= ( P @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1152_segment__bound__lemma,axiom,
! [B2: real,X: real,Y: real,U: real] :
( ( ord_less_eq_real @ B2 @ X )
=> ( ( ord_less_eq_real @ B2 @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ U @ one_one_real )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U ) @ X ) @ ( times_times_real @ U @ Y ) ) ) ) ) ) ) ).
% segment_bound_lemma
thf(fact_1153_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1154_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1155_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1156_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1157_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1158_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1159_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1160_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1161_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1162_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1163_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1164_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1165_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1166_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1167_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1168_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1169_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1170_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( P @ N )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1171_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A4: real,B3: real,C5: real] :
( ( P @ A4 @ B3 )
=> ( ( P @ B3 @ C5 )
=> ( ( ord_less_eq_real @ A4 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C5 )
=> ( P @ A4 @ C5 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [A4: real,B3: real] :
( ( ( ord_less_eq_real @ A4 @ X4 )
& ( ord_less_eq_real @ X4 @ B3 )
& ( ord_less_real @ ( minus_minus_real @ B3 @ A4 ) @ D4 ) )
=> ( P @ A4 @ B3 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1172_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J3: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1173_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1174_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1175_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1176_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1177_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1178_add__lessD1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ K )
=> ( ord_less_nat @ I3 @ K ) ) ).
% add_lessD1
thf(fact_1179_add__less__mono,axiom,
! [I3: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1180_not__add__less1,axiom,
! [I3: nat,J3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ I3 ) ).
% not_add_less1
thf(fact_1181_not__add__less2,axiom,
! [J3: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J3 @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_1182_add__less__mono1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% add_less_mono1
thf(fact_1183_trans__less__add1,axiom,
! [I3: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_less_add1
thf(fact_1184_trans__less__add2,axiom,
! [I3: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_less_add2
thf(fact_1185_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1186_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1187_less__imp__add__positive,axiom,
! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I3 @ K3 )
= J3 ) ) ) ).
% less_imp_add_positive
thf(fact_1188_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1189_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1190_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1191_mult__less__mono2,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J3 ) ) ) ) ).
% mult_less_mono2
thf(fact_1192_mult__less__mono1,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J3 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1193_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1194_kuhn__lemma,axiom,
! [P3: nat,N2: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P3 )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P3 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( ( Label @ X4 @ I2 )
= zero_zero_nat )
| ( ( Label @ X4 @ I2 )
= one_one_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P3 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( ( X4 @ I2 )
= zero_zero_nat )
=> ( ( Label @ X4 @ I2 )
= zero_zero_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I4 ) @ P3 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ( ( X4 @ I2 )
= P3 )
=> ( ( Label @ X4 @ I2 )
= one_one_nat ) ) ) )
=> ~ ! [Q2: nat > nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( ord_less_nat @ ( Q2 @ I4 ) @ P3 ) )
=> ~ ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ? [R3: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q2 @ J4 ) @ ( R3 @ J4 ) )
& ( ord_less_eq_nat @ ( R3 @ J4 ) @ ( plus_plus_nat @ ( Q2 @ J4 ) @ one_one_nat ) ) ) )
& ? [S4: nat > nat] :
( ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q2 @ J4 ) @ ( S4 @ J4 ) )
& ( ord_less_eq_nat @ ( S4 @ J4 ) @ ( plus_plus_nat @ ( Q2 @ J4 ) @ one_one_nat ) ) ) )
& ( ( Label @ R3 @ I4 )
!= ( Label @ S4 @ I4 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1195_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1196_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1197_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1198_diff__diff__cancel,axiom,
! [I3: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1199_diff__diff__left,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% diff_diff_left
thf(fact_1200_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1201_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1202_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_1203_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J3 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1204_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1205_Nat_Odiff__diff__right,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J3 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1206_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1207_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M )
= zero_zero_nat )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1208_eq__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1209_le__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1210_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1211_diff__le__mono,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1212_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_1213_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1214_diff__le__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1215_diff__less,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).
