TPTP Problem File: SLH0720^1.p
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%------------------------------------------------------------------------------
% 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 : Hales_Jewett/0002_Hales_Jewett/prob_00967_040069__5755656_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1523 ( 507 unt; 248 typ; 0 def)
% Number of atoms : 3957 (1318 equ; 0 cnn)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 12359 ( 360 ~; 57 |; 304 &;9855 @)
% ( 0 <=>;1783 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 30 ( 29 usr)
% Number of type conns : 3566 (3566 >; 0 *; 0 +; 0 <<)
% Number of symbols : 222 ( 219 usr; 25 con; 0-6 aty)
% Number of variables : 4263 ( 428 ^;3671 !; 164 ?;4263 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:46:01.474
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_na8175506400003264433at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_na6857298508006588994at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_na3764207730537033026at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_na8778986904112484418at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_na8843485148432118594at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
set_na7938001796681673538at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_se3022870823424313865at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
set_na2445831480116662482_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_nat_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat_nat3: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat_nat4: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat_nat5: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_set_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
set_nat_nat_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J_J,type,
set_nat_nat_nat_o2: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_nat_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
set_nat_nat_o: $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__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $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__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $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__Nat__Onat,type,
nat: $tType ).
% Explicit typings (219)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
comple2115216063353097951_nat_o: set_na2445831480116662482_nat_o > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
comple8231226574009213710_nat_o: set_nat_nat_nat_o2 > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
comple3396693796109600270_nat_o: set_nat_nat_nat_o > ( nat > nat > nat ) > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
comple8312177224774716605_nat_o: set_nat_nat_o > ( nat > nat ) > $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__Nat__Onat_Mt__Nat__Onat_J,type,
comple2450677804321093138at_nat: set_nat_nat > nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
comple2605510978757769510at_nat: set_se3022870823424313865at_nat > set_nat_nat_nat_nat3 ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
comple1667856448326461495at_nat: set_set_nat_nat_nat2 > set_nat_nat_nat2 ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
comple8167887107183641911at_nat: set_set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
comple5448282615319421384at_nat: set_set_nat_nat > set_nat_nat ).
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__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001t__Nat__Onat_001t__Nat__Onat,type,
disjoi6798895846410478970at_nat: ( nat > set_nat ) > set_nat > $o ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
fun_upd_nat_nat_nat: ( nat > nat > nat ) > nat > ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
the_in5300466440149791684at_nat: set_nat_nat > ( ( nat > nat ) > nat ) > nat > nat > nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na6564615839001774232at_nat: set_nat_nat_nat_nat3 > ( ( ( nat > nat ) > nat > nat ) > set_nat_nat ) > set_na8175506400003264433at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
piE_na4548495224246695081at_nat: set_nat_nat_nat_nat3 > ( ( ( nat > nat ) > nat > nat ) > set_nat ) > set_na7938001796681673538at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na6840239867990089257at_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_nat_nat ) > set_na8843485148432118594at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
piE_nat_nat_nat_nat: set_nat_nat_nat2 > ( ( ( nat > nat ) > nat ) > set_nat ) > set_nat_nat_nat_nat5 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_na7122919648973241129at_nat: set_nat_nat_nat > ( ( nat > nat > nat ) > set_nat_nat ) > set_na8778986904112484418at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
piE_nat_nat_nat_nat2: set_nat_nat_nat > ( ( nat > nat > nat ) > set_nat ) > set_nat_nat_nat_nat4 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_na8678869062391380393at_nat: set_nat_nat > ( ( nat > nat ) > set_nat_nat_nat ) > set_na3764207730537033026at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_nat_nat_nat_nat3: set_nat_nat > ( ( nat > nat ) > set_nat_nat ) > set_nat_nat_nat_nat3 ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
piE_nat_nat_nat: set_nat_nat > ( ( nat > nat ) > set_nat ) > set_nat_nat_nat2 ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_na2748089427378204713at_nat: set_nat > ( nat > set_nat_nat_nat_nat3 ) > set_na6857298508006588994at_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
piE_nat_nat_nat_nat4: set_nat > ( nat > set_nat_nat_nat2 ) > set_nat_nat_nat_nat2 ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_nat_nat_nat_nat5: set_nat > ( nat > set_nat_nat_nat ) > set_nat_nat_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_nat_nat_nat2: set_nat > ( nat > set_nat_nat ) > set_nat_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Nat__Onat,type,
piE_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
piE_nat_set_nat: set_nat > ( nat > set_set_nat ) > set_nat_set_nat ).
thf(sy_c_FuncSet_Orestrict_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
restri4446420529079022766at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
restrict_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > ( nat > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
restrict_nat_nat_nat2: ( nat > nat > nat ) > set_nat > nat > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001t__Nat__Onat,type,
restrict_nat_nat: ( nat > nat ) > set_nat > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
restrict_nat_set_nat: ( nat > set_nat ) > set_nat > nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_Oundefined_001t__Nat__Onat,type,
undefined_nat: nat ).
thf(sy_c_Hales__Jewett_Oclasses,type,
hales_classes: nat > nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ocube,type,
hales_cube: nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ohj,type,
hales_hj: nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__line,type,
hales_is_line: ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__subspace,type,
hales_is_subspace: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ojoin_001t__Nat__Onat,type,
hales_join_nat: ( nat > nat ) > ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Hales__Jewett_Olayered__subspace_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
hales_4783935871306402712at_nat: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > ( nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Hales__Jewett_Olayered__subspace_001t__Nat__Onat,type,
hales_4261547300027266985ce_nat: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > nat > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Hales__Jewett_Olhj,type,
hales_lhj: nat > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Oset__incr,type,
hales_set_incr: nat > set_nat > set_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inf_in2949407623404935909at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
inf_in7997761893158376566at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > set_nat_nat_nat2 ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inf_in5274420515160781174at_nat: set_nat_nat_nat > set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inf_inf_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
sup_su3836648520750444671at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
sup_su6057362541959223568at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > set_nat_nat_nat2 ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
sup_su3334021163961628176at_nat: set_nat_nat_nat > set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_sup_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bot_nat_nat_o: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo3618716324728726597at_nat: set_na8175506400003264433at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
bot_bo3386126977483763158at_nat: set_na7938001796681673538at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo4291610329234208214at_nat: set_na8843485148432118594at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
bot_bo4508028030728203495at_nat: set_nat_nat_nat_nat5 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo4227112084914574038at_nat: set_na8778986904112484418at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
bot_bo3013702615682746855at_nat: set_nat_nat_nat_nat4 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo3919185967433191911at_nat: set_nat_nat_nat_nat3 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
bot_bo945813143650711160at_nat: set_nat_nat_nat2 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo7445843802507891576at_nat: set_nat_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo7376149671870096959at_nat: set_set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le4961065272816086430_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le8812218136411540557_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le3977685358511927117_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le4629963735342356977at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_less_nat_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
ord_less_nat_nat_nat: ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_nat_nat_nat2: ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le6177938698872215975at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le371403230139555384at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le6871433888996735800at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le5430825838364970130_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le996020443555834177_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le5384859702510996545_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le747776305331315197at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le7366121074344172400_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
ord_le2017632242545079438at_nat: ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3127000006974329230at_nat: ( nat > nat > nat ) > ( nat > nat > nat ) > $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__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $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__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le9041126503034175505at_nat: set_na8175506400003264433at_nat > set_na8175506400003264433at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3190276326201062306at_nat: set_na8843485148432118594at_nat > set_na8843485148432118594at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3125778081881428130at_nat: set_na8778986904112484418at_nat > set_na8778986904112484418at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le5260717879541182899at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le5934964663421696068at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3211623285424100676at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $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__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4954213926817602059at_nat: set_set_nat_nat > set_set_nat_nat > $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_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec3567154360959927026at_nat: ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat3 ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
collect_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_nat_nat_nat2: ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collect_set_nat_nat: ( set_nat_nat > $o ) > set_set_nat_nat ).
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_Oimage_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_4065302347126311296at_nat: ( ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat3 ) > set_na2445831480116662482_nat_o > set_se3022870823424313865at_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
image_5425260358592644672at_nat: ( ( ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 ) > set_nat_nat_nat_o2 > set_set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_3610001086604609088at_nat: ( ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat ) > set_nat_nat_nat_o > set_set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_8194121248528334964at_nat: ( ( ( nat > nat ) > nat > nat ) > nat ) > set_nat_nat_nat_nat3 > set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6952571752803954585at_nat: ( ( ( nat > nat ) > nat > nat ) > set_nat_nat ) > set_nat_nat_nat_nat3 > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1946857609996606506et_nat: ( ( ( nat > nat ) > nat > nat ) > set_nat ) > set_nat_nat_nat_nat3 > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7977807581451749376at_nat: ( ( ( nat > nat ) > $o ) > set_nat_nat ) > set_nat_nat_o > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1262493855416953332at_nat: ( ( ( nat > nat ) > nat ) > nat > nat ) > set_nat_nat_nat2 > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_7809927846809980933at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat2 > set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_2070201431993601450at_nat: ( ( ( nat > nat ) > nat ) > set_nat_nat ) > set_nat_nat_nat2 > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7565631143590340539et_nat: ( ( ( nat > nat ) > nat ) > set_nat ) > set_nat_nat_nat2 > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1545173636400105204at_nat: ( ( nat > nat > nat ) > nat > nat ) > set_nat_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_913610194320715013at_nat: ( ( nat > nat > nat ) > nat ) > set_nat_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7416711816588782250at_nat: ( ( nat > nat > nat ) > set_nat_nat ) > set_nat_nat_nat > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_6782468043973903547et_nat: ( ( nat > nat > nat ) > set_nat ) > set_nat_nat_nat > set_set_nat ).
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__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_3205354838064109189at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_470123710477037866at_nat: ( ( nat > nat ) > set_nat_nat_nat ) > set_nat_nat > set_set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6905811865970898491at_nat: ( ( nat > nat ) > set_nat_nat ) > set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7432509271690132940et_nat: ( ( nat > nat ) > set_nat ) > set_nat_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6393715451659844596at_nat: ( nat > ( nat > nat ) > nat > nat ) > set_nat > set_nat_nat_nat_nat3 ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_5809701139083627781at_nat: ( nat > ( nat > nat ) > nat ) > set_nat > set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6919068903512877573at_nat: ( nat > nat > nat > nat ) > set_nat > set_nat_nat_nat ).
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_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_nat ).
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__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_3332361743537024938at_nat: ( nat > set_nat_nat_nat_nat3 ) > set_nat > set_se3022870823424313865at_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
image_8854229838293529787at_nat: ( nat > set_nat_nat_nat2 ) > set_nat > set_set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
image_6130888460295934395at_nat: ( nat > set_nat_nat_nat ) > set_nat > set_set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7301343469591561292at_nat: ( nat > set_nat_nat ) > set_nat > set_set_nat_nat ).
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__Set__Oset_It__Nat__Onat_J_J,type,
image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
image_4040409651686222360_nat_o: ( set_nat_nat_nat_nat3 > ( ( nat > nat ) > nat > nat ) > $o ) > set_se3022870823424313865at_nat > set_na2445831480116662482_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001_Eo,type,
image_7580978635682194622_nat_o: ( set_nat_nat_nat_nat3 > $o ) > set_se3022870823424313865at_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
image_6357918107393578614_nat_o: ( set_nat_nat_nat2 > ( ( nat > nat ) > nat ) > $o ) > set_set_nat_nat_nat2 > set_nat_nat_nat_o2 ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_Eo,type,
image_8774134582277556973_nat_o: ( set_nat_nat_nat2 > $o ) > set_set_nat_nat_nat2 > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
image_2840114971476761718_nat_o: ( set_nat_nat_nat > ( nat > nat > nat ) > $o ) > set_set_nat_nat_nat > set_nat_nat_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001_Eo,type,
image_5198217506544545261_nat_o: ( set_nat_nat_nat > $o ) > set_set_nat_nat_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
image_1242417779249009364_nat_o: ( set_nat_nat > ( nat > nat ) > $o ) > set_set_nat_nat > set_nat_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
image_set_nat_nat_o: ( set_nat_nat > $o ) > set_set_nat_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_3832368097948589297at_nat: ( set_nat_nat > set_nat_nat ) > set_set_nat_nat > set_set_nat_nat ).
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_o2: ( set_nat > nat > $o ) > set_set_nat > set_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_8569768528772619084at_nat: ( set_nat > nat > nat ) > set_set_nat > set_nat_nat ).
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_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7054278410236665602at_nat: ( set_nat > set_nat_nat ) > set_set_nat > set_set_nat_nat ).
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__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_Oinsert_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
insert1655528249163143117at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 ).
thf(sy_c_Set_Oinsert_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
insert_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > set_nat_nat_nat2 ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
insert_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert_nat_nat: ( nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or3591701359631937174at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
set_or5033131092550408871at_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or6142498856979658663at_nat: ( nat > nat > nat ) > set_nat_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or9140604705432621368at_nat: ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or250740698829186286at_nat: set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or7562748684798938298at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
set_or2699333443382148811at_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or3808701207811398603at_nat: ( nat > nat > nat ) > set_nat_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or1140352010380016476at_nat: ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
fChoic52552927678224201at_nat: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( nat > nat ) > nat > nat ).
thf(sy_c_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
fChoice_nat_nat_nat: ( ( ( nat > nat ) > nat ) > $o ) > ( nat > nat ) > nat ).
thf(sy_c_fChoice_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
fChoice_nat_nat_nat2: ( ( nat > nat > nat ) > $o ) > nat > nat > nat ).
thf(sy_c_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
fChoice_nat_nat: ( ( nat > nat ) > $o ) > nat > nat ).
thf(sy_c_fChoice_001t__Nat__Onat,type,
fChoice_nat: ( nat > $o ) > nat ).
thf(sy_c_member_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member2991261302380110260at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat_nat5 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
member5318315686745620148at_nat: ( ( nat > nat > nat ) > nat ) > set_nat_nat_nat_nat4 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member1679187572556404771at_nat: ( ( nat > nat ) > nat > nat > nat ) > set_na3764207730537033026at_nat > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member952132173341509300at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat3 > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8743709692935548195at_nat: ( nat > ( nat > nat ) > nat > nat ) > set_na6857298508006588994at_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member2740455936716430260at_nat: ( nat > ( nat > nat ) > nat ) > set_nat_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member17114558718834868at_nat: ( nat > nat > nat > nat ) > set_nat_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member_nat_set_nat: ( nat > set_nat ) > set_nat_set_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_B____,type,
b: nat > set_nat ).
thf(sy_v_L_H____,type,
l: nat > nat > nat ).
thf(sy_v_L____,type,
l2: nat > nat > nat ).
thf(sy_v_N_H____,type,
n: nat ).
thf(sy_v_N____,type,
n2: nat ).
thf(sy_v_S1____,type,
s1: ( nat > nat ) > nat > nat ).
thf(sy_v__092_060chi_062____,type,
chi: ( nat > nat ) > nat ).
thf(sy_v_f____,type,
f: nat > nat ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_j____,type,
j: nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (1266)
thf(fact_0_that,axiom,
ord_less_nat @ i @ t ).
% that
thf(fact_1__C1_C,axiom,
member_nat @ j @ ( b @ one_one_nat ) ).
% "1"
thf(fact_2__092_060open_062j_A_060_AN_H_092_060close_062,axiom,
ord_less_nat @ j @ n ).
% \<open>j < N'\<close>
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062j_A_092_060in_062_AB_A1_A_092_060Longrightarrow_062_Athesis_059_Aj_A_092_060in_062_AB_A0_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
( ~ ( member_nat @ j @ ( b @ one_one_nat ) )
=> ( member_nat @ j @ ( b @ zero_zero_nat ) ) ) ).
% \<open>\<And>thesis. \<lbrakk>j \<in> B 1 \<Longrightarrow> thesis; j \<in> B 0 \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_4_calculation_I1_J,axiom,
! [Y: nat > nat] :
( ( member_nat_nat @ Y @ ( hales_cube @ one_one_nat @ t ) )
=> ( ( restri4446420529079022766at_nat
@ ^ [Y2: nat > nat] : ( l2 @ ( Y2 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ t )
@ Y
@ j )
= ( f @ j ) ) ) ).
% calculation(1)
thf(fact_5_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ t ).
% assms(1)
thf(fact_6_assms_I2_J,axiom,
! [R: nat] : ( hales_hj @ R @ t ) ).
% assms(2)
thf(fact_7_calculation_I3_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ t )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ one_one_nat @ t ) )
& ( ( X @ zero_zero_nat )
= I )
& ! [Y3: nat > nat] :
( ( ( member_nat_nat @ Y3 @ ( hales_cube @ one_one_nat @ t ) )
& ( ( Y3 @ zero_zero_nat )
= I ) )
=> ( Y3 = X ) ) ) ) ).
% calculation(3)
thf(fact_8_calculation_I2_J,axiom,
! [Y: nat > nat] :
( ( member_nat_nat @ Y @ ( hales_cube @ one_one_nat @ t ) )
=> ( ( ord_less_nat @ ( Y @ zero_zero_nat ) @ t )
& ! [Y3: nat] :
( ( ( ord_less_nat @ Y3 @ t )
& ( ( Y @ zero_zero_nat )
= Y3 ) )
=> ( Y3
= ( Y @ zero_zero_nat ) ) ) ) ) ).
% calculation(2)
thf(fact_9_cube__props_I1_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ one_one_nat @ T ) )
& ( ( X @ zero_zero_nat )
= S ) ) ) ).
% cube_props(1)
thf(fact_10__092_060open_062is__subspace_A_I_092_060lambda_062y_092_060in_062cube_A1_At_O_AL_A_Iy_A0_J_J_A1_AN_H_At_092_060close_062,axiom,
( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y2: nat > nat] : ( l2 @ ( Y2 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ t ) )
@ one_one_nat
@ n
@ t ) ).
% \<open>is_subspace (\<lambda>y\<in>cube 1 t. L (y 0)) 1 N' t\<close>
thf(fact_11_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_12_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_13_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_14_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_15_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_16_S1__def,axiom,
( s1
= ( restri4446420529079022766at_nat
@ ^ [Y2: nat > nat] : ( l @ ( Y2 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% S1_def
thf(fact_17_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_18_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_19_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_20_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_21_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_22_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_23_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_24_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_25_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_26_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_27_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_28_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_29_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_30_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_31_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_32_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_33_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_34_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_35_A2,axiom,
? [J: nat] :
( ( ord_less_nat @ J @ n )
& ! [S2: nat] :
( ( ord_less_nat @ S2 @ ( plus_plus_nat @ t @ one_one_nat ) )
=> ( ( l @ S2 @ J )
= S2 ) ) ) ).
% A2
thf(fact_36_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_37_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_38_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_39_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_40_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_41_mem__Collect__eq,axiom,
! [A: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A @ ( collect_nat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
! [A: nat > nat > nat,P: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ A @ ( collect_nat_nat_nat2 @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: ( nat > nat ) > nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ A @ ( collect_nat_nat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: ( nat > nat ) > nat > nat,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( member952132173341509300at_nat @ A @ ( collec3567154360959927026at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat] :
( ( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat2] :
( ( collect_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_51_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_52_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_53_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_54_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_55_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_56_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_57_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_58_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_59_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_60_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_61_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_62_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_63_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_64_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_65_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_66_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_67_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_68_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_69_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_70_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_71_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_72_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_73_trans__less__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_74_trans__less__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_75_add__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_76_not__add__less2,axiom,
! [J2: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).
% not_add_less2
thf(fact_77_not__add__less1,axiom,
! [I: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).
% not_add_less1
thf(fact_78_add__less__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_79_add__lessD1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_80_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_81_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_82_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_83_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_84_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_85_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_86_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_87_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_88_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_89_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_90_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_91_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_92_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_93_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_94_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_95_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_96_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_97_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_98_dim0__subspace__ex,axiom,
! [T: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_is_subspace @ S3 @ zero_zero_nat @ N @ T ) ) ).
% dim0_subspace_ex
thf(fact_99_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_100_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_101_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_102_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_103_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_104_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_105_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_106_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_107_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_108_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_109_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_110_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_111_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_112_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_113_join__cubes,axiom,
! [F: nat > nat,N: nat,T: nat,G: nat > nat,M: nat] :
( ( member_nat_nat @ F @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) )
=> ( ( member_nat_nat @ G @ ( hales_cube @ M @ ( plus_plus_nat @ T @ one_one_nat ) ) )
=> ( member_nat_nat @ ( hales_join_nat @ F @ G @ N @ M ) @ ( hales_cube @ ( plus_plus_nat @ N @ M ) @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ) ) ).
% join_cubes
thf(fact_114_line__is__dim1__subspace__t__ge__1,axiom,
! [N: nat,T: nat,L2: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ one_one_nat @ T )
=> ( ( hales_is_line @ L2 @ N @ T )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y2: nat > nat] : ( L2 @ ( Y2 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace_t_ge_1
thf(fact_115_line__is__dim1__subspace__t__1,axiom,
! [N: nat,L2: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( hales_is_line @ L2 @ N @ one_one_nat )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y2: nat > nat] : ( L2 @ ( Y2 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ one_one_nat ) )
@ one_one_nat
@ N
@ one_one_nat ) ) ) ).
% line_is_dim1_subspace_t_1
thf(fact_116_line__is__dim1__subspace,axiom,
! [N: nat,T: nat,L2: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_is_line @ L2 @ N @ T )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y2: nat > nat] : ( L2 @ ( Y2 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace
thf(fact_117_asm_I1_J,axiom,
ord_less_eq_nat @ n2 @ n ).
% asm(1)
thf(fact_118_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_119_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_120_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_121_A1,axiom,
( member_nat_nat_nat2 @ l
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat] : ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% A1
thf(fact_122_lessThan__eq__iff,axiom,
! [X2: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X2 )
= ( set_ord_lessThan_nat @ Y ) )
= ( X2 = Y ) ) ).
% lessThan_eq_iff
thf(fact_123_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_124_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_125_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_126_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_127_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_128_lessThan__subset__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X2 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_129_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_130_lessThan__iff,axiom,
! [I: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I @ ( set_or1140352010380016476at_nat @ K ) )
= ( ord_less_nat_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_131_lessThan__iff,axiom,
! [I: nat > nat > nat,K: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ ( set_or3808701207811398603at_nat @ K ) )
= ( ord_less_nat_nat_nat2 @ I @ K ) ) ).
% lessThan_iff
thf(fact_132_lessThan__iff,axiom,
! [I: ( nat > nat ) > nat,K: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ ( set_or2699333443382148811at_nat @ K ) )
= ( ord_less_nat_nat_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_133_lessThan__iff,axiom,
! [I: ( nat > nat ) > nat > nat,K: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I @ ( set_or7562748684798938298at_nat @ K ) )
= ( ord_le4629963735342356977at_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_134_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_135_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_136_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_137_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_138_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_139_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_140_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_141_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_142_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_143_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_144_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_145_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_146_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_147_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_148_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M3: nat] :
( ( P @ X2 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_149_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat,B3: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ B3 )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X
@ ( piE_nat_nat @ A2
@ ^ [I2: nat] : B3 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_150_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat,B3: set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B3 )
=> ? [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I2: nat] : B3 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_151_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat,B3: set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B3 )
=> ? [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X
@ ( piE_nat_nat_nat @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_152_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat > nat,B3: set_nat_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B3 )
=> ? [X: nat > nat > nat > nat] :
( ( member17114558718834868at_nat @ X
@ ( piE_nat_nat_nat_nat5 @ A2
@ ^ [I2: nat] : B3 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_153_fun__ex,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat,B3: set_nat_nat_nat2] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat @ B @ B3 )
=> ? [X: nat > ( nat > nat ) > nat] :
( ( member2740455936716430260at_nat @ X
@ ( piE_nat_nat_nat_nat4 @ A2
@ ^ [I2: nat] : B3 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_154_fun__ex,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B: nat,B3: set_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( member_nat @ B @ B3 )
=> ? [X: ( nat > nat > nat ) > nat] :
( ( member5318315686745620148at_nat @ X
@ ( piE_nat_nat_nat_nat2 @ A2
@ ^ [I2: nat > nat > nat] : B3 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_155_fun__ex,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,B3: set_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B3 )
=> ? [X: ( ( nat > nat ) > nat ) > nat] :
( ( member2991261302380110260at_nat @ X
@ ( piE_nat_nat_nat_nat @ A2
@ ^ [I2: ( nat > nat ) > nat] : B3 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_156_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat,B3: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B3 )
=> ? [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_157_fun__ex,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat > nat,B3: set_nat_nat_nat_nat3] :
( ( member_nat @ A @ A2 )
=> ( ( member952132173341509300at_nat @ B @ B3 )
=> ? [X: nat > ( nat > nat ) > nat > nat] :
( ( member8743709692935548195at_nat @ X
@ ( piE_na2748089427378204713at_nat @ A2
@ ^ [I2: nat] : B3 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_158_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat > nat,B3: set_nat_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B3 )
=> ? [X: ( nat > nat ) > nat > nat > nat] :
( ( member1679187572556404771at_nat @ X
@ ( piE_na8678869062391380393at_nat @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ( ( X @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_159_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_160_line__points__in__cube__unfolded,axiom,
! [L2: nat > nat > nat,N: nat,T: nat,S: nat,J2: nat] :
( ( hales_is_line @ L2 @ N @ T )
=> ( ( ord_less_nat @ S @ T )
=> ( ( ord_less_nat @ J2 @ N )
=> ( member_nat @ ( L2 @ S @ J2 ) @ ( set_ord_lessThan_nat @ T ) ) ) ) ) ).
% line_points_in_cube_unfolded
thf(fact_161_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_162_is__line__def,axiom,
( hales_is_line
= ( ^ [L3: nat > nat > nat,N3: nat,T2: nat] :
( ( member_nat_nat_nat2 @ L3
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ T2 )
@ ^ [I2: nat] : ( hales_cube @ N3 @ T2 ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ N3 )
=> ( ! [X3: nat] :
( ( ord_less_nat @ X3 @ T2 )
=> ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ T2 )
=> ( ( L3 @ X3 @ J3 )
= ( L3 @ Y2 @ J3 ) ) ) )
| ! [S4: nat] :
( ( ord_less_nat @ S4 @ T2 )
=> ( ( L3 @ S4 @ J3 )
= S4 ) ) ) )
& ? [J3: nat] :
( ( ord_less_nat @ J3 @ N3 )
& ! [S4: nat] :
( ( ord_less_nat @ S4 @ T2 )
=> ( ( L3 @ S4 @ J3 )
= S4 ) ) ) ) ) ) ).
% is_line_def
thf(fact_163_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_nat @ X3 @ U ) ) ) ) ).
% lessThan_def
thf(fact_164_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_165_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_166_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_167_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_168_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_169_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_170_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_171_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_172_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_173_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_174_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_175_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_176_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_177_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_178_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_179_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_180_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_181_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_182_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_183_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_184_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_185_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_186_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_187_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_188_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_189_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_190_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_191_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_192_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_193_add__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_194_add__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_195_trans__le__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_196_trans__le__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_197_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_198_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_199_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_200_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_201_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_202_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_203_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_204_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_205_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_206_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_207_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_208_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_209_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_210_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_211_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_212_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_213_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_214_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_215_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_216_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_217_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_218_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_219_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_220_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_221_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_222_line__points__in__cube,axiom,
! [L2: nat > nat > nat,N: nat,T: nat,S: nat] :
( ( hales_is_line @ L2 @ N @ T )
=> ( ( ord_less_nat @ S @ T )
=> ( member_nat_nat @ ( L2 @ S ) @ ( hales_cube @ N @ T ) ) ) ) ).
% line_points_in_cube
thf(fact_223_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ A2 @ B3 ) )
=> ( ( restri4446420529079022766at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_224_PiE__restrict,axiom,
! [F: nat > nat,A2: set_nat,B3: nat > set_nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ A2 @ B3 ) )
=> ( ( restrict_nat_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_225_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B3: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ A2 @ B3 ) )
=> ( ( restrict_nat_nat_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_226_PiE__restrict,axiom,
! [F: nat > nat > nat,A2: set_nat,B3: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ A2 @ B3 ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_227_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_228_order__refl,axiom,
! [X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_229_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_230_dual__order_Orefl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_231_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat > nat,I5: set_nat_nat,X5: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ ( restri4446420529079022766at_nat @ F @ I5 ) @ ( piE_nat_nat_nat_nat3 @ I5 @ X5 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ( member_nat_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_232_restrict__PiE__iff,axiom,
! [F: nat > nat,I5: set_nat,X5: nat > set_nat] :
( ( member_nat_nat @ ( restrict_nat_nat @ F @ I5 ) @ ( piE_nat_nat @ I5 @ X5 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( member_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_233_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat,I5: set_nat_nat,X5: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ I5 ) @ ( piE_nat_nat_nat @ I5 @ X5 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ( member_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_234_restrict__PiE__iff,axiom,
! [F: nat > nat > nat,I5: set_nat,X5: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ I5 ) @ ( piE_nat_nat_nat2 @ I5 @ X5 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( member_nat_nat @ ( F @ X3 ) @ ( X5 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_235_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_236_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_237_PiE__mono,axiom,
! [A2: set_nat_nat,B3: ( nat > nat ) > set_nat,C4: ( nat > nat ) > set_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le5934964663421696068at_nat @ ( piE_nat_nat_nat @ A2 @ B3 ) @ ( piE_nat_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_238_PiE__mono,axiom,
! [A2: set_nat,B3: nat > set_nat,C4: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ A2 @ B3 ) @ ( piE_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_239_PiE__mono,axiom,
! [A2: set_nat_nat_nat,B3: ( nat > nat > nat ) > set_nat_nat,C4: ( nat > nat > nat ) > set_nat_nat] :
( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le3125778081881428130at_nat @ ( piE_na7122919648973241129at_nat @ A2 @ B3 ) @ ( piE_na7122919648973241129at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_240_PiE__mono,axiom,
! [A2: set_nat_nat_nat2,B3: ( ( nat > nat ) > nat ) > set_nat_nat,C4: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le3190276326201062306at_nat @ ( piE_na6840239867990089257at_nat @ A2 @ B3 ) @ ( piE_na6840239867990089257at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_241_PiE__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B3: ( ( nat > nat ) > nat > nat ) > set_nat_nat,C4: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le9041126503034175505at_nat @ ( piE_na6564615839001774232at_nat @ A2 @ B3 ) @ ( piE_na6564615839001774232at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_242_PiE__mono,axiom,
! [A2: set_nat,B3: nat > set_nat_nat,C4: nat > set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ A2 @ B3 ) @ ( piE_nat_nat_nat2 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_243_PiE__mono,axiom,
! [A2: set_nat_nat,B3: ( nat > nat ) > set_nat_nat,C4: ( nat > nat ) > set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X ) @ ( C4 @ X ) ) )
=> ( ord_le5260717879541182899at_nat @ ( piE_nat_nat_nat_nat3 @ A2 @ B3 ) @ ( piE_nat_nat_nat_nat3 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_244_cube__def,axiom,
( hales_cube
= ( ^ [N3: nat,T2: nat] :
( piE_nat_nat @ ( set_ord_lessThan_nat @ N3 )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T2 ) ) ) ) ).
% cube_def
thf(fact_245_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_246_order__antisym__conv,axiom,
! [Y: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_247_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_248_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_249_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_250_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_251_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_252_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_253_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_254_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_255_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_256_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_257_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_258_order__eq__refl,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( X2 = Y )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_259_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_260_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_261_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_262_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_263_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_264_order__subst1,axiom,
! [A: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_265_order__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_266_order__subst1,axiom,
! [A: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_267_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_268_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B2 )
& ( ord_le9059583361652607317at_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_269_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_270_antisym,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_271_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_272_dual__order_Otrans,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_le9059583361652607317at_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_273_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_274_dual__order_Oantisym,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_275_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_276_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A3 )
& ( ord_le9059583361652607317at_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_277_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_278_order__trans,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_279_order__trans,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ Z3 )
=> ( ord_le9059583361652607317at_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_280_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_281_order_Otrans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_282_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_283_order__antisym,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_284_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_285_ord__le__eq__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_286_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_287_ord__eq__le__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A = B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_288_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_289_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
& ( ord_le9059583361652607317at_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_290_le__cases3,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_291_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_292_cube__subset,axiom,
! [N: nat,T: nat] : ( ord_le9059583361652607317at_nat @ ( hales_cube @ N @ T ) @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% cube_subset
thf(fact_293_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_294_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_295_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_296_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_297_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_298_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_299_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_300_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_301_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_302_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_303_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_304_order__less__trans,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_305_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_306_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_307_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_308_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_309_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_310_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_311_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_312_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_313_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_314_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_315_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_316_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_317_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_318_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_319_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_320_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_321_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_322_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_323_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_324_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_325_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_326_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_327_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_328_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_329_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_330_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_331_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_332_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_333_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_334_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_335_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_336_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_337_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_338_cube__restrict,axiom,
! [J2: nat,N: nat,Y: nat > nat,T: nat] :
( ( ord_less_nat @ J2 @ N )
=> ( ( member_nat_nat @ Y @ ( hales_cube @ N @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ Y @ ( set_ord_lessThan_nat @ J2 ) ) @ ( hales_cube @ J2 @ T ) ) ) ) ).
% cube_restrict
thf(fact_339_PiE__cong,axiom,
! [I5: set_nat,A2: nat > set_nat_nat,B3: nat > set_nat_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( A2 @ I3 )
= ( B3 @ I3 ) ) )
=> ( ( piE_nat_nat_nat2 @ I5 @ A2 )
= ( piE_nat_nat_nat2 @ I5 @ B3 ) ) ) ).
% PiE_cong
thf(fact_340_PiE__cong,axiom,
! [I5: set_nat,A2: nat > set_nat,B3: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( A2 @ I3 )
= ( B3 @ I3 ) ) )
=> ( ( piE_nat_nat @ I5 @ A2 )
= ( piE_nat_nat @ I5 @ B3 ) ) ) ).
% PiE_cong
thf(fact_341_PiE__cong,axiom,
! [I5: set_nat_nat,A2: ( nat > nat ) > set_nat,B3: ( nat > nat ) > set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( A2 @ I3 )
= ( B3 @ I3 ) ) )
=> ( ( piE_nat_nat_nat @ I5 @ A2 )
= ( piE_nat_nat_nat @ I5 @ B3 ) ) ) ).
% PiE_cong
thf(fact_342_PiE__cong,axiom,
! [I5: set_nat_nat,A2: ( nat > nat ) > set_nat_nat,B3: ( nat > nat ) > set_nat_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( A2 @ I3 )
= ( B3 @ I3 ) ) )
=> ( ( piE_nat_nat_nat_nat3 @ I5 @ A2 )
= ( piE_nat_nat_nat_nat3 @ I5 @ B3 ) ) ) ).
% PiE_cong
thf(fact_343_PiE__mem,axiom,
! [F: nat > nat,S5: set_nat,T3: nat > set_nat,X2: nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ S5 @ T3 ) )
=> ( ( member_nat @ X2 @ S5 )
=> ( member_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_344_PiE__mem,axiom,
! [F: nat > nat > nat,S5: set_nat,T3: nat > set_nat_nat,X2: nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ S5 @ T3 ) )
=> ( ( member_nat @ X2 @ S5 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_345_PiE__mem,axiom,
! [F: ( nat > nat ) > nat,S5: set_nat_nat,T3: ( nat > nat ) > set_nat,X2: nat > nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ S5 @ T3 ) )
=> ( ( member_nat_nat @ X2 @ S5 )
=> ( member_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_346_PiE__mem,axiom,
! [F: nat > nat > nat > nat,S5: set_nat,T3: nat > set_nat_nat_nat,X2: nat] :
( ( member17114558718834868at_nat @ F @ ( piE_nat_nat_nat_nat5 @ S5 @ T3 ) )
=> ( ( member_nat @ X2 @ S5 )
=> ( member_nat_nat_nat2 @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_347_PiE__mem,axiom,
! [F: nat > ( nat > nat ) > nat,S5: set_nat,T3: nat > set_nat_nat_nat2,X2: nat] :
( ( member2740455936716430260at_nat @ F @ ( piE_nat_nat_nat_nat4 @ S5 @ T3 ) )
=> ( ( member_nat @ X2 @ S5 )
=> ( member_nat_nat_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_348_PiE__mem,axiom,
! [F: ( nat > nat > nat ) > nat,S5: set_nat_nat_nat,T3: ( nat > nat > nat ) > set_nat,X2: nat > nat > nat] :
( ( member5318315686745620148at_nat @ F @ ( piE_nat_nat_nat_nat2 @ S5 @ T3 ) )
=> ( ( member_nat_nat_nat2 @ X2 @ S5 )
=> ( member_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_349_PiE__mem,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,S5: set_nat_nat_nat2,T3: ( ( nat > nat ) > nat ) > set_nat,X2: ( nat > nat ) > nat] :
( ( member2991261302380110260at_nat @ F @ ( piE_nat_nat_nat_nat @ S5 @ T3 ) )
=> ( ( member_nat_nat_nat @ X2 @ S5 )
=> ( member_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_350_PiE__mem,axiom,
! [F: ( nat > nat ) > nat > nat,S5: set_nat_nat,T3: ( nat > nat ) > set_nat_nat,X2: nat > nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ S5 @ T3 ) )
=> ( ( member_nat_nat @ X2 @ S5 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_351_PiE__mem,axiom,
! [F: nat > ( nat > nat ) > nat > nat,S5: set_nat,T3: nat > set_nat_nat_nat_nat3,X2: nat] :
( ( member8743709692935548195at_nat @ F @ ( piE_na2748089427378204713at_nat @ S5 @ T3 ) )
=> ( ( member_nat @ X2 @ S5 )
=> ( member952132173341509300at_nat @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_352_PiE__mem,axiom,
! [F: ( nat > nat ) > nat > nat > nat,S5: set_nat_nat,T3: ( nat > nat ) > set_nat_nat_nat,X2: nat > nat] :
( ( member1679187572556404771at_nat @ F @ ( piE_na8678869062391380393at_nat @ S5 @ T3 ) )
=> ( ( member_nat_nat @ X2 @ S5 )
=> ( member_nat_nat_nat2 @ ( F @ X2 ) @ ( T3 @ X2 ) ) ) ) ).
% PiE_mem
thf(fact_353_PiE__ext,axiom,
! [X2: nat > nat > nat,K: set_nat,S: nat > set_nat_nat,Y: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ ( piE_nat_nat_nat2 @ K @ S ) )
=> ( ( member_nat_nat_nat2 @ Y @ ( piE_nat_nat_nat2 @ K @ S ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ( X2 @ I3 )
= ( Y @ I3 ) ) )
=> ( X2 = Y ) ) ) ) ).
% PiE_ext
thf(fact_354_PiE__ext,axiom,
! [X2: nat > nat,K: set_nat,S: nat > set_nat,Y: nat > nat] :
( ( member_nat_nat @ X2 @ ( piE_nat_nat @ K @ S ) )
=> ( ( member_nat_nat @ Y @ ( piE_nat_nat @ K @ S ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ K )
=> ( ( X2 @ I3 )
= ( Y @ I3 ) ) )
=> ( X2 = Y ) ) ) ) ).
% PiE_ext
thf(fact_355_PiE__ext,axiom,
! [X2: ( nat > nat ) > nat,K: set_nat_nat,S: ( nat > nat ) > set_nat,Y: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ ( piE_nat_nat_nat @ K @ S ) )
=> ( ( member_nat_nat_nat @ Y @ ( piE_nat_nat_nat @ K @ S ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ K )
=> ( ( X2 @ I3 )
= ( Y @ I3 ) ) )
=> ( X2 = Y ) ) ) ) ).
% PiE_ext
thf(fact_356_PiE__ext,axiom,
! [X2: ( nat > nat ) > nat > nat,K: set_nat_nat,S: ( nat > nat ) > set_nat_nat,Y: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ ( piE_nat_nat_nat_nat3 @ K @ S ) )
=> ( ( member952132173341509300at_nat @ Y @ ( piE_nat_nat_nat_nat3 @ K @ S ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ K )
=> ( ( X2 @ I3 )
= ( Y @ I3 ) ) )
=> ( X2 = Y ) ) ) ) ).
% PiE_ext
thf(fact_357_split__cube_I1_J,axiom,
! [X2: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ X2 @ ( set_ord_lessThan_nat @ one_one_nat ) ) @ ( hales_cube @ one_one_nat @ T ) ) ) ).
% split_cube(1)
thf(fact_358_split__cube_I2_J,axiom,
! [X2: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [Y2: nat] : ( X2 @ ( plus_plus_nat @ Y2 @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K ) )
@ ( hales_cube @ K @ T ) ) ) ).
% split_cube(2)
thf(fact_359_restrict__apply_H,axiom,
! [X2: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( ( restri4446420529079022766at_nat @ F @ A2 @ X2 )
= ( F @ X2 ) ) ) ).
% restrict_apply'
thf(fact_360_restrict__apply_H,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( restrict_nat_nat @ F @ A2 @ X2 )
= ( F @ X2 ) ) ) ).
% restrict_apply'
thf(fact_361_restrict__apply_H,axiom,
! [X2: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( ( restrict_nat_nat_nat @ F @ A2 @ X2 )
= ( F @ X2 ) ) ) ).
% restrict_apply'
thf(fact_362_restrict__apply_H,axiom,
! [X2: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 @ X2 )
= ( F @ X2 ) ) ) ).
% restrict_apply'
thf(fact_363_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( restri4446420529079022766at_nat @ F @ A2 )
= ( restri4446420529079022766at_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_364_restrict__ext,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( restrict_nat_nat @ F @ A2 )
= ( restrict_nat_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_365_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,G: ( nat > nat ) > nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( restrict_nat_nat_nat @ F @ A2 )
= ( restrict_nat_nat_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_366_restrict__ext,axiom,
! [A2: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 )
= ( restrict_nat_nat_nat2 @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_367_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_368_order__le__imp__less__or__eq,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
=> ( ( ord_less_set_nat_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_369_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_370_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_371_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_set_nat_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_372_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_373_order__less__le__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_less_set_nat_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_374_order__less__le__subst1,axiom,
! [A: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_375_order__less__le__subst1,axiom,
! [A: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_376_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_377_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_378_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_379_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_380_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_381_order__le__less__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_set_nat_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_382_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_383_order__less__le__trans,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z3: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ Z3 )
=> ( ord_less_set_nat_nat @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_384_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_385_order__le__less__trans,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
=> ( ( ord_less_set_nat_nat @ Y @ Z3 )
=> ( ord_less_set_nat_nat @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_386_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_387_order__neq__le__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A != B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_388_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_389_order__le__neq__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_390_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_391_order__less__imp__le,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ Y )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_392_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_393_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_394_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_395_order__less__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_396_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_397_order__le__less,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_398_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_399_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_400_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_401_order_Ostrict__implies__order,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_402_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_403_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [B2: set_nat_nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A3 )
& ~ ( ord_le9059583361652607317at_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_404_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_405_dual__order_Ostrict__trans2,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_less_set_nat_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_406_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_407_dual__order_Ostrict__trans1,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_less_set_nat_nat @ C @ B )
=> ( ord_less_set_nat_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_408_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_409_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [B2: set_nat_nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_410_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_411_dual__order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B2: set_nat_nat,A3: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_412_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_413_order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B2 )
& ~ ( ord_le9059583361652607317at_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_414_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_415_order_Ostrict__trans2,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_416_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_417_order_Ostrict__trans1,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_418_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_419_order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_420_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_421_order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_422_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_423_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_424_less__le__not__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
& ~ ( ord_le9059583361652607317at_nat @ Y2 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_425_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_426_antisym__conv2,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_set_nat_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_427_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_428_antisym__conv1,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ~ ( ord_less_set_nat_nat @ X2 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_429_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_430_nless__le,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ~ ( ord_less_set_nat_nat @ A @ B ) )
= ( ~ ( ord_le9059583361652607317at_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_431_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_432_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_433_leD,axiom,
! [Y: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X2 )
=> ~ ( ord_less_set_nat_nat @ X2 @ Y ) ) ).
% leD
thf(fact_434_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_435_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_436_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_437_asm_I2_J,axiom,
( member_nat_nat_nat @ chi
@ ( piE_nat_nat_nat @ ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% asm(2)
thf(fact_438_psubsetI,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_nat_nat @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_439_N__def,axiom,
( ( ord_less_nat @ zero_zero_nat @ n2 )
& ! [N4: nat] :
( ( ord_less_eq_nat @ n2 @ N4 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N4 @ t )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [L4: nat > nat > nat,C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ( hales_is_line @ L4 @ N4 @ t )
& ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ L4 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( Chi @ X4 )
= C2 ) ) ) ) ) ) ).
% N_def
thf(fact_440_lhj__def,axiom,
( hales_lhj
= ( ^ [R2: nat,T2: nat,K3: nat] :
? [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
& ! [N6: nat] :
( ( ord_less_eq_nat @ N5 @ N6 )
=> ! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N6 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) )
=> ? [S6: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S6 @ K3 @ N6 @ T2 @ R2 @ Chi2 ) ) ) ) ) ) ).
% lhj_def
thf(fact_441_dim0__layered__subspace__ex,axiom,
! [Chi3: ( nat > nat ) > nat,N: nat,T: nat,R3: nat] :
( ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R3 ) ) )
=> ? [S3: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S3 @ zero_zero_nat @ N @ T @ R3 @ Chi3 ) ) ).
% dim0_layered_subspace_ex
thf(fact_442_hj__def,axiom,
( hales_hj
= ( ^ [R2: nat,T2: nat] :
? [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
& ! [N6: nat] :
( ( ord_less_eq_nat @ N5 @ N6 )
=> ! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N6 @ T2 )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) )
=> ? [L3: nat > nat > nat,C3: nat] :
( ( ord_less_nat @ C3 @ R2 )
& ( hales_is_line @ L3 @ N6 @ T2 )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ L3 @ ( set_ord_lessThan_nat @ T2 ) ) )
=> ( ( Chi2 @ X3 )
= C3 ) ) ) ) ) ) ) ) ).
% hj_def
thf(fact_443_subsetI,axiom,
! [A2: set_nat,B3: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B3 ) )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_444_subsetI,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat_nat_nat2 @ X @ B3 ) )
=> ( ord_le3211623285424100676at_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_445_subsetI,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat_nat_nat @ X @ B3 ) )
=> ( ord_le5934964663421696068at_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_446_subsetI,axiom,
! [A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A2 )
=> ( member952132173341509300at_nat @ X @ B3 ) )
=> ( ord_le5260717879541182899at_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_447_subsetI,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ X @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_448_image__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_449_image__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_450_image__eqI,axiom,
! [B: nat > nat,F: nat > nat > nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_451_image__eqI,axiom,
! [B: nat,F: ( nat > nat ) > nat,X2: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_452_image__eqI,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_453_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_454_image__eqI,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,X2: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_455_image__eqI,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,X2: nat > nat > nat,A2: set_nat_nat_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat_nat2 @ X2 @ A2 )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_456_image__eqI,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,X2: ( nat > nat ) > nat,A2: set_nat_nat_nat2] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat_nat @ X2 @ A2 )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_457_image__eqI,axiom,
! [B: ( nat > nat ) > nat > nat,F: nat > ( nat > nat ) > nat > nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member952132173341509300at_nat @ B @ ( image_6393715451659844596at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_458_subset__antisym,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_459_image__ident,axiom,
! [Y6: set_nat_nat] :
( ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : X3
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_460_image__ident,axiom,
! [Y6: set_nat] :
( ( image_nat_nat
@ ^ [X3: nat] : X3
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_461_L__def,axiom,
( ( hales_is_line @ l2 @ n @ t )
& ? [C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ l2 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( restrict_nat_nat_nat @ chi @ ( hales_cube @ n @ t ) @ X4 )
= C2 ) ) ) ) ).
% L_def
thf(fact_462_image__restrict__eq,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( image_nat_set_nat @ ( restrict_nat_set_nat @ F @ A2 ) @ A2 )
= ( image_nat_set_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_463_image__restrict__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( image_3205354838064109189at_nat @ ( restri4446420529079022766at_nat @ F @ A2 ) @ A2 )
= ( image_3205354838064109189at_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_464_image__restrict__eq,axiom,
! [F: nat > nat,A2: set_nat] :
( ( image_nat_nat @ ( restrict_nat_nat @ F @ A2 ) @ A2 )
= ( image_nat_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_465_image__restrict__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ A2 ) @ A2 )
= ( image_nat_nat_nat @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_466_image__restrict__eq,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ A2 ) @ A2 )
= ( image_nat_nat_nat2 @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_467_image__add__0,axiom,
! [S5: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S5 )
= S5 ) ).
% image_add_0
thf(fact_468__092_060open_062restrict_A_092_060chi_062_A_Icube_AN_H_At_J_A_092_060in_062_Acube_AN_H_At_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_092_060close_062,axiom,
( member_nat_nat_nat @ ( restrict_nat_nat_nat @ chi @ ( hales_cube @ n @ t ) )
@ ( piE_nat_nat_nat @ ( hales_cube @ n @ t )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% \<open>restrict \<chi> (cube N' t) \<in> cube N' t \<rightarrow>\<^sub>E {..<r}\<close>
thf(fact_469__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062N_O_A0_A_060_AN_A_092_060and_062_A_I_092_060forall_062N_H_092_060ge_062N_O_A_092_060forall_062_092_060chi_062_O_A_092_060chi_062_A_092_060in_062_Acube_AN_H_At_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_A_092_060longrightarrow_062_A_I_092_060exists_062L_Ac_O_Ac_A_060_Ar_A_092_060and_062_Ais__line_AL_AN_H_At_A_092_060and_062_A_I_092_060forall_062y_092_060in_062L_A_096_A_123_O_O_060t_125_O_A_092_060chi_062_Ay_A_061_Ac_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N7: nat] :
~ ( ( ord_less_nat @ zero_zero_nat @ N7 )
& ! [N4: nat] :
( ( ord_less_eq_nat @ N7 @ N4 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N4 @ t )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [L4: nat > nat > nat,C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ( hales_is_line @ L4 @ N4 @ t )
& ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ L4 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( Chi @ X4 )
= C2 ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>N. 0 < N \<and> (\<forall>N'\<ge>N. \<forall>\<chi>. \<chi> \<in> cube N' t \<rightarrow>\<^sub>E {..<r} \<longrightarrow> (\<exists>L c. c < r \<and> is_line L N' t \<and> (\<forall>y\<in>L ` {..<t}. \<chi> y = c))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_470_N_H__props,axiom,
( ( ord_less_nat @ zero_zero_nat @ n )
& ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ n @ t )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [L4: nat > nat > nat,C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ( hales_is_line @ L4 @ n @ t )
& ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ L4 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( Chi @ X4 )
= C2 ) ) ) ) ) ).
% N'_props
thf(fact_471__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062L_O_Ais__line_AL_AN_H_At_A_092_060and_062_A_I_092_060exists_062c_060r_O_A_092_060forall_062y_092_060in_062L_A_096_A_123_O_O_060t_125_O_Arestrict_A_092_060chi_062_A_Icube_AN_H_At_J_Ay_A_061_Ac_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [L4: nat > nat > nat] :
~ ( ( hales_is_line @ L4 @ n @ t )
& ? [C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ L4 @ ( set_ord_lessThan_nat @ t ) ) )
=> ( ( restrict_nat_nat_nat @ chi @ ( hales_cube @ n @ t ) @ X4 )
= C2 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>L. is_line L N' t \<and> (\<exists>c<r. \<forall>y\<in>L ` {..<t}. restrict \<chi> (cube N' t) y = c) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_472_image__mono,axiom,
! [A2: set_nat,B3: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_473_image__mono,axiom,
! [A2: set_nat,B3: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_474_image__mono,axiom,
! [A2: set_nat,B3: set_nat,F: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ ( image_nat_nat_nat2 @ F @ B3 ) ) ) ).
% image_mono
thf(fact_475_image__mono,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ ( image_3205354838064109189at_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_476_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B3: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_477_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_nat,B3: set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B3 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_478_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,B3: set_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_479_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat,B3: set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_480_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat > nat,B3: set_nat_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ B3 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_481_image__subsetI,axiom,
! [A2: set_nat,F: nat > ( nat > nat ) > nat,B3: set_nat_nat_nat2] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat @ ( F @ X ) @ B3 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_482_image__subsetI,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat,B3: set_nat] :
( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_913610194320715013at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_483_image__subsetI,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,B3: set_nat] :
( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_484_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,B3: set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_485_image__subsetI,axiom,
! [A2: set_nat,F: nat > ( nat > nat ) > nat > nat,B3: set_nat_nat_nat_nat3] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member952132173341509300at_nat @ ( F @ X ) @ B3 ) )
=> ( ord_le5260717879541182899at_nat @ ( image_6393715451659844596at_nat @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_486_subset__imageE,axiom,
! [B3: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B3
!= ( image_nat_set_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_487_subset__imageE,axiom,
! [B3: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B3
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_488_subset__imageE,axiom,
! [B3: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B3
!= ( image_nat_nat_nat2 @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_489_subset__imageE,axiom,
! [B3: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A2 )
=> ( B3
!= ( image_3205354838064109189at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_490_image__subset__iff,axiom,
! [F: nat > set_nat,A2: set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_491_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_492_image__subset__iff,axiom,
! [F: nat > nat > nat,A2: set_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_493_image__subset__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B3 )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_494_subset__image__iff,axiom,
! [B3: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B3
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_495_subset__image__iff,axiom,
! [B3: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B3
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_496_subset__image__iff,axiom,
! [B3: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B3
= ( image_nat_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_497_subset__image__iff,axiom,
! [B3: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A2 )
& ( B3
= ( image_3205354838064109189at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_498_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_499_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_500_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ).
% imageI
thf(fact_501_imageI,axiom,
! [X2: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_502_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat > nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X2 ) @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_503_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat_nat @ ( F @ X2 ) @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_504_imageI,axiom,
! [X2: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_505_imageI,axiom,
! [X2: nat > nat > nat,A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_506_imageI,axiom,
! [X2: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_507_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > ( nat > nat ) > nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member952132173341509300at_nat @ ( F @ X2 ) @ ( image_6393715451659844596at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_508_image__iff,axiom,
! [Z3: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ Z3 @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_509_image__iff,axiom,
! [Z3: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_510_image__iff,axiom,
! [Z3: nat > nat,F: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ Z3 @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_511_image__iff,axiom,
! [Z3: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ Z3 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_512_bex__imageD,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_513_bex__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_514_bex__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 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_515_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_516_image__cong,axiom,
! [M3: set_nat,N8: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ( M3 = N8 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N8 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat_nat2 @ F @ M3 )
= ( image_nat_nat_nat2 @ G @ N8 ) ) ) ) ).
% image_cong
thf(fact_517_image__cong,axiom,
! [M3: set_nat,N8: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M3 = N8 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N8 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_set_nat @ F @ M3 )
= ( image_nat_set_nat @ G @ N8 ) ) ) ) ).
% image_cong
thf(fact_518_image__cong,axiom,
! [M3: set_nat,N8: set_nat,F: nat > nat,G: nat > nat] :
( ( M3 = N8 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N8 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat @ F @ M3 )
= ( image_nat_nat @ G @ N8 ) ) ) ) ).
% image_cong
thf(fact_519_image__cong,axiom,
! [M3: set_nat_nat,N8: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ( M3 = N8 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ N8 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_3205354838064109189at_nat @ F @ M3 )
= ( image_3205354838064109189at_nat @ G @ N8 ) ) ) ) ).
% image_cong
thf(fact_520_ball__imageD,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_521_ball__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_522_ball__imageD,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_523_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_524_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_525_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_526_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_527_rev__image__eqI,axiom,
! [X2: nat > nat,A2: set_nat_nat,B: nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_528_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: nat > nat > nat,F: nat > nat > nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_529_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_530_rev__image__eqI,axiom,
! [X2: nat > nat,A2: set_nat_nat,B: nat > nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_531_rev__image__eqI,axiom,
! [X2: nat > nat > nat,A2: set_nat_nat_nat,B: nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_532_rev__image__eqI,axiom,
! [X2: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_533_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B: ( nat > nat ) > nat > nat,F: nat > ( nat > nat ) > nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member952132173341509300at_nat @ B @ ( image_6393715451659844596at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_534_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_535_imageE,axiom,
! [B: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_536_imageE,axiom,
! [B: nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) )
=> ~ ! [X: nat > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_537_imageE,axiom,
! [B: nat > nat,F: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_538_imageE,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat] :
( ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) )
=> ~ ! [X: nat > nat > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat_nat_nat2 @ X @ A2 ) ) ) ).
% imageE
thf(fact_539_imageE,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2] :
( ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) )
=> ~ ! [X: ( nat > nat ) > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat_nat_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_540_imageE,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ~ ! [X: nat > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_541_imageE,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,A2: set_nat] :
( ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_542_imageE,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,A2: set_nat] :
( ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_543_imageE,axiom,
! [B: nat,F: ( ( nat > nat ) > nat > nat ) > nat,A2: set_nat_nat_nat_nat3] :
( ( member_nat @ B @ ( image_8194121248528334964at_nat @ F @ A2 ) )
=> ~ ! [X: ( nat > nat ) > nat > nat] :
( ( B
= ( F @ X ) )
=> ~ ( member952132173341509300at_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_544_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_545_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
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_546_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
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_547_image__image,axiom,
! [F: ( nat > nat ) > nat,G: nat > nat > nat,A2: set_nat] :
( ( image_nat_nat_nat @ F @ ( image_nat_nat_nat2 @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_548_image__image,axiom,
! [F: set_nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( image_nat_set_nat @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_549_image__image,axiom,
! [F: nat > nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_550_image__image,axiom,
! [F: ( nat > nat ) > set_nat,G: nat > nat > nat,A2: set_nat] :
( ( image_7432509271690132940et_nat @ F @ ( image_nat_nat_nat2 @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_551_image__image,axiom,
! [F: set_nat > nat > nat,G: nat > set_nat,A2: set_nat] :
( ( image_8569768528772619084at_nat @ F @ ( image_nat_set_nat @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_552_image__image,axiom,
! [F: nat > nat > nat,G: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat_nat2 @ F @ ( image_nat_nat_nat @ G @ A2 ) )
= ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_553_image__image,axiom,
! [F: ( nat > nat ) > nat > nat,G: nat > nat > nat,A2: set_nat] :
( ( image_3205354838064109189at_nat @ F @ ( image_nat_nat_nat2 @ G @ A2 ) )
= ( image_nat_nat_nat2
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_554_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_555_Compr__image__eq,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_set_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_556_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat_nat @ F
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_557_Compr__image__eq,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat_nat2 @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_558_Compr__image__eq,axiom,
! [F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_913610194320715013at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_913610194320715013at_nat @ F
@ ( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_559_Compr__image__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_7809927846809980933at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_7809927846809980933at_nat @ F
@ ( collect_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_560_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_3205354838064109189at_nat @ F
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_561_Compr__image__eq,axiom,
! [F: nat > nat > nat > nat,A2: set_nat,P: ( nat > nat > nat ) > $o] :
( ( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ ( image_6919068903512877573at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_6919068903512877573at_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_562_Compr__image__eq,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( collect_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ ( image_5809701139083627781at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_5809701139083627781at_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_563_Compr__image__eq,axiom,
! [F: ( ( nat > nat ) > nat > nat ) > nat,A2: set_nat_nat_nat_nat3,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_8194121248528334964at_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_8194121248528334964at_nat @ F
@ ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_564_PiE__uniqueness,axiom,
! [F: nat > set_nat,A2: set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B3 )
=> ? [X: nat > set_nat] :
( ( member_nat_set_nat @ X
@ ( piE_nat_set_nat @ A2
@ ^ [I2: nat] : B3 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y3: nat > set_nat] :
( ( ( member_nat_set_nat @ Y3
@ ( piE_nat_set_nat @ A2
@ ^ [I2: nat] : B3 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y3 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y3 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_565_PiE__uniqueness,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B3 )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X
@ ( piE_nat_nat @ A2
@ ^ [I2: nat] : B3 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y3: nat > nat] :
( ( ( member_nat_nat @ Y3
@ ( piE_nat_nat @ A2
@ ^ [I2: nat] : B3 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y3 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y3 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_566_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B3 )
=> ? [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X
@ ( piE_nat_nat_nat @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y3: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ Y3
@ ( piE_nat_nat_nat @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y3 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y3 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_567_PiE__uniqueness,axiom,
! [F: nat > nat > nat,A2: set_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B3 )
=> ? [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I2: nat] : B3 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y3: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y3
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I2: nat] : B3 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y3 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y3 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_568_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B3 )
=> ? [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X @ Xa )
= ( F @ Xa ) ) )
& ! [Y3: ( nat > nat ) > nat > nat] :
( ( ( member952132173341509300at_nat @ Y3
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I2: nat > nat] : B3 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y3 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y3 = X ) ) ) ) ).
% PiE_uniqueness
thf(fact_569_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_570_set__eq__subset,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_571_subset__trans,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ C4 )
=> ( ord_le9059583361652607317at_nat @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_572_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_573_subset__refl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_574_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_575_subset__iff,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
! [T2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ T2 @ A6 )
=> ( member_nat_nat_nat2 @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_576_subset__iff,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
! [T2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ T2 @ A6 )
=> ( member_nat_nat_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_577_subset__iff,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A6: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
! [T2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ T2 @ A6 )
=> ( member952132173341509300at_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_578_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
! [T2: nat > nat] :
( ( member_nat_nat @ T2 @ A6 )
=> ( member_nat_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_579_equalityD2,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( A2 = B3 )
=> ( ord_le9059583361652607317at_nat @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_580_equalityD1,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( A2 = B3 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_581_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( member_nat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_582_subset__eq,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A6 )
=> ( member_nat_nat_nat2 @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_583_subset__eq,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A6 )
=> ( member_nat_nat_nat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_584_subset__eq,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A6: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A6 )
=> ( member952132173341509300at_nat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_585_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A6 )
=> ( member_nat_nat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_586_equalityE,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( A2 = B3 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ~ ( ord_le9059583361652607317at_nat @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_587_subsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_588_subsetD,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% subsetD
thf(fact_589_subsetD,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_590_subsetD,axiom,
! [A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3,C: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B3 )
=> ( ( member952132173341509300at_nat @ C @ A2 )
=> ( member952132173341509300at_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_591_subsetD,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_592_in__mono,axiom,
! [A2: set_nat,B3: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_593_in__mono,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,X2: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat2 @ X2 @ A2 )
=> ( member_nat_nat_nat2 @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_594_in__mono,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,X2: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat @ X2 @ A2 )
=> ( member_nat_nat_nat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_595_in__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3,X2: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B3 )
=> ( ( member952132173341509300at_nat @ X2 @ A2 )
=> ( member952132173341509300at_nat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_596_in__mono,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,X2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_597_psubsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_598_psubsetD,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C: nat > nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_599_psubsetD,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le6871433888996735800at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_600_psubsetD,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le371403230139555384at_nat @ A2 @ B3 )
=> ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_601_psubsetD,axiom,
! [A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3,C: ( nat > nat ) > nat > nat] :
( ( ord_le6177938698872215975at_nat @ A2 @ B3 )
=> ( ( member952132173341509300at_nat @ C @ A2 )
=> ( member952132173341509300at_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_602_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_603_Collect__subset,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_604_Collect__subset,axiom,
! [A2: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ord_le5934964663421696068at_nat
@ ( collect_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_605_Collect__subset,axiom,
! [A2: set_nat_nat_nat_nat3,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ord_le5260717879541182899at_nat
@ ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_606_Collect__subset,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_607_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_608_less__eq__set__def,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
( ord_le5384859702510996545_nat_o
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ A6 )
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_609_less__eq__set__def,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
( ord_le996020443555834177_nat_o
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ A6 )
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_610_less__eq__set__def,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A6: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
( ord_le5430825838364970130_nat_o
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A6 )
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_611_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A6 )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_612_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_613_less__set__def,axiom,
( ord_less_set_nat_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ord_less_nat_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A6 )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_614_less__set__def,axiom,
( ord_le6871433888996735800at_nat
= ( ^ [A6: set_nat_nat_nat,B6: set_nat_nat_nat] :
( ord_le3977685358511927117_nat_o
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ A6 )
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_615_less__set__def,axiom,
( ord_le371403230139555384at_nat
= ( ^ [A6: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
( ord_le8812218136411540557_nat_o
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ A6 )
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_616_less__set__def,axiom,
( ord_le6177938698872215975at_nat
= ( ^ [A6: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
( ord_le4961065272816086430_nat_o
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A6 )
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_617_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ( ord_less_set_nat_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_618_subset__psubset__trans,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( ord_less_set_nat_nat @ B3 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_619_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ B6 )
& ~ ( ord_le9059583361652607317at_nat @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_620_psubset__subset__trans,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_621_psubset__imp__subset,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_622_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A6: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_623_psubsetE,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B3 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_624_classes__subset__cube,axiom,
! [N: nat,T: nat,I: nat] : ( ord_le9059583361652607317at_nat @ ( hales_classes @ N @ T @ I ) @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% classes_subset_cube
thf(fact_625_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > set_nat,B3: set_set_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_set_nat @ ( F @ X ) @ B3 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( collect_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_626_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B3: set_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat @ ( F @ X ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_627_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat > nat,B3: set_nat_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat_nat @ ( F @ X ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ ( collect_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_628_image__Collect__subsetI,axiom,
! [P: ( nat > nat ) > $o,F: ( nat > nat ) > nat > nat,B3: set_nat_nat] :
( ! [X: nat > nat] :
( ( P @ X )
=> ( member_nat_nat @ ( F @ X ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ ( collect_nat_nat @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_629_dim1__subspace__is__line,axiom,
! [T: nat,S5: ( nat > nat ) > nat > nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_is_subspace @ S5 @ one_one_nat @ N @ T )
=> ( hales_is_line
@ ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S5
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ T ) )
@ N
@ T ) ) ) ).
% dim1_subspace_is_line
thf(fact_630_dim1__layered__subspace__as__line,axiom,
! [T: nat,S5: ( nat > nat ) > nat > nat,N: nat,R3: nat,Chi3: ( nat > nat ) > nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4261547300027266985ce_nat @ S5 @ one_one_nat @ N @ T @ R3 @ Chi3 )
=> ? [C1: nat,C22: nat] :
( ( ord_less_nat @ C1 @ R3 )
& ( ord_less_nat @ C22 @ R3 )
& ! [S2: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( ( Chi3
@ ( S5
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= S2 ) ) ) ) )
= C1 ) )
& ( ( Chi3
@ ( S5
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= T ) ) ) ) )
= C22 ) ) ) ) ).
% dim1_layered_subspace_as_line
thf(fact_631_dim1__layered__subspace__mono__line,axiom,
! [T: nat,S5: ( nat > nat ) > nat > nat,N: nat,R3: nat,Chi3: ( nat > nat ) > nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_4261547300027266985ce_nat @ S5 @ one_one_nat @ N @ T @ R3 @ Chi3 )
=> ! [S2: nat] :
( ( ord_less_nat @ S2 @ T )
=> ! [L5: nat] :
( ( ord_less_nat @ L5 @ T )
=> ( ( ( Chi3
@ ( S5
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= S2 ) ) ) ) )
= ( Chi3
@ ( S5
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= L5 ) ) ) ) ) )
& ( ord_less_nat
@ ( Chi3
@ ( S5
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= S2 ) ) ) ) )
@ R3 ) ) ) ) ) ) ).
% dim1_layered_subspace_mono_line
thf(fact_632_verit__sko__forall__indirect2,axiom,
! [X2: nat > nat,P: ( nat > nat ) > $o,P4: ( nat > nat ) > $o] :
( ( X2
= ( fChoice_nat_nat
@ ^ [X3: nat > nat] :
~ ( P @ X3 ) ) )
=> ( ! [X: nat > nat] :
( ( P @ X )
= ( P4 @ X ) )
=> ( ( ! [X7: nat > nat] : ( P4 @ X7 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_633_verit__sko__forall__indirect,axiom,
! [X2: nat > nat,P: ( nat > nat ) > $o] :
( ( X2
= ( fChoice_nat_nat
@ ^ [X3: nat > nat] :
~ ( P @ X3 ) ) )
=> ( ( ! [X7: nat > nat] : ( P @ X7 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_forall_indirect
thf(fact_634_verit__sko__ex__indirect2,axiom,
! [X2: nat > nat,P: ( nat > nat ) > $o,P4: ( nat > nat ) > $o] :
( ( X2
= ( fChoice_nat_nat @ P ) )
=> ( ! [X: nat > nat] :
( ( P @ X )
= ( P4 @ X ) )
=> ( ( ? [X7: nat > nat] : ( P4 @ X7 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_635_verit__sko__ex__indirect,axiom,
! [X2: nat > nat,P: ( nat > nat ) > $o] :
( ( X2
= ( fChoice_nat_nat @ P ) )
=> ( ( ? [X7: nat > nat] : ( P @ X7 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_ex_indirect
thf(fact_636_verit__sko__forall_H_H,axiom,
! [B3: nat > nat,A2: nat > nat,P: ( nat > nat ) > $o] :
( ( B3 = A2 )
=> ( ( ( fChoice_nat_nat @ P )
= A2 )
= ( ( fChoice_nat_nat @ P )
= B3 ) ) ) ).
% verit_sko_forall''
thf(fact_637_verit__sko__forall_H,axiom,
! [P: ( nat > nat ) > $o,A2: $o] :
( ( ( P
@ ( fChoice_nat_nat
@ ^ [X3: nat > nat] :
~ ( P @ X3 ) ) )
= A2 )
=> ( ( ! [X7: nat > nat] : ( P @ X7 ) )
= A2 ) ) ).
% verit_sko_forall'
thf(fact_638_verit__sko__forall,axiom,
( ( ^ [P2: ( nat > nat ) > $o] :
! [X6: nat > nat] : ( P2 @ X6 ) )
= ( ^ [P3: ( nat > nat ) > $o] :
( P3
@ ( fChoice_nat_nat
@ ^ [X3: nat > nat] :
~ ( P3 @ X3 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_639_verit__sko__ex_H,axiom,
! [P: ( nat > nat ) > $o,A2: $o] :
( ( ( P @ ( fChoice_nat_nat @ P ) )
= A2 )
=> ( ( ? [X7: nat > nat] : ( P @ X7 ) )
= A2 ) ) ).
% verit_sko_ex'
thf(fact_640_subspace__elems__embed,axiom,
! [S5: ( nat > nat ) > nat > nat,K: nat,N: nat,T: nat] :
( ( hales_is_subspace @ S5 @ K @ N @ T )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ S5 @ ( hales_cube @ K @ T ) ) @ ( hales_cube @ N @ T ) ) ) ).
% subspace_elems_embed
thf(fact_641_cube__props_I2_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S ) )
@ zero_zero_nat )
= S ) ) ).
% cube_props(2)
thf(fact_642_cube__props_I4_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( member_nat_nat
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S ) ) )
@ ( hales_cube @ one_one_nat @ T ) ) ) ).
% cube_props(4)
thf(fact_643_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_644_Collect__restrict,axiom,
! [X5: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_645_Collect__restrict,axiom,
! [X5: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ord_le5934964663421696068at_nat
@ ( collect_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_646_Collect__restrict,axiom,
! [X5: set_nat_nat_nat_nat3,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ord_le5260717879541182899at_nat
@ ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_647_Collect__restrict,axiom,
! [X5: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_648_prop__restrict,axiom,
! [X2: nat,Z5: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_649_prop__restrict,axiom,
! [X2: nat > nat > nat,Z5: set_nat_nat_nat,X5: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ X2 @ Z5 )
=> ( ( ord_le3211623285424100676at_nat @ Z5
@ ( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_650_prop__restrict,axiom,
! [X2: ( nat > nat ) > nat,Z5: set_nat_nat_nat2,X5: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ X2 @ Z5 )
=> ( ( ord_le5934964663421696068at_nat @ Z5
@ ( collect_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_651_prop__restrict,axiom,
! [X2: ( nat > nat ) > nat > nat,Z5: set_nat_nat_nat_nat3,X5: set_nat_nat_nat_nat3,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( member952132173341509300at_nat @ X2 @ Z5 )
=> ( ( ord_le5260717879541182899at_nat @ Z5
@ ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_652_prop__restrict,axiom,
! [X2: nat > nat,Z5: set_nat_nat,X5: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X2 @ Z5 )
=> ( ( ord_le9059583361652607317at_nat @ Z5
@ ( collect_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_653_cube__props_I3_J,axiom,
! [S: nat,T: nat,S5: ( nat > nat ) > nat] :
( ( ord_less_nat @ S @ T )
=> ( ( restrict_nat_nat
@ ^ [S4: nat] :
( S5
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ S )
= ( restrict_nat_nat
@ ^ [S4: nat] :
( S5
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S ) )
@ zero_zero_nat ) ) ) ) ).
% cube_props(3)
thf(fact_654_cube__props_I3_J,axiom,
! [S: nat,T: nat,S5: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ S @ T )
=> ( ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S5
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ S )
= ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S5
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ T )
@ ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S ) )
@ zero_zero_nat ) ) ) ) ).
% cube_props(3)
thf(fact_655_some__sym__eq__trivial,axiom,
! [X2: nat > nat] :
( ( fChoice_nat_nat
@ ( ^ [Y5: nat > nat,Z2: nat > nat] : ( Y5 = Z2 )
@ X2 ) )
= X2 ) ).
% some_sym_eq_trivial
thf(fact_656_some__eq__trivial,axiom,
! [X2: nat > nat] :
( ( fChoice_nat_nat
@ ^ [Y2: nat > nat] : ( Y2 = X2 ) )
= X2 ) ).
% some_eq_trivial
thf(fact_657_some__equality,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat] :
( ( P @ A )
=> ( ! [X: nat > nat] :
( ( P @ X )
=> ( X = A ) )
=> ( ( fChoice_nat_nat @ P )
= A ) ) ) ).
% some_equality
thf(fact_658_layered__subspace__def,axiom,
( hales_4783935871306402712at_nat
= ( ^ [S6: ( nat > nat ) > nat > nat,K3: nat,N3: nat,T2: nat,R2: nat > nat,Chi2: ( nat > nat ) > nat > nat] :
( ( hales_is_subspace @ S6 @ K3 @ N3 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_ord_atMost_nat @ K3 ) )
=> ? [C3: nat > nat] :
( ( ord_less_nat_nat @ C3 @ R2 )
& ! [Y2: nat > nat] :
( ( member_nat_nat @ Y2 @ ( hales_classes @ K3 @ T2 @ X3 ) )
=> ( ( Chi2 @ ( S6 @ Y2 ) )
= C3 ) ) ) )
& ( member952132173341509300at_nat @ Chi2
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ N3 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_or1140352010380016476at_nat @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_659_layered__subspace__def,axiom,
( hales_4261547300027266985ce_nat
= ( ^ [S6: ( nat > nat ) > nat > nat,K3: nat,N3: nat,T2: nat,R2: nat,Chi2: ( nat > nat ) > nat] :
( ( hales_is_subspace @ S6 @ K3 @ N3 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_ord_atMost_nat @ K3 ) )
=> ? [C3: nat] :
( ( ord_less_nat @ C3 @ R2 )
& ! [Y2: nat > nat] :
( ( member_nat_nat @ Y2 @ ( hales_classes @ K3 @ T2 @ X3 ) )
=> ( ( Chi2 @ ( S6 @ Y2 ) )
= C3 ) ) ) )
& ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N3 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_660_pred__subset__eq,axiom,
! [R4: set_nat,S5: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R4 )
@ ^ [X3: nat] : ( member_nat @ X3 @ S5 ) )
= ( ord_less_eq_set_nat @ R4 @ S5 ) ) ).
% pred_subset_eq
thf(fact_661_pred__subset__eq,axiom,
! [R4: set_nat_nat_nat,S5: set_nat_nat_nat] :
( ( ord_le5384859702510996545_nat_o
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ R4 )
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ S5 ) )
= ( ord_le3211623285424100676at_nat @ R4 @ S5 ) ) ).
% pred_subset_eq
thf(fact_662_pred__subset__eq,axiom,
! [R4: set_nat_nat_nat2,S5: set_nat_nat_nat2] :
( ( ord_le996020443555834177_nat_o
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ R4 )
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ S5 ) )
= ( ord_le5934964663421696068at_nat @ R4 @ S5 ) ) ).
% pred_subset_eq
thf(fact_663_pred__subset__eq,axiom,
! [R4: set_nat_nat_nat_nat3,S5: set_nat_nat_nat_nat3] :
( ( ord_le5430825838364970130_nat_o
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ R4 )
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ S5 ) )
= ( ord_le5260717879541182899at_nat @ R4 @ S5 ) ) ).
% pred_subset_eq
thf(fact_664_pred__subset__eq,axiom,
! [R4: set_nat_nat,S5: set_nat_nat] :
( ( ord_le7366121074344172400_nat_o
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ R4 )
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ S5 ) )
= ( ord_le9059583361652607317at_nat @ R4 @ S5 ) ) ).
% pred_subset_eq
thf(fact_665_atMost__eq__iff,axiom,
! [X2: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X2 )
= ( set_ord_atMost_nat @ Y ) )
= ( X2 = Y ) ) ).
% atMost_eq_iff
thf(fact_666_atMost__iff,axiom,
! [I: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I @ ( set_or9140604705432621368at_nat @ K ) )
= ( ord_less_eq_nat_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_667_atMost__iff,axiom,
! [I: nat > nat > nat,K: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ ( set_or6142498856979658663at_nat @ K ) )
= ( ord_le3127000006974329230at_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_668_atMost__iff,axiom,
! [I: ( nat > nat ) > nat,K: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ ( set_or5033131092550408871at_nat @ K ) )
= ( ord_le2017632242545079438at_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_669_atMost__iff,axiom,
! [I: ( nat > nat ) > nat > nat,K: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I @ ( set_or3591701359631937174at_nat @ K ) )
= ( ord_le747776305331315197at_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_670_atMost__iff,axiom,
! [I: set_nat_nat,K: set_nat_nat] :
( ( member_set_nat_nat @ I @ ( set_or250740698829186286at_nat @ K ) )
= ( ord_le9059583361652607317at_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_671_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_672_atMost__subset__iff,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ ( set_or250740698829186286at_nat @ X2 ) @ ( set_or250740698829186286at_nat @ Y ) )
= ( ord_le9059583361652607317at_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_673_atMost__subset__iff,axiom,
! [X2: nat > nat,Y: nat > nat] :
( ( ord_le9059583361652607317at_nat @ ( set_or9140604705432621368at_nat @ X2 ) @ ( set_or9140604705432621368at_nat @ Y ) )
= ( ord_less_eq_nat_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_674_atMost__subset__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_675_atMost__def,axiom,
( set_or250740698829186286at_nat
= ( ^ [U: set_nat_nat] :
( collect_set_nat_nat
@ ^ [X3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_676_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U ) ) ) ) ).
% atMost_def
thf(fact_677_layered__eq__classes,axiom,
! [S5: ( nat > nat ) > nat > nat,K: nat,N: nat,T: nat,R3: nat,Chi3: ( nat > nat ) > nat] :
( ( hales_4261547300027266985ce_nat @ S5 @ K @ N @ T @ R3 @ Chi3 )
=> ! [X4: nat] :
( ( member_nat @ X4 @ ( set_ord_atMost_nat @ K ) )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ ( hales_classes @ K @ T @ X4 ) )
=> ! [Xb: nat > nat] :
( ( member_nat_nat @ Xb @ ( hales_classes @ K @ T @ X4 ) )
=> ( ( Chi3 @ ( S5 @ Xa ) )
= ( Chi3 @ ( S5 @ Xb ) ) ) ) ) ) ) ).
% layered_eq_classes
thf(fact_678_Iic__subset__Iio__iff,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_679_someI2,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat,Q: ( nat > nat ) > $o] :
( ( P @ A )
=> ( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_nat_nat @ P ) ) ) ) ).
% someI2
thf(fact_680_someI__ex,axiom,
! [P: ( nat > nat ) > $o] :
( ? [X_12: nat > nat] : ( P @ X_12 )
=> ( P @ ( fChoice_nat_nat @ P ) ) ) ).
% someI_ex
thf(fact_681_someI2__ex,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ? [X_12: nat > nat] : ( P @ X_12 )
=> ( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_nat_nat @ P ) ) ) ) ).
% someI2_ex
thf(fact_682_someI2__bex,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X: nat] :
( ( ( member_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_683_someI2__bex,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ? [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat_nat_nat2
@ ^ [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_684_someI2__bex,axiom,
! [A2: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ? [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_685_someI2__bex,axiom,
! [A2: set_nat_nat_nat_nat3,P: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
( ? [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X: ( nat > nat ) > nat > nat] :
( ( ( member952132173341509300at_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoic52552927678224201at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_686_someI2__bex,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
& ( P @ X4 ) )
=> ( ! [X: nat > nat] :
( ( ( member_nat_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat_nat
@ ^ [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_687_some__eq__ex,axiom,
! [P: ( nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat @ P ) )
= ( ? [X7: nat > nat] : ( P @ X7 ) ) ) ).
% some_eq_ex
thf(fact_688_some1__equality,axiom,
! [P: ( nat > nat ) > $o,A: nat > nat] :
( ? [X4: nat > nat] :
( ( P @ X4 )
& ! [Y4: nat > nat] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ( P @ A )
=> ( ( fChoice_nat_nat @ P )
= A ) ) ) ).
% some1_equality
thf(fact_689_Bf__defs,axiom,
( ( disjoi6798895846410478970at_nat @ b @ ( set_ord_atMost_nat @ one_one_nat ) )
& ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ b @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ n ) )
& ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ b @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
& ( member_nat_nat @ f
@ ( piE_nat_nat @ ( b @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ t ) ) )
& ( member952132173341509300at_nat
@ ( restri4446420529079022766at_nat
@ ^ [Y2: nat > nat] : ( l2 @ ( Y2 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ t ) )
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ t )
@ ^ [I2: nat > nat] : ( hales_cube @ n @ t ) ) )
& ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ t ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( b @ one_one_nat ) )
=> ( ( restri4446420529079022766at_nat
@ ^ [Y2: nat > nat] : ( l2 @ ( Y2 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ t )
@ X4
@ Xa )
= ( f @ Xa ) ) )
& ! [J4: nat] :
( ( ord_less_nat @ J4 @ one_one_nat )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( b @ J4 ) )
=> ( ( restri4446420529079022766at_nat
@ ^ [Y2: nat > nat] : ( l2 @ ( Y2 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ t )
@ X4
@ Xa )
= ( X4 @ J4 ) ) ) ) ) ) ) ).
% Bf_defs
thf(fact_690_some__inv__into__2,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P5 @ zero_zero_nat )
= S ) ) )
= ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ S ) ) ) ).
% some_inv_into_2
thf(fact_691_some__inv__into,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P5: nat > nat] :
( ( member_nat_nat @ P5 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P5 @ zero_zero_nat )
= S ) ) )
= ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ S ) ) ) ).
% some_inv_into
thf(fact_692_all__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ! [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_nat_set_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_693_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_694_all__subset__image,axiom,
! [F: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A2 )
=> ( P @ ( image_nat_nat_nat2 @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_695_all__subset__image,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: set_nat_nat > $o] :
( ( ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ A2 )
=> ( P @ ( image_3205354838064109189at_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_696_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D2: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ D2 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_697_empty__iff,axiom,
! [C: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ C @ bot_bo7445843802507891576at_nat ) ).
% empty_iff
thf(fact_698_empty__iff,axiom,
! [C: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ C @ bot_bo945813143650711160at_nat ) ).
% empty_iff
thf(fact_699_empty__iff,axiom,
! [C: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ C @ bot_bo3919185967433191911at_nat ) ).
% empty_iff
thf(fact_700_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_701_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_702_all__not__in__conv,axiom,
! [A2: set_nat_nat_nat] :
( ( ! [X3: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ X3 @ A2 ) )
= ( A2 = bot_bo7445843802507891576at_nat ) ) ).
% all_not_in_conv
thf(fact_703_all__not__in__conv,axiom,
! [A2: set_nat_nat_nat2] :
( ( ! [X3: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ X3 @ A2 ) )
= ( A2 = bot_bo945813143650711160at_nat ) ) ).
% all_not_in_conv
thf(fact_704_all__not__in__conv,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ( ! [X3: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ X3 @ A2 ) )
= ( A2 = bot_bo3919185967433191911at_nat ) ) ).
% all_not_in_conv
thf(fact_705_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_706_all__not__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ! [X3: nat > nat] :
~ ( member_nat_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_707_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_708_Collect__empty__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( ! [X3: nat > nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_709_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_710_empty__Collect__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P ) )
= ( ! [X3: nat > nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_711_image__is__empty,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( ( image_nat_set_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_712_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_713_image__is__empty,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( ( image_nat_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_714_image__is__empty,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( ( image_nat_nat_nat2 @ F @ A2 )
= bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_715_image__is__empty,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ( image_3205354838064109189at_nat @ F @ A2 )
= bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_716_empty__is__image,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_717_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_718_empty__is__image,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_719_empty__is__image,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( bot_bot_set_nat_nat
= ( image_nat_nat_nat2 @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_720_empty__is__image,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_721_image__empty,axiom,
! [F: nat > set_nat] :
( ( image_nat_set_nat @ F @ bot_bot_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_722_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_723_image__empty,axiom,
! [F: nat > nat > nat] :
( ( image_nat_nat_nat2 @ F @ bot_bot_set_nat )
= bot_bot_set_nat_nat ) ).
% image_empty
thf(fact_724_image__empty,axiom,
! [F: ( nat > nat ) > nat] :
( ( image_nat_nat_nat @ F @ bot_bot_set_nat_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_725_image__empty,axiom,
! [F: ( nat > nat ) > nat > nat] :
( ( image_3205354838064109189at_nat @ F @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% image_empty
thf(fact_726_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_727_empty__subsetI,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% empty_subsetI
thf(fact_728_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_729_subset__empty,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% subset_empty
thf(fact_730_Sup__atMost,axiom,
! [Y: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_731_PiE__empty__range,axiom,
! [I: nat > nat > nat,I5: set_nat_nat_nat,F3: ( nat > nat > nat ) > set_nat] :
( ( member_nat_nat_nat2 @ I @ I5 )
=> ( ( ( F3 @ I )
= bot_bot_set_nat )
=> ( ( piE_nat_nat_nat_nat2 @ I5 @ F3 )
= bot_bo3013702615682746855at_nat ) ) ) ).
% PiE_empty_range
thf(fact_732_PiE__empty__range,axiom,
! [I: ( nat > nat ) > nat,I5: set_nat_nat_nat2,F3: ( ( nat > nat ) > nat ) > set_nat] :
( ( member_nat_nat_nat @ I @ I5 )
=> ( ( ( F3 @ I )
= bot_bot_set_nat )
=> ( ( piE_nat_nat_nat_nat @ I5 @ F3 )
= bot_bo4508028030728203495at_nat ) ) ) ).
% PiE_empty_range
thf(fact_733_PiE__empty__range,axiom,
! [I: ( nat > nat ) > nat > nat,I5: set_nat_nat_nat_nat3,F3: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ( member952132173341509300at_nat @ I @ I5 )
=> ( ( ( F3 @ I )
= bot_bot_set_nat )
=> ( ( piE_na4548495224246695081at_nat @ I5 @ F3 )
= bot_bo3386126977483763158at_nat ) ) ) ).
% PiE_empty_range
thf(fact_734_PiE__empty__range,axiom,
! [I: nat > nat,I5: set_nat_nat,F3: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ I @ I5 )
=> ( ( ( F3 @ I )
= bot_bot_set_nat )
=> ( ( piE_nat_nat_nat @ I5 @ F3 )
= bot_bo945813143650711160at_nat ) ) ) ).
% PiE_empty_range
thf(fact_735_PiE__empty__range,axiom,
! [I: nat,I5: set_nat,F3: nat > set_nat] :
( ( member_nat @ I @ I5 )
=> ( ( ( F3 @ I )
= bot_bot_set_nat )
=> ( ( piE_nat_nat @ I5 @ F3 )
= bot_bot_set_nat_nat ) ) ) ).
% PiE_empty_range
thf(fact_736_PiE__empty__range,axiom,
! [I: nat > nat > nat,I5: set_nat_nat_nat,F3: ( nat > nat > nat ) > set_nat_nat] :
( ( member_nat_nat_nat2 @ I @ I5 )
=> ( ( ( F3 @ I )
= bot_bot_set_nat_nat )
=> ( ( piE_na7122919648973241129at_nat @ I5 @ F3 )
= bot_bo4227112084914574038at_nat ) ) ) ).
% PiE_empty_range
thf(fact_737_PiE__empty__range,axiom,
! [I: ( nat > nat ) > nat,I5: set_nat_nat_nat2,F3: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ( member_nat_nat_nat @ I @ I5 )
=> ( ( ( F3 @ I )
= bot_bot_set_nat_nat )
=> ( ( piE_na6840239867990089257at_nat @ I5 @ F3 )
= bot_bo4291610329234208214at_nat ) ) ) ).
% PiE_empty_range
thf(fact_738_PiE__empty__range,axiom,
! [I: ( nat > nat ) > nat > nat,I5: set_nat_nat_nat_nat3,F3: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ I @ I5 )
=> ( ( ( F3 @ I )
= bot_bot_set_nat_nat )
=> ( ( piE_na6564615839001774232at_nat @ I5 @ F3 )
= bot_bo3618716324728726597at_nat ) ) ) ).
% PiE_empty_range
thf(fact_739_PiE__empty__range,axiom,
! [I: nat,I5: set_nat,F3: nat > set_nat_nat] :
( ( member_nat @ I @ I5 )
=> ( ( ( F3 @ I )
= bot_bot_set_nat_nat )
=> ( ( piE_nat_nat_nat2 @ I5 @ F3 )
= bot_bo7445843802507891576at_nat ) ) ) ).
% PiE_empty_range
thf(fact_740_PiE__empty__range,axiom,
! [I: nat > nat,I5: set_nat_nat,F3: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ I @ I5 )
=> ( ( ( F3 @ I )
= bot_bot_set_nat_nat )
=> ( ( piE_nat_nat_nat_nat3 @ I5 @ F3 )
= bot_bo3919185967433191911at_nat ) ) ) ).
% PiE_empty_range
thf(fact_741_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_742_cSUP__const,axiom,
! [A2: set_nat,C: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_743_cSUP__const,axiom,
! [A2: set_nat_nat,C: set_nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [X3: nat > nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_744_cSUP__const,axiom,
! [A2: set_nat,C: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X3: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_745_cSUP__const,axiom,
! [A2: set_nat_nat,C: nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat_nat
@ ^ [X3: nat > nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_746_cSUP__least,axiom,
! [A2: set_nat,F: nat > nat,M3: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_747_cSUP__least,axiom,
! [A2: set_nat,F: nat > set_nat,M3: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ M3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_748_cSUP__least,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,M3: nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_749_cSUP__least,axiom,
! [A2: set_nat,F: nat > set_nat_nat,M3: set_nat_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ M3 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_750_cSUP__least,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat,M3: set_nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ M3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_751_cSUP__least,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat,M3: nat] :
( ( A2 != bot_bo7445843802507891576at_nat )
=> ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_913610194320715013at_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_752_cSUP__least,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,M3: nat] :
( ( A2 != bot_bo945813143650711160at_nat )
=> ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_7809927846809980933at_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_753_cSUP__least,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat_nat,M3: set_nat_nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ M3 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_754_cSUP__least,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat,M3: set_nat] :
( ( A2 != bot_bo7445843802507891576at_nat )
=> ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ M3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_755_cSUP__least,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat,M3: set_nat] :
( ( A2 != bot_bo945813143650711160at_nat )
=> ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ M3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_756_PiE__eq__empty__iff,axiom,
! [I5: set_nat_nat,F3: ( nat > nat ) > set_nat] :
( ( ( piE_nat_nat_nat @ I5 @ F3 )
= bot_bo945813143650711160at_nat )
= ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
& ( ( F3 @ X3 )
= bot_bot_set_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_757_PiE__eq__empty__iff,axiom,
! [I5: set_nat,F3: nat > set_nat_nat] :
( ( ( piE_nat_nat_nat2 @ I5 @ F3 )
= bot_bo7445843802507891576at_nat )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ I5 )
& ( ( F3 @ X3 )
= bot_bot_set_nat_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_758_PiE__eq__empty__iff,axiom,
! [I5: set_nat_nat,F3: ( nat > nat ) > set_nat_nat] :
( ( ( piE_nat_nat_nat_nat3 @ I5 @ F3 )
= bot_bo3919185967433191911at_nat )
= ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
& ( ( F3 @ X3 )
= bot_bot_set_nat_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_759_PiE__eq__empty__iff,axiom,
! [I5: set_nat,F3: nat > set_nat] :
( ( ( piE_nat_nat @ I5 @ F3 )
= bot_bot_set_nat_nat )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ I5 )
& ( ( F3 @ X3 )
= bot_bot_set_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_760_subset__emptyI,axiom,
! [A2: set_nat_nat_nat] :
( ! [X: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ X @ A2 )
=> ( ord_le3211623285424100676at_nat @ A2 @ bot_bo7445843802507891576at_nat ) ) ).
% subset_emptyI
thf(fact_761_subset__emptyI,axiom,
! [A2: set_nat_nat_nat2] :
( ! [X: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ X @ A2 )
=> ( ord_le5934964663421696068at_nat @ A2 @ bot_bo945813143650711160at_nat ) ) ).
% subset_emptyI
thf(fact_762_subset__emptyI,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ! [X: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ X @ A2 )
=> ( ord_le5260717879541182899at_nat @ A2 @ bot_bo3919185967433191911at_nat ) ) ).
% subset_emptyI
thf(fact_763_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_764_subset__emptyI,axiom,
! [A2: set_nat_nat] :
( ! [X: nat > nat] :
~ ( member_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat ) ) ).
% subset_emptyI
thf(fact_765_emptyE,axiom,
! [A: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ A @ bot_bo7445843802507891576at_nat ) ).
% emptyE
thf(fact_766_emptyE,axiom,
! [A: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ A @ bot_bo945813143650711160at_nat ) ).
% emptyE
thf(fact_767_emptyE,axiom,
! [A: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ A @ bot_bo3919185967433191911at_nat ) ).
% emptyE
thf(fact_768_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_769_emptyE,axiom,
! [A: nat > nat] :
~ ( member_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_770_equals0D,axiom,
! [A2: set_nat_nat_nat,A: nat > nat > nat] :
( ( A2 = bot_bo7445843802507891576at_nat )
=> ~ ( member_nat_nat_nat2 @ A @ A2 ) ) ).
% equals0D
thf(fact_771_equals0D,axiom,
! [A2: set_nat_nat_nat2,A: ( nat > nat ) > nat] :
( ( A2 = bot_bo945813143650711160at_nat )
=> ~ ( member_nat_nat_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_772_equals0D,axiom,
! [A2: set_nat_nat_nat_nat3,A: ( nat > nat ) > nat > nat] :
( ( A2 = bot_bo3919185967433191911at_nat )
=> ~ ( member952132173341509300at_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_773_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_774_equals0D,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( A2 = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_775_equals0I,axiom,
! [A2: set_nat_nat_nat] :
( ! [Y4: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ Y4 @ A2 )
=> ( A2 = bot_bo7445843802507891576at_nat ) ) ).
% equals0I
thf(fact_776_equals0I,axiom,
! [A2: set_nat_nat_nat2] :
( ! [Y4: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ Y4 @ A2 )
=> ( A2 = bot_bo945813143650711160at_nat ) ) ).
% equals0I
thf(fact_777_equals0I,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ! [Y4: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ Y4 @ A2 )
=> ( A2 = bot_bo3919185967433191911at_nat ) ) ).
% equals0I
thf(fact_778_equals0I,axiom,
! [A2: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_779_equals0I,axiom,
! [A2: set_nat_nat] :
( ! [Y4: nat > nat] :
~ ( member_nat_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_780_ex__in__conv,axiom,
! [A2: set_nat_nat_nat] :
( ( ? [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ A2 ) )
= ( A2 != bot_bo7445843802507891576at_nat ) ) ).
% ex_in_conv
thf(fact_781_ex__in__conv,axiom,
! [A2: set_nat_nat_nat2] :
( ( ? [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ A2 ) )
= ( A2 != bot_bo945813143650711160at_nat ) ) ).
% ex_in_conv
thf(fact_782_ex__in__conv,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ( ? [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A2 ) )
= ( A2 != bot_bo3919185967433191911at_nat ) ) ).
% ex_in_conv
thf(fact_783_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_784_ex__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ? [X3: nat > nat] : ( member_nat_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_785_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_786_empty__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat
@ ^ [X3: nat > nat] : $false ) ) ).
% empty_def
thf(fact_787_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_788_less__cSupE,axiom,
! [Y: nat,X5: set_nat] :
( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X5 ) )
=> ( ( X5 != bot_bot_set_nat )
=> ~ ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ~ ( ord_less_nat @ Y @ X ) ) ) ) ).
% less_cSupE
thf(fact_789_less__cSupD,axiom,
! [X5: set_nat,Z3: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z3 @ ( complete_Sup_Sup_nat @ X5 ) )
=> ? [X: nat] :
( ( member_nat @ X @ X5 )
& ( ord_less_nat @ Z3 @ X ) ) ) ) ).
% less_cSupD
thf(fact_790_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_791_bot_Oextremum__strict,axiom,
! [A: set_nat_nat] :
~ ( ord_less_set_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% bot.extremum_strict
thf(fact_792_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_793_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_794_bot_Onot__eq__extremum,axiom,
! [A: set_nat_nat] :
( ( A != bot_bot_set_nat_nat )
= ( ord_less_set_nat_nat @ bot_bot_set_nat_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_795_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_796_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_797_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_798_bot_Oextremum,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% bot.extremum
thf(fact_799_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_800_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_801_bot_Oextremum__unique,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% bot.extremum_unique
thf(fact_802_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_803_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_804_bot_Oextremum__uniqueI,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
=> ( A = bot_bot_set_nat_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_805_cSup__eq__non__empty,axiom,
! [X5: set_set_nat_nat,A: set_nat_nat] :
( ( X5 != bot_bo7376149671870096959at_nat )
=> ( ! [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ X5 )
=> ( ord_le9059583361652607317at_nat @ X @ A ) )
=> ( ! [Y4: set_nat_nat] :
( ! [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ X5 )
=> ( ord_le9059583361652607317at_nat @ X4 @ Y4 ) )
=> ( ord_le9059583361652607317at_nat @ A @ Y4 ) )
=> ( ( comple5448282615319421384at_nat @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_806_cSup__eq__non__empty,axiom,
! [X5: set_set_nat,A: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X5 )
=> ( ord_less_eq_set_nat @ X @ A ) )
=> ( ! [Y4: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ Y4 ) )
=> ( ord_less_eq_set_nat @ A @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_807_cSup__eq__non__empty,axiom,
! [X5: set_nat,A: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ X @ A ) )
=> ( ! [Y4: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Y4 ) )
=> ( ord_less_eq_nat @ A @ Y4 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_808_cSup__eq__maximum,axiom,
! [Z3: set_nat_nat,X5: set_set_nat_nat] :
( ( member_set_nat_nat @ Z3 @ X5 )
=> ( ! [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ X5 )
=> ( ord_le9059583361652607317at_nat @ X @ Z3 ) )
=> ( ( comple5448282615319421384at_nat @ X5 )
= Z3 ) ) ) ).
% cSup_eq_maximum
thf(fact_809_cSup__eq__maximum,axiom,
! [Z3: set_nat,X5: set_set_nat] :
( ( member_set_nat @ Z3 @ X5 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X5 )
=> ( ord_less_eq_set_nat @ X @ Z3 ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= Z3 ) ) ) ).
% cSup_eq_maximum
thf(fact_810_cSup__eq__maximum,axiom,
! [Z3: nat,X5: set_nat] :
( ( member_nat @ Z3 @ X5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z3 ) ) ) ).
% cSup_eq_maximum
thf(fact_811_cSup__least,axiom,
! [X5: set_set_nat_nat,Z3: set_nat_nat] :
( ( X5 != bot_bo7376149671870096959at_nat )
=> ( ! [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ X5 )
=> ( ord_le9059583361652607317at_nat @ X @ Z3 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ X5 ) @ Z3 ) ) ) ).
% cSup_least
thf(fact_812_cSup__least,axiom,
! [X5: set_set_nat,Z3: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X5 )
=> ( ord_less_eq_set_nat @ X @ Z3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X5 ) @ Z3 ) ) ) ).
% cSup_least
thf(fact_813_cSup__least,axiom,
! [X5: set_nat,Z3: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z3 ) ) ) ).
% cSup_least
thf(fact_814_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_815_not__psubset__empty,axiom,
! [A2: set_nat_nat] :
~ ( ord_less_set_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% not_psubset_empty
thf(fact_816_not__empty__eq__Iic__eq__empty,axiom,
! [H: nat > nat] :
( bot_bot_set_nat_nat
!= ( set_or9140604705432621368at_nat @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_817_not__empty__eq__Iic__eq__empty,axiom,
! [H: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_818_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat_nat,F3: ( nat > nat > nat ) > set_nat,F4: ( nat > nat > nat ) > set_nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat_nat2 @ I5 @ F3 )
= ( piE_nat_nat_nat_nat2 @ I5 @ F4 ) )
= ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_819_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat_nat2,F3: ( ( nat > nat ) > nat ) > set_nat,F4: ( ( nat > nat ) > nat ) > set_nat] :
( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat_nat @ I5 @ F3 )
= ( piE_nat_nat_nat_nat @ I5 @ F4 ) )
= ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_820_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat_nat_nat3,F3: ( ( nat > nat ) > nat > nat ) > set_nat,F4: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_na4548495224246695081at_nat @ I5 @ F3 )
= ( piE_na4548495224246695081at_nat @ I5 @ F4 ) )
= ( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_821_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat,F3: nat > set_nat,F4: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat @ I5 @ F3 )
= ( piE_nat_nat @ I5 @ F4 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_822_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat,F3: ( nat > nat ) > set_nat,F4: ( nat > nat ) > set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat @ I5 @ F3 )
= ( piE_nat_nat_nat @ I5 @ F4 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_823_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat_nat,F3: ( nat > nat > nat ) > set_nat_nat,F4: ( nat > nat > nat ) > set_nat_nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na7122919648973241129at_nat @ I5 @ F3 )
= ( piE_na7122919648973241129at_nat @ I5 @ F4 ) )
= ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_824_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat_nat2,F3: ( ( nat > nat ) > nat ) > set_nat_nat,F4: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na6840239867990089257at_nat @ I5 @ F3 )
= ( piE_na6840239867990089257at_nat @ I5 @ F4 ) )
= ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_825_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat_nat_nat3,F3: ( ( nat > nat ) > nat > nat ) > set_nat_nat,F4: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na6564615839001774232at_nat @ I5 @ F3 )
= ( piE_na6564615839001774232at_nat @ I5 @ F4 ) )
= ( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_826_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat,F3: nat > set_nat_nat,F4: nat > set_nat_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat2 @ I5 @ F3 )
= ( piE_nat_nat_nat2 @ I5 @ F4 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_827_PiE__eq__iff__not__empty,axiom,
! [I5: set_nat_nat,F3: ( nat > nat ) > set_nat_nat,F4: ( nat > nat ) > set_nat_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat_nat3 @ I5 @ F3 )
= ( piE_nat_nat_nat_nat3 @ I5 @ F4 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_828_PiE__eq__iff,axiom,
! [I5: set_nat,F3: nat > set_nat,F4: nat > set_nat] :
( ( ( piE_nat_nat @ I5 @ F3 )
= ( piE_nat_nat @ I5 @ F4 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) )
| ( ? [X3: nat] :
( ( member_nat @ X3 @ I5 )
& ( ( F3 @ X3 )
= bot_bot_set_nat ) )
& ? [X3: nat] :
( ( member_nat @ X3 @ I5 )
& ( ( F4 @ X3 )
= bot_bot_set_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_829_PiE__eq__iff,axiom,
! [I5: set_nat_nat,F3: ( nat > nat ) > set_nat,F4: ( nat > nat ) > set_nat] :
( ( ( piE_nat_nat_nat @ I5 @ F3 )
= ( piE_nat_nat_nat @ I5 @ F4 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) )
| ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
& ( ( F3 @ X3 )
= bot_bot_set_nat ) )
& ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
& ( ( F4 @ X3 )
= bot_bot_set_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_830_PiE__eq__iff,axiom,
! [I5: set_nat,F3: nat > set_nat_nat,F4: nat > set_nat_nat] :
( ( ( piE_nat_nat_nat2 @ I5 @ F3 )
= ( piE_nat_nat_nat2 @ I5 @ F4 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) )
| ( ? [X3: nat] :
( ( member_nat @ X3 @ I5 )
& ( ( F3 @ X3 )
= bot_bot_set_nat_nat ) )
& ? [X3: nat] :
( ( member_nat @ X3 @ I5 )
& ( ( F4 @ X3 )
= bot_bot_set_nat_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_831_PiE__eq__iff,axiom,
! [I5: set_nat_nat,F3: ( nat > nat ) > set_nat_nat,F4: ( nat > nat ) > set_nat_nat] :
( ( ( piE_nat_nat_nat_nat3 @ I5 @ F3 )
= ( piE_nat_nat_nat_nat3 @ I5 @ F4 ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ( ( F3 @ X3 )
= ( F4 @ X3 ) ) )
| ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
& ( ( F3 @ X3 )
= bot_bot_set_nat_nat ) )
& ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
& ( ( F4 @ X3 )
= bot_bot_set_nat_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_832_some__in__eq,axiom,
! [A2: set_nat_nat_nat] :
( ( member_nat_nat_nat2
@ ( fChoice_nat_nat_nat2
@ ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ A2 ) )
@ A2 )
= ( A2 != bot_bo7445843802507891576at_nat ) ) ).
% some_in_eq
thf(fact_833_some__in__eq,axiom,
! [A2: set_nat_nat_nat2] :
( ( member_nat_nat_nat
@ ( fChoice_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ A2 ) )
@ A2 )
= ( A2 != bot_bo945813143650711160at_nat ) ) ).
% some_in_eq
thf(fact_834_some__in__eq,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat
@ ( fChoic52552927678224201at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A2 ) )
@ A2 )
= ( A2 != bot_bo3919185967433191911at_nat ) ) ).
% some_in_eq
thf(fact_835_some__in__eq,axiom,
! [A2: set_nat] :
( ( member_nat
@ ( fChoice_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_nat ) ) ).
% some_in_eq
thf(fact_836_some__in__eq,axiom,
! [A2: set_nat_nat] :
( ( member_nat_nat
@ ( fChoice_nat_nat
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_nat_nat ) ) ).
% some_in_eq
thf(fact_837_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_838_PiE__eq__subset,axiom,
! [I5: set_nat_nat_nat,F3: ( nat > nat > nat ) > set_nat,F4: ( nat > nat > nat ) > set_nat,I: nat > nat > nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat_nat2 @ I5 @ F3 )
= ( piE_nat_nat_nat_nat2 @ I5 @ F4 ) )
=> ( ( member_nat_nat_nat2 @ I @ I5 )
=> ( ord_less_eq_set_nat @ ( F3 @ I ) @ ( F4 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_839_PiE__eq__subset,axiom,
! [I5: set_nat_nat_nat2,F3: ( ( nat > nat ) > nat ) > set_nat,F4: ( ( nat > nat ) > nat ) > set_nat,I: ( nat > nat ) > nat] :
( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat_nat @ I5 @ F3 )
= ( piE_nat_nat_nat_nat @ I5 @ F4 ) )
=> ( ( member_nat_nat_nat @ I @ I5 )
=> ( ord_less_eq_set_nat @ ( F3 @ I ) @ ( F4 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_840_PiE__eq__subset,axiom,
! [I5: set_nat_nat_nat_nat3,F3: ( ( nat > nat ) > nat > nat ) > set_nat,F4: ( ( nat > nat ) > nat > nat ) > set_nat,I: ( nat > nat ) > nat > nat] :
( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_na4548495224246695081at_nat @ I5 @ F3 )
= ( piE_na4548495224246695081at_nat @ I5 @ F4 ) )
=> ( ( member952132173341509300at_nat @ I @ I5 )
=> ( ord_less_eq_set_nat @ ( F3 @ I ) @ ( F4 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_841_PiE__eq__subset,axiom,
! [I5: set_nat,F3: nat > set_nat,F4: nat > set_nat,I: nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat @ I5 @ F3 )
= ( piE_nat_nat @ I5 @ F4 ) )
=> ( ( member_nat @ I @ I5 )
=> ( ord_less_eq_set_nat @ ( F3 @ I ) @ ( F4 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_842_PiE__eq__subset,axiom,
! [I5: set_nat_nat,F3: ( nat > nat ) > set_nat,F4: ( nat > nat ) > set_nat,I: nat > nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat @ I5 @ F3 )
= ( piE_nat_nat_nat @ I5 @ F4 ) )
=> ( ( member_nat_nat @ I @ I5 )
=> ( ord_less_eq_set_nat @ ( F3 @ I ) @ ( F4 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_843_PiE__eq__subset,axiom,
! [I5: set_nat_nat_nat,F3: ( nat > nat > nat ) > set_nat_nat,F4: ( nat > nat > nat ) > set_nat_nat,I: nat > nat > nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na7122919648973241129at_nat @ I5 @ F3 )
= ( piE_na7122919648973241129at_nat @ I5 @ F4 ) )
=> ( ( member_nat_nat_nat2 @ I @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( F3 @ I ) @ ( F4 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_844_PiE__eq__subset,axiom,
! [I5: set_nat_nat_nat2,F3: ( ( nat > nat ) > nat ) > set_nat_nat,F4: ( ( nat > nat ) > nat ) > set_nat_nat,I: ( nat > nat ) > nat] :
( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na6840239867990089257at_nat @ I5 @ F3 )
= ( piE_na6840239867990089257at_nat @ I5 @ F4 ) )
=> ( ( member_nat_nat_nat @ I @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( F3 @ I ) @ ( F4 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_845_PiE__eq__subset,axiom,
! [I5: set_nat_nat_nat_nat3,F3: ( ( nat > nat ) > nat > nat ) > set_nat_nat,F4: ( ( nat > nat ) > nat > nat ) > set_nat_nat,I: ( nat > nat ) > nat > nat] :
( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na6564615839001774232at_nat @ I5 @ F3 )
= ( piE_na6564615839001774232at_nat @ I5 @ F4 ) )
=> ( ( member952132173341509300at_nat @ I @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( F3 @ I ) @ ( F4 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_846_PiE__eq__subset,axiom,
! [I5: set_nat,F3: nat > set_nat_nat,F4: nat > set_nat_nat,I: nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat2 @ I5 @ F3 )
= ( piE_nat_nat_nat2 @ I5 @ F4 ) )
=> ( ( member_nat @ I @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( F3 @ I ) @ ( F4 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_847_PiE__eq__subset,axiom,
! [I5: set_nat_nat,F3: ( nat > nat ) > set_nat_nat,F4: ( nat > nat ) > set_nat_nat,I: nat > nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat_nat3 @ I5 @ F3 )
= ( piE_nat_nat_nat_nat3 @ I5 @ F4 ) )
=> ( ( member_nat_nat @ I @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( F3 @ I ) @ ( F4 @ I ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_848_is__subspace__def,axiom,
( hales_is_subspace
= ( ^ [S6: ( nat > nat ) > nat > nat,K3: nat,N3: nat,T2: nat] :
? [B6: nat > set_nat] :
( ( disjoi6798895846410478970at_nat @ B6 @ ( set_ord_atMost_nat @ K3 ) )
& ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B6 @ ( set_ord_atMost_nat @ K3 ) ) )
= ( set_ord_lessThan_nat @ N3 ) )
& ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B6 @ ( set_ord_lessThan_nat @ K3 ) ) )
& ? [F2: nat > nat] :
( ( member_nat_nat @ F2
@ ( piE_nat_nat @ ( B6 @ K3 )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T2 ) ) )
& ( member952132173341509300at_nat @ S6
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ K3 @ T2 )
@ ^ [I2: nat > nat] : ( hales_cube @ N3 @ T2 ) ) )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ K3 @ T2 ) )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B6 @ K3 ) )
=> ( ( S6 @ X3 @ Y2 )
= ( F2 @ Y2 ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ K3 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B6 @ J3 ) )
=> ( ( S6 @ X3 @ Y2 )
= ( X3 @ J3 ) ) ) ) ) ) ) ) ) ) ).
% is_subspace_def
thf(fact_849_dim1__subspace__elims_I3_J,axiom,
! [B3: nat > set_nat,N: nat,F: nat > nat,T: nat,S5: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B3 @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S5
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ T )
@ ^ [I2: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ one_one_nat ) )
=> ( ( S5 @ X @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ J ) )
=> ( ( S5 @ X @ Xa2 )
= ( X @ J ) ) ) ) ) )
=> ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( B3 @ one_one_nat ) )
=> ( ( S5 @ X4 @ Xa )
= ( F @ Xa ) ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( B3 @ zero_zero_nat ) )
=> ( ( S5 @ X4 @ Xa )
= ( X4 @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ).
% dim1_subspace_elims(3)
thf(fact_850_dim1__subspace__elims_I4_J,axiom,
! [B3: nat > set_nat,N: nat,F: nat > nat,T: nat,S5: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B3 @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S5
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ T )
@ ^ [I2: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ one_one_nat ) )
=> ( ( S5 @ X @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ J ) )
=> ( ( S5 @ X @ Xa2 )
= ( X @ J ) ) ) ) ) )
=> ( ( B3 @ zero_zero_nat )
!= bot_bot_set_nat ) ) ) ) ) ) ) ).
% dim1_subspace_elims(4)
thf(fact_851_inv__into__cube__props_I1_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( member_nat_nat
@ ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ S )
@ ( hales_cube @ one_one_nat @ T ) ) ) ).
% inv_into_cube_props(1)
thf(fact_852_inv__into__cube__props_I2_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ S
@ zero_zero_nat )
= S ) ) ).
% inv_into_cube_props(2)
thf(fact_853_SUP__apply,axiom,
! [F: nat > nat > nat,A2: set_nat,X2: nat] :
( ( comple2450677804321093138at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ X2 )
= ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [Y2: nat] : ( F @ Y2 @ X2 )
@ A2 ) ) ) ).
% SUP_apply
thf(fact_854_SUP__apply,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,X2: nat] :
( ( comple2450677804321093138at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ X2 )
= ( complete_Sup_Sup_nat
@ ( image_nat_nat_nat
@ ^ [Y2: nat > nat] : ( F @ Y2 @ X2 )
@ A2 ) ) ) ).
% SUP_apply
thf(fact_855_UN__constant,axiom,
! [A2: set_nat,C: set_nat_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple5448282615319421384at_nat
@ ( image_7301343469591561292at_nat
@ ^ [Y2: nat] : C
@ A2 ) )
= bot_bot_set_nat_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple5448282615319421384at_nat
@ ( image_7301343469591561292at_nat
@ ^ [Y2: nat] : C
@ A2 ) )
= C ) ) ) ).
% UN_constant
thf(fact_856_UN__constant,axiom,
! [A2: set_nat_nat,C: set_nat_nat] :
( ( ( A2 = bot_bot_set_nat_nat )
=> ( ( comple5448282615319421384at_nat
@ ( image_6905811865970898491at_nat
@ ^ [Y2: nat > nat] : C
@ A2 ) )
= bot_bot_set_nat_nat ) )
& ( ( A2 != bot_bot_set_nat_nat )
=> ( ( comple5448282615319421384at_nat
@ ( image_6905811865970898491at_nat
@ ^ [Y2: nat > nat] : C
@ A2 ) )
= C ) ) ) ).
% UN_constant
thf(fact_857_UN__constant,axiom,
! [A2: set_nat,C: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C
@ A2 ) )
= C ) ) ) ).
% UN_constant
thf(fact_858_UN__constant,axiom,
! [A2: set_nat_nat,C: set_nat] :
( ( ( A2 = bot_bot_set_nat_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [Y2: nat > nat] : C
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [Y2: nat > nat] : C
@ A2 ) )
= C ) ) ) ).
% UN_constant
thf(fact_859_SUP__const,axiom,
! [A2: set_nat,F: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_860_SUP__const,axiom,
! [A2: set_nat_nat,F: set_nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [I2: nat > nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_861_SUP__bot__conv_I2_J,axiom,
! [B3: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B3 @ X3 )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_862_ball__UN,axiom,
! [B3: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B3 @ X3 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_863_bex__UN,axiom,
! [B3: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( B3 @ X3 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_864_UN__ball__bex__simps_I2_J,axiom,
! [B3: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B3 @ X3 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_865_UN__ball__bex__simps_I4_J,axiom,
! [B3: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( B3 @ X3 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_866_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_867_SUP__identity__eq,axiom,
! [A2: set_nat_nat] :
( ( comple2450677804321093138at_nat
@ ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : X3
@ A2 ) )
= ( comple2450677804321093138at_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_868_SUP__identity__eq,axiom,
! [A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X3: set_nat] : X3
@ A2 ) )
= ( comple7399068483239264473et_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_869_SUP__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_870_UN__iff,axiom,
! [B: nat,B3: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( member_nat @ B @ ( B3 @ X3 ) ) ) ) ) ).
% UN_iff
thf(fact_871_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat,B3: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B3 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_872_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat > nat,B3: nat > set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ ( B3 @ A ) )
=> ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_873_UN__I,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat,B3: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B3 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_874_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat > nat > nat,B3: nat > set_nat_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ ( B3 @ A ) )
=> ( member_nat_nat_nat2 @ B @ ( comple8167887107183641911at_nat @ ( image_6130888460295934395at_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_875_UN__I,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat,B3: nat > set_nat_nat_nat2] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat @ B @ ( B3 @ A ) )
=> ( member_nat_nat_nat @ B @ ( comple1667856448326461495at_nat @ ( image_8854229838293529787at_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_876_UN__I,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat,B3: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ ( B3 @ A ) )
=> ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_877_UN__I,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B: nat,B3: ( nat > nat > nat ) > set_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( member_nat @ B @ ( B3 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_878_UN__I,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,B3: ( ( nat > nat ) > nat ) > set_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B3 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_879_UN__I,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat > nat,B3: nat > set_nat_nat_nat_nat3] :
( ( member_nat @ A @ A2 )
=> ( ( member952132173341509300at_nat @ B @ ( B3 @ A ) )
=> ( member952132173341509300at_nat @ B @ ( comple2605510978757769510at_nat @ ( image_3332361743537024938at_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_880_UN__I,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat > nat,B3: ( nat > nat ) > set_nat_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ ( B3 @ A ) )
=> ( member_nat_nat_nat2 @ B @ ( comple8167887107183641911at_nat @ ( image_470123710477037866at_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_881_SUP__bot,axiom,
! [A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : bot_bot_set_nat
@ A2 ) )
= bot_bot_set_nat ) ).
% SUP_bot
thf(fact_882_SUP__bot__conv_I1_J,axiom,
! [B3: nat > set_nat,A2: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B3 @ X3 )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_883_Sup__SUP__eq,axiom,
( comple8312177224774716605_nat_o
= ( ^ [S6: set_nat_nat_o,X3: nat > nat] : ( member_nat_nat @ X3 @ ( comple5448282615319421384at_nat @ ( image_7977807581451749376at_nat @ collect_nat_nat @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_884_Sup__SUP__eq,axiom,
( comple3396693796109600270_nat_o
= ( ^ [S6: set_nat_nat_nat_o,X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ ( comple8167887107183641911at_nat @ ( image_3610001086604609088at_nat @ collect_nat_nat_nat2 @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_885_Sup__SUP__eq,axiom,
( comple8231226574009213710_nat_o
= ( ^ [S6: set_nat_nat_nat_o2,X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ ( comple1667856448326461495at_nat @ ( image_5425260358592644672at_nat @ collect_nat_nat_nat @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_886_Sup__SUP__eq,axiom,
( comple2115216063353097951_nat_o
= ( ^ [S6: set_na2445831480116662482_nat_o,X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ ( comple2605510978757769510at_nat @ ( image_4065302347126311296at_nat @ collec3567154360959927026at_nat @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_887_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S6: set_nat_o,X3: nat] : ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_888_Sup__set__def,axiom,
( comple5448282615319421384at_nat
= ( ^ [A6: set_set_nat_nat] :
( collect_nat_nat
@ ^ [X3: nat > nat] : ( complete_Sup_Sup_o @ ( image_set_nat_nat_o @ ( member_nat_nat @ X3 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_889_Sup__set__def,axiom,
( comple8167887107183641911at_nat
= ( ^ [A6: set_set_nat_nat_nat] :
( collect_nat_nat_nat2
@ ^ [X3: nat > nat > nat] : ( complete_Sup_Sup_o @ ( image_5198217506544545261_nat_o @ ( member_nat_nat_nat2 @ X3 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_890_Sup__set__def,axiom,
( comple1667856448326461495at_nat
= ( ^ [A6: set_set_nat_nat_nat2] :
( collect_nat_nat_nat
@ ^ [X3: ( nat > nat ) > nat] : ( complete_Sup_Sup_o @ ( image_8774134582277556973_nat_o @ ( member_nat_nat_nat @ X3 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_891_Sup__set__def,axiom,
( comple2605510978757769510at_nat
= ( ^ [A6: set_se3022870823424313865at_nat] :
( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] : ( complete_Sup_Sup_o @ ( image_7580978635682194622_nat_o @ ( member952132173341509300at_nat @ X3 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_892_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A6: set_set_nat] :
( collect_nat
@ ^ [X3: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X3 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_893_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_894_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_895_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_896_all__PiE__elements,axiom,
! [I5: set_nat,S5: nat > set_nat_nat,P: nat > ( nat > nat ) > $o] :
( ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ ( piE_nat_nat_nat2 @ I5 @ S5 ) )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ I5 )
=> ( P @ Y2 @ ( X3 @ Y2 ) ) ) ) )
= ( ( ( piE_nat_nat_nat2 @ I5 @ S5 )
= bot_bo7445843802507891576at_nat )
| ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ! [Y2: nat > nat] :
( ( member_nat_nat @ Y2 @ ( S5 @ X3 ) )
=> ( P @ X3 @ Y2 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_897_all__PiE__elements,axiom,
! [I5: set_nat_nat,S5: ( nat > nat ) > set_nat,P: ( nat > nat ) > nat > $o] :
( ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ ( piE_nat_nat_nat @ I5 @ S5 ) )
=> ! [Y2: nat > nat] :
( ( member_nat_nat @ Y2 @ I5 )
=> ( P @ Y2 @ ( X3 @ Y2 ) ) ) ) )
= ( ( ( piE_nat_nat_nat @ I5 @ S5 )
= bot_bo945813143650711160at_nat )
| ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( S5 @ X3 ) )
=> ( P @ X3 @ Y2 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_898_all__PiE__elements,axiom,
! [I5: set_nat_nat,S5: ( nat > nat ) > set_nat_nat,P: ( nat > nat ) > ( nat > nat ) > $o] :
( ( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ ( piE_nat_nat_nat_nat3 @ I5 @ S5 ) )
=> ! [Y2: nat > nat] :
( ( member_nat_nat @ Y2 @ I5 )
=> ( P @ Y2 @ ( X3 @ Y2 ) ) ) ) )
= ( ( ( piE_nat_nat_nat_nat3 @ I5 @ S5 )
= bot_bo3919185967433191911at_nat )
| ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ! [Y2: nat > nat] :
( ( member_nat_nat @ Y2 @ ( S5 @ X3 ) )
=> ( P @ X3 @ Y2 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_899_all__PiE__elements,axiom,
! [I5: set_nat,S5: nat > set_nat,P: nat > nat > $o] :
( ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( piE_nat_nat @ I5 @ S5 ) )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ I5 )
=> ( P @ Y2 @ ( X3 @ Y2 ) ) ) ) )
= ( ( ( piE_nat_nat @ I5 @ S5 )
= bot_bot_set_nat_nat )
| ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( S5 @ X3 ) )
=> ( P @ X3 @ Y2 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_900_PiE__eq,axiom,
! [I5: set_nat,S5: nat > set_nat_nat,T3: nat > set_nat_nat] :
( ( ( piE_nat_nat_nat2 @ I5 @ S5 )
= ( piE_nat_nat_nat2 @ I5 @ T3 ) )
= ( ( ( ( piE_nat_nat_nat2 @ I5 @ S5 )
= bot_bo7445843802507891576at_nat )
& ( ( piE_nat_nat_nat2 @ I5 @ T3 )
= bot_bo7445843802507891576at_nat ) )
| ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ( S5 @ X3 )
= ( T3 @ X3 ) ) ) ) ) ).
% PiE_eq
thf(fact_901_PiE__eq,axiom,
! [I5: set_nat_nat,S5: ( nat > nat ) > set_nat,T3: ( nat > nat ) > set_nat] :
( ( ( piE_nat_nat_nat @ I5 @ S5 )
= ( piE_nat_nat_nat @ I5 @ T3 ) )
= ( ( ( ( piE_nat_nat_nat @ I5 @ S5 )
= bot_bo945813143650711160at_nat )
& ( ( piE_nat_nat_nat @ I5 @ T3 )
= bot_bo945813143650711160at_nat ) )
| ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ( ( S5 @ X3 )
= ( T3 @ X3 ) ) ) ) ) ).
% PiE_eq
thf(fact_902_PiE__eq,axiom,
! [I5: set_nat_nat,S5: ( nat > nat ) > set_nat_nat,T3: ( nat > nat ) > set_nat_nat] :
( ( ( piE_nat_nat_nat_nat3 @ I5 @ S5 )
= ( piE_nat_nat_nat_nat3 @ I5 @ T3 ) )
= ( ( ( ( piE_nat_nat_nat_nat3 @ I5 @ S5 )
= bot_bo3919185967433191911at_nat )
& ( ( piE_nat_nat_nat_nat3 @ I5 @ T3 )
= bot_bo3919185967433191911at_nat ) )
| ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ( ( S5 @ X3 )
= ( T3 @ X3 ) ) ) ) ) ).
% PiE_eq
thf(fact_903_PiE__eq,axiom,
! [I5: set_nat,S5: nat > set_nat,T3: nat > set_nat] :
( ( ( piE_nat_nat @ I5 @ S5 )
= ( piE_nat_nat @ I5 @ T3 ) )
= ( ( ( ( piE_nat_nat @ I5 @ S5 )
= bot_bot_set_nat_nat )
& ( ( piE_nat_nat @ I5 @ T3 )
= bot_bot_set_nat_nat ) )
| ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ( S5 @ X3 )
= ( T3 @ X3 ) ) ) ) ) ).
% PiE_eq
thf(fact_904_SUP__Sup__eq,axiom,
! [S5: set_set_nat_nat] :
( ( comple8312177224774716605_nat_o
@ ( image_1242417779249009364_nat_o
@ ^ [I2: set_nat_nat,X3: nat > nat] : ( member_nat_nat @ X3 @ I2 )
@ S5 ) )
= ( ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ ( comple5448282615319421384at_nat @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_905_SUP__Sup__eq,axiom,
! [S5: set_set_nat_nat_nat] :
( ( comple3396693796109600270_nat_o
@ ( image_2840114971476761718_nat_o
@ ^ [I2: set_nat_nat_nat,X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ I2 )
@ S5 ) )
= ( ^ [X3: nat > nat > nat] : ( member_nat_nat_nat2 @ X3 @ ( comple8167887107183641911at_nat @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_906_SUP__Sup__eq,axiom,
! [S5: set_set_nat_nat_nat2] :
( ( comple8231226574009213710_nat_o
@ ( image_6357918107393578614_nat_o
@ ^ [I2: set_nat_nat_nat2,X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ I2 )
@ S5 ) )
= ( ^ [X3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X3 @ ( comple1667856448326461495at_nat @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_907_SUP__Sup__eq,axiom,
! [S5: set_se3022870823424313865at_nat] :
( ( comple2115216063353097951_nat_o
@ ( image_4040409651686222360_nat_o
@ ^ [I2: set_nat_nat_nat_nat3,X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ I2 )
@ S5 ) )
= ( ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ ( comple2605510978757769510at_nat @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_908_SUP__Sup__eq,axiom,
! [S5: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o2
@ ^ [I2: set_nat,X3: nat] : ( member_nat @ X3 @ I2 )
@ S5 ) )
= ( ^ [X3: nat] : ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_909_SUP__UN__eq,axiom,
! [R3: nat > set_nat,S5: set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_nat_nat_o
@ ^ [I2: nat,X3: nat] : ( member_nat @ X3 @ ( R3 @ I2 ) )
@ S5 ) )
= ( ^ [X3: nat] : ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ R3 @ S5 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_910_Inf_OINF__cong,axiom,
! [A2: set_nat,B3: set_nat,C4: nat > nat > nat,D3: nat > nat > nat,Inf: set_nat_nat > nat > nat] :
( ( A2 = B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( Inf @ ( image_nat_nat_nat2 @ C4 @ A2 ) )
= ( Inf @ ( image_nat_nat_nat2 @ D3 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_911_Inf_OINF__cong,axiom,
! [A2: set_nat,B3: set_nat,C4: nat > set_nat,D3: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C4 @ A2 ) )
= ( Inf @ ( image_nat_set_nat @ D3 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_912_Inf_OINF__cong,axiom,
! [A2: set_nat,B3: set_nat,C4: nat > nat,D3: nat > nat,Inf: set_nat > nat] :
( ( A2 = B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( Inf @ ( image_nat_nat @ C4 @ A2 ) )
= ( Inf @ ( image_nat_nat @ D3 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_913_Inf_OINF__cong,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: ( nat > nat ) > nat > nat,D3: ( nat > nat ) > nat > nat,Inf: set_nat_nat > nat > nat] :
( ( A2 = B3 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( Inf @ ( image_3205354838064109189at_nat @ C4 @ A2 ) )
= ( Inf @ ( image_3205354838064109189at_nat @ D3 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_914_Sup_OSUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C4: nat > nat > nat,D3: nat > nat > nat,Sup: set_nat_nat > nat > nat] :
( ( A2 = B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( Sup @ ( image_nat_nat_nat2 @ C4 @ A2 ) )
= ( Sup @ ( image_nat_nat_nat2 @ D3 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_915_Sup_OSUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C4: nat > set_nat,D3: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C4 @ A2 ) )
= ( Sup @ ( image_nat_set_nat @ D3 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_916_Sup_OSUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C4: nat > nat,D3: nat > nat,Sup: set_nat > nat] :
( ( A2 = B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( Sup @ ( image_nat_nat @ C4 @ A2 ) )
= ( Sup @ ( image_nat_nat @ D3 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_917_Sup_OSUP__cong,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: ( nat > nat ) > nat > nat,D3: ( nat > nat ) > nat > nat,Sup: set_nat_nat > nat > nat] :
( ( A2 = B3 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( Sup @ ( image_3205354838064109189at_nat @ C4 @ A2 ) )
= ( Sup @ ( image_3205354838064109189at_nat @ D3 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_918_subset__PiE,axiom,
! [I5: set_nat_nat,S5: ( nat > nat ) > set_nat,T3: ( nat > nat ) > set_nat] :
( ( ord_le5934964663421696068at_nat @ ( piE_nat_nat_nat @ I5 @ S5 ) @ ( piE_nat_nat_nat @ I5 @ T3 ) )
= ( ( ( piE_nat_nat_nat @ I5 @ S5 )
= bot_bo945813143650711160at_nat )
| ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ( ord_less_eq_set_nat @ ( S5 @ X3 ) @ ( T3 @ X3 ) ) ) ) ) ).
% subset_PiE
thf(fact_919_subset__PiE,axiom,
! [I5: set_nat,S5: nat > set_nat_nat,T3: nat > set_nat_nat] :
( ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ I5 @ S5 ) @ ( piE_nat_nat_nat2 @ I5 @ T3 ) )
= ( ( ( piE_nat_nat_nat2 @ I5 @ S5 )
= bot_bo7445843802507891576at_nat )
| ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( S5 @ X3 ) @ ( T3 @ X3 ) ) ) ) ) ).
% subset_PiE
thf(fact_920_subset__PiE,axiom,
! [I5: set_nat_nat,S5: ( nat > nat ) > set_nat_nat,T3: ( nat > nat ) > set_nat_nat] :
( ( ord_le5260717879541182899at_nat @ ( piE_nat_nat_nat_nat3 @ I5 @ S5 ) @ ( piE_nat_nat_nat_nat3 @ I5 @ T3 ) )
= ( ( ( piE_nat_nat_nat_nat3 @ I5 @ S5 )
= bot_bo3919185967433191911at_nat )
| ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ( ord_le9059583361652607317at_nat @ ( S5 @ X3 ) @ ( T3 @ X3 ) ) ) ) ) ).
% subset_PiE
thf(fact_921_subset__PiE,axiom,
! [I5: set_nat,S5: nat > set_nat,T3: nat > set_nat] :
( ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ I5 @ S5 ) @ ( piE_nat_nat @ I5 @ T3 ) )
= ( ( ( piE_nat_nat @ I5 @ S5 )
= bot_bot_set_nat_nat )
| ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ord_less_eq_set_nat @ ( S5 @ X3 ) @ ( T3 @ X3 ) ) ) ) ) ).
% subset_PiE
thf(fact_922_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat_nat > nat > nat,A2: set_nat_nat] :
( ( Inf
@ ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : X3
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_923_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_924_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat_nat > nat > nat,A2: set_nat_nat] :
( ( Sup
@ ( image_3205354838064109189at_nat
@ ^ [X3: nat > nat] : X3
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_925_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_926_Sup__eqI,axiom,
! [A2: set_set_nat_nat,X2: set_nat_nat] :
( ! [Y4: set_nat_nat] :
( ( member_set_nat_nat @ Y4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ Y4 @ X2 ) )
=> ( ! [Y4: set_nat_nat] :
( ! [Z4: set_nat_nat] :
( ( member_set_nat_nat @ Z4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ Z4 @ Y4 ) )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y4 ) )
=> ( ( comple5448282615319421384at_nat @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_927_Sup__eqI,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ A2 )
=> ( ord_less_eq_set_nat @ Y4 @ X2 ) )
=> ( ! [Y4: set_nat] :
( ! [Z4: set_nat] :
( ( member_set_nat @ Z4 @ A2 )
=> ( ord_less_eq_set_nat @ Z4 @ Y4 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_928_Sup__mono,axiom,
! [A2: set_set_nat_nat,B3: set_set_nat_nat] :
( ! [A4: set_nat_nat] :
( ( member_set_nat_nat @ A4 @ A2 )
=> ? [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ B3 )
& ( ord_le9059583361652607317at_nat @ A4 @ X4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B3 ) ) ) ).
% Sup_mono
thf(fact_929_Sup__mono,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ A2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ A4 @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B3 ) ) ) ).
% Sup_mono
thf(fact_930_Sup__least,axiom,
! [A2: set_set_nat_nat,Z3: set_nat_nat] :
( ! [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ X @ Z3 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ Z3 ) ) ).
% Sup_least
thf(fact_931_Sup__least,axiom,
! [A2: set_set_nat,Z3: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ Z3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ Z3 ) ) ).
% Sup_least
thf(fact_932_Sup__upper,axiom,
! [X2: set_nat_nat,A2: set_set_nat_nat] :
( ( member_set_nat_nat @ X2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( comple5448282615319421384at_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_933_Sup__upper,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_934_Sup__le__iff,axiom,
! [A2: set_set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ B )
= ( ! [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_935_Sup__le__iff,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ B )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_936_Sup__upper2,axiom,
! [U2: set_nat_nat,A2: set_set_nat_nat,V: set_nat_nat] :
( ( member_set_nat_nat @ U2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ V @ U2 )
=> ( ord_le9059583361652607317at_nat @ V @ ( comple5448282615319421384at_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_937_Sup__upper2,axiom,
! [U2: set_nat,A2: set_set_nat,V: set_nat] :
( ( member_set_nat @ U2 @ A2 )
=> ( ( ord_less_eq_set_nat @ V @ U2 )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_938_SUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C4: nat > nat,D3: nat > nat] :
( ( A2 = B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D3 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_939_SUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C4: nat > set_nat,D3: nat > set_nat] :
( ( A2 = B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D3 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_940_SUP__cong,axiom,
! [A2: set_nat,B3: set_nat,C4: nat > nat > nat,D3: nat > nat > nat] :
( ( A2 = B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( comple2450677804321093138at_nat @ ( image_nat_nat_nat2 @ C4 @ A2 ) )
= ( comple2450677804321093138at_nat @ ( image_nat_nat_nat2 @ D3 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_941_SUP__cong,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: ( nat > nat ) > nat,D3: ( nat > nat ) > nat] :
( ( A2 = B3 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ D3 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_942_SUP__cong,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: ( nat > nat ) > set_nat,D3: ( nat > nat ) > set_nat] :
( ( A2 = B3 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ C4 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ D3 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_943_SUP__cong,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: ( nat > nat ) > nat > nat,D3: ( nat > nat ) > nat > nat] :
( ( A2 = B3 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( comple2450677804321093138at_nat @ ( image_3205354838064109189at_nat @ C4 @ A2 ) )
= ( comple2450677804321093138at_nat @ ( image_3205354838064109189at_nat @ D3 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_944_SUP__cong,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,C4: ( nat > nat > nat ) > nat,D3: ( nat > nat > nat ) > nat] :
( ( A2 = B3 )
=> ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_913610194320715013at_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_913610194320715013at_nat @ D3 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_945_SUP__cong,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,C4: ( ( nat > nat ) > nat ) > nat,D3: ( ( nat > nat ) > nat ) > nat] :
( ( A2 = B3 )
=> ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_7809927846809980933at_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_7809927846809980933at_nat @ D3 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_946_SUP__cong,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,C4: ( nat > nat > nat ) > set_nat,D3: ( nat > nat > nat ) > set_nat] :
( ( A2 = B3 )
=> ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ C4 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ D3 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_947_SUP__cong,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,C4: ( ( nat > nat ) > nat ) > set_nat,D3: ( ( nat > nat ) > nat ) > set_nat] :
( ( A2 = B3 )
=> ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ B3 )
=> ( ( C4 @ X )
= ( D3 @ X ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ C4 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ D3 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_948_Union__mono,axiom,
! [A2: set_set_nat_nat,B3: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B3 ) ) ) ).
% Union_mono
thf(fact_949_Union__mono,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B3 ) ) ) ).
% Union_mono
thf(fact_950_Union__least,axiom,
! [A2: set_set_nat_nat,C4: set_nat_nat] :
( ! [X8: set_nat_nat] :
( ( member_set_nat_nat @ X8 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X8 @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ C4 ) ) ).
% Union_least
thf(fact_951_Union__least,axiom,
! [A2: set_set_nat,C4: set_nat] :
( ! [X8: set_nat] :
( ( member_set_nat @ X8 @ A2 )
=> ( ord_less_eq_set_nat @ X8 @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ C4 ) ) ).
% Union_least
thf(fact_952_Union__upper,axiom,
! [B3: set_nat_nat,A2: set_set_nat_nat] :
( ( member_set_nat_nat @ B3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ B3 @ ( comple5448282615319421384at_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_953_Union__upper,axiom,
! [B3: set_nat,A2: set_set_nat] :
( ( member_set_nat @ B3 @ A2 )
=> ( ord_less_eq_set_nat @ B3 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_954_Union__subsetI,axiom,
! [A2: set_set_nat_nat,B3: set_set_nat_nat] :
( ! [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A2 )
=> ? [Y3: set_nat_nat] :
( ( member_set_nat_nat @ Y3 @ B3 )
& ( ord_le9059583361652607317at_nat @ X @ Y3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B3 ) ) ) ).
% Union_subsetI
thf(fact_955_Union__subsetI,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ? [Y3: set_nat] :
( ( member_set_nat @ Y3 @ B3 )
& ( ord_less_eq_set_nat @ X @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B3 ) ) ) ).
% Union_subsetI
thf(fact_956_SUP__commute,axiom,
! [F: nat > nat > set_nat,B3: set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I2 ) @ B3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J3: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( F @ I2 @ J3 )
@ A2 ) )
@ B3 ) ) ) ).
% SUP_commute
thf(fact_957_image__Union,axiom,
! [F: ( nat > nat ) > nat > nat,S5: set_set_nat_nat] :
( ( image_3205354838064109189at_nat @ F @ ( comple5448282615319421384at_nat @ S5 ) )
= ( comple5448282615319421384at_nat @ ( image_3832368097948589297at_nat @ ( image_3205354838064109189at_nat @ F ) @ S5 ) ) ) ).
% image_Union
thf(fact_958_image__Union,axiom,
! [F: nat > nat > nat,S5: set_set_nat] :
( ( image_nat_nat_nat2 @ F @ ( comple7399068483239264473et_nat @ S5 ) )
= ( comple5448282615319421384at_nat @ ( image_7054278410236665602at_nat @ ( image_nat_nat_nat2 @ F ) @ S5 ) ) ) ).
% image_Union
thf(fact_959_image__Union,axiom,
! [F: nat > set_nat,S5: set_set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ S5 ) )
= ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ S5 ) ) ) ).
% image_Union
thf(fact_960_image__Union,axiom,
! [F: nat > nat,S5: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S5 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S5 ) ) ) ).
% image_Union
thf(fact_961_UN__UN__flatten,axiom,
! [C4: nat > set_nat,B3: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( B3 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_962_UN__E,axiom,
! [B: nat,B3: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ~ ( member_nat @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_963_UN__E,axiom,
! [B: nat > nat,B3: nat > set_nat_nat,A2: set_nat] :
( ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B3 @ A2 ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ~ ( member_nat_nat @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_964_UN__E,axiom,
! [B: nat,B3: ( nat > nat ) > set_nat,A2: set_nat_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B3 @ A2 ) ) )
=> ~ ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ~ ( member_nat @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_965_UN__E,axiom,
! [B: nat > nat,B3: ( nat > nat ) > set_nat_nat,A2: set_nat_nat] :
( ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ B3 @ A2 ) ) )
=> ~ ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ~ ( member_nat_nat @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_966_UN__E,axiom,
! [B: nat > nat > nat,B3: nat > set_nat_nat_nat,A2: set_nat] :
( ( member_nat_nat_nat2 @ B @ ( comple8167887107183641911at_nat @ ( image_6130888460295934395at_nat @ B3 @ A2 ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ~ ( member_nat_nat_nat2 @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_967_UN__E,axiom,
! [B: ( nat > nat ) > nat,B3: nat > set_nat_nat_nat2,A2: set_nat] :
( ( member_nat_nat_nat @ B @ ( comple1667856448326461495at_nat @ ( image_8854229838293529787at_nat @ B3 @ A2 ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ~ ( member_nat_nat_nat @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_968_UN__E,axiom,
! [B: nat,B3: ( nat > nat > nat ) > set_nat,A2: set_nat_nat_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ B3 @ A2 ) ) )
=> ~ ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ~ ( member_nat @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_969_UN__E,axiom,
! [B: nat,B3: ( ( nat > nat ) > nat ) > set_nat,A2: set_nat_nat_nat2] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ B3 @ A2 ) ) )
=> ~ ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ~ ( member_nat @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_970_UN__E,axiom,
! [B: nat > nat,B3: ( nat > nat > nat ) > set_nat_nat,A2: set_nat_nat_nat] :
( ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ B3 @ A2 ) ) )
=> ~ ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ~ ( member_nat_nat @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_971_UN__E,axiom,
! [B: nat > nat,B3: ( ( nat > nat ) > nat ) > set_nat_nat,A2: set_nat_nat_nat2] :
( ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ B3 @ A2 ) ) )
=> ~ ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ~ ( member_nat_nat @ B @ ( B3 @ X ) ) ) ) ).
% UN_E
thf(fact_972_UN__extend__simps_I8_J,axiom,
! [B3: nat > set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y2: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ Y2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_973_UN__extend__simps_I9_J,axiom,
! [C4: nat > set_nat,B3: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( B3 @ X3 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_974_SUP__eq,axiom,
! [A2: set_nat,B3: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_975_SUP__eq,axiom,
! [A2: set_nat,B3: set_nat,F: nat > set_nat_nat,G: nat > set_nat_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_le9059583361652607317at_nat @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) )
= ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_976_SUP__eq,axiom,
! [A2: set_nat,B3: set_nat_nat,F: nat > set_nat,G: ( nat > nat ) > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: nat > nat] :
( ( member_nat_nat @ J @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_977_SUP__eq,axiom,
! [A2: set_nat_nat,B3: set_nat,F: ( nat > nat ) > set_nat,G: nat > set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B3 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_978_SUP__eq,axiom,
! [A2: set_nat,B3: set_nat_nat,F: nat > set_nat_nat,G: ( nat > nat ) > set_nat_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B3 )
& ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: nat > nat] :
( ( member_nat_nat @ J @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_le9059583361652607317at_nat @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) )
= ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_979_SUP__eq,axiom,
! [A2: set_nat_nat,B3: set_nat,F: ( nat > nat ) > set_nat_nat,G: nat > set_nat_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B3 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
& ( ord_le9059583361652607317at_nat @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) )
= ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_980_SUP__eq,axiom,
! [A2: set_nat,B3: set_nat_nat_nat,F: nat > set_nat,G: ( nat > nat > nat ) > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: nat > nat > nat] :
( ( member_nat_nat_nat2 @ J @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_981_SUP__eq,axiom,
! [A2: set_nat,B3: set_nat_nat_nat2,F: nat > set_nat,G: ( ( nat > nat ) > nat ) > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ? [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ J @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_982_SUP__eq,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,F: ( nat > nat ) > set_nat,G: ( nat > nat ) > set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ A2 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: nat > nat] :
( ( member_nat_nat @ J @ B3 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_983_SUP__eq,axiom,
! [A2: set_nat_nat_nat,B3: set_nat,F: ( nat > nat > nat ) > set_nat,G: nat > set_nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B3 )
=> ? [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_984_less__eq__Sup,axiom,
! [A2: set_set_nat_nat,U2: set_nat_nat] :
( ! [V2: set_nat_nat] :
( ( member_set_nat_nat @ V2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ U2 @ V2 ) )
=> ( ( A2 != bot_bo7376149671870096959at_nat )
=> ( ord_le9059583361652607317at_nat @ U2 @ ( comple5448282615319421384at_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_985_less__eq__Sup,axiom,
! [A2: set_set_nat,U2: set_nat] :
( ! [V2: set_nat] :
( ( member_set_nat @ V2 @ A2 )
=> ( ord_less_eq_set_nat @ U2 @ V2 ) )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_986_Sup__subset__mono,axiom,
! [A2: set_set_nat_nat,B3: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ A2 @ B3 )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B3 ) ) ) ).
% Sup_subset_mono
thf(fact_987_Sup__subset__mono,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B3 ) ) ) ).
% Sup_subset_mono
thf(fact_988_SUP__eq__const,axiom,
! [I5: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat,X2: set_nat] :
( ( I5 != bot_bo7445843802507891576at_nat )
=> ( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ I5 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_989_SUP__eq__const,axiom,
! [I5: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat,X2: set_nat] :
( ( I5 != bot_bo945813143650711160at_nat )
=> ( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ I5 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_990_SUP__eq__const,axiom,
! [I5: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat,X2: set_nat] :
( ( I5 != bot_bo3919185967433191911at_nat )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ I5 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_991_SUP__eq__const,axiom,
! [I5: set_nat,F: nat > set_nat,X2: set_nat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I5 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_992_SUP__eq__const,axiom,
! [I5: set_nat_nat,F: ( nat > nat ) > set_nat,X2: set_nat] :
( ( I5 != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ I5 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_993_SUP__upper2,axiom,
! [I: nat,A2: set_nat,U2: set_nat_nat,F: nat > set_nat_nat] :
( ( member_nat @ I @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ U2 @ ( F @ I ) )
=> ( ord_le9059583361652607317at_nat @ U2 @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_994_SUP__upper2,axiom,
! [I: nat > nat,A2: set_nat_nat,U2: set_nat_nat,F: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ I @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ U2 @ ( F @ I ) )
=> ( ord_le9059583361652607317at_nat @ U2 @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_995_SUP__upper2,axiom,
! [I: nat > nat > nat,A2: set_nat_nat_nat,U2: set_nat_nat,F: ( nat > nat > nat ) > set_nat_nat] :
( ( member_nat_nat_nat2 @ I @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ U2 @ ( F @ I ) )
=> ( ord_le9059583361652607317at_nat @ U2 @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_996_SUP__upper2,axiom,
! [I: ( nat > nat ) > nat,A2: set_nat_nat_nat2,U2: set_nat_nat,F: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ( member_nat_nat_nat @ I @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ U2 @ ( F @ I ) )
=> ( ord_le9059583361652607317at_nat @ U2 @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_997_SUP__upper2,axiom,
! [I: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,U2: set_nat_nat,F: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ I @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ U2 @ ( F @ I ) )
=> ( ord_le9059583361652607317at_nat @ U2 @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_998_SUP__upper2,axiom,
! [I: nat,A2: set_nat,U2: set_nat,F: nat > set_nat] :
( ( member_nat @ I @ A2 )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_999_SUP__upper2,axiom,
! [I: nat > nat,A2: set_nat_nat,U2: set_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ I @ A2 )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1000_SUP__upper2,axiom,
! [I: nat > nat > nat,A2: set_nat_nat_nat,U2: set_nat,F: ( nat > nat > nat ) > set_nat] :
( ( member_nat_nat_nat2 @ I @ A2 )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1001_SUP__upper2,axiom,
! [I: ( nat > nat ) > nat,A2: set_nat_nat_nat2,U2: set_nat,F: ( ( nat > nat ) > nat ) > set_nat] :
( ( member_nat_nat_nat @ I @ A2 )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1002_SUP__upper2,axiom,
! [I: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,U2: set_nat,F: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ( member952132173341509300at_nat @ I @ A2 )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1003_SUP__le__iff,axiom,
! [F: nat > set_nat,A2: set_nat,U2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ U2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ U2 ) ) ) ) ).
% SUP_le_iff
thf(fact_1004_SUP__upper,axiom,
! [I: nat,A2: set_nat,F: nat > set_nat_nat] :
( ( member_nat @ I @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I ) @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1005_SUP__upper,axiom,
! [I: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ I @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I ) @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1006_SUP__upper,axiom,
! [I: nat > nat > nat,A2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat_nat] :
( ( member_nat_nat_nat2 @ I @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I ) @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1007_SUP__upper,axiom,
! [I: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ( member