% diff_less
thf(fact_1216_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1217_less__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1218_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_1219_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1220_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_1221_less__not__refl3,axiom,
! [S3: nat,T2: nat] :
( ( ord_less_nat @ S3 @ T2 )
=> ( S3 != T2 ) ) ).
% less_not_refl3
thf(fact_1222_diff__less__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1223_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1224_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( P @ M5 ) )
=> ( P @ N ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_1225_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N: nat] :
( ~ ( P @ N )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ~ ( P @ M5 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_1226_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1227_less__imp__diff__less,axiom,
! [J3: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J3 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1228_add__diff__inverse__nat,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less_nat @ M @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1229_less__diff__conv,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J3 ) ) ).
% less_diff_conv
thf(fact_1230_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1231_diff__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% diff_cancel2
thf(fact_1232_diff__add__inverse,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
= M ) ).
% diff_add_inverse
thf(fact_1233_diff__add__inverse2,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
= M ) ).
% diff_add_inverse2
thf(fact_1234_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1235_diff__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1236_diff__add__0,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1237_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1238_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1239_le__diff__conv,axiom,
! [J3: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J3 @ K ) @ I3 )
= ( ord_less_eq_nat @ J3 @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_1240_Nat_Ole__diff__conv2,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J3 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1241_Nat_Odiff__add__assoc,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J3 ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J3 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1242_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1243_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ( minus_minus_nat @ J3 @ I3 )
= K )
= ( J3
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1244_less__diff__conv2,axiom,
! [K: nat,J3: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J3 @ K ) @ I3 )
= ( ord_less_nat @ J3 @ ( plus_plus_nat @ I3 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1245_nat__eq__add__iff1,axiom,
! [J3: nat,I3: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J3 @ I3 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J3 ) @ U ) @ M )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1246_nat__eq__add__iff2,axiom,
! [I3: nat,J3: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N2 ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J3 @ I3 ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1247_nat__le__add__iff1,axiom,
! [J3: nat,I3: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J3 @ I3 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J3 ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1248_nat__le__add__iff2,axiom,
! [I3: nat,J3: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J3 @ I3 ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1249_nat__diff__add__eq1,axiom,
! [J3: nat,I3: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J3 @ I3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J3 ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1250_nat__diff__add__eq2,axiom,
! [I3: nat,J3: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N2 ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J3 @ I3 ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1251_nat__less__add__iff1,axiom,
! [J3: nat,I3: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J3 @ I3 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J3 ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1252_nat__less__add__iff2,axiom,
! [I3: nat,J3: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N2 ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J3 @ I3 ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1253_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1254_square__bound__lemma,axiom,
! [X: real] : ( ord_less_real @ X @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X ) @ ( plus_plus_real @ one_one_real @ X ) ) ) ).
% square_bound_lemma
thf(fact_1255_ex__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ( P @ M4 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1256_diff__commute,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J3 ) ) ).
% diff_commute
thf(fact_1257_all__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1258_seq__mono__lemma,axiom,
! [M: nat,D2: nat > real,E2: nat > real] :
( ! [N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_real @ ( D2 @ N ) @ ( E2 @ N ) ) )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_real @ ( E2 @ N ) @ ( E2 @ M ) ) )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M @ N5 )
=> ( ord_less_real @ ( D2 @ N5 ) @ ( E2 @ M ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_1259_complete__real,axiom,
! [S: set_real] :
( ? [X6: real] : ( member_real @ X6 @ S )
=> ( ? [Z2: real] :
! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z2 ) )
=> ? [Y3: real] :
( ! [X6: real] :
( ( member_real @ X6 @ S )
=> ( ord_less_eq_real @ X6 @ Y3 ) )
& ! [Z2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z2 ) )
=> ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ) ) ).
% complete_real
thf(fact_1260_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_1261_eq__diff__eq_H,axiom,
! [X: real,Y: real,Z: real] :
( ( X
= ( minus_minus_real @ Y @ Z ) )
= ( Y
= ( plus_plus_real @ X @ Z ) ) ) ).
% eq_diff_eq'
thf(fact_1262_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N2 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1263_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1264_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ ( times_times_nat @ n @ m ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ a @ ( set_ord_lessThan_nat @ n ) ) ) ).
%------------------------------------------------------------------------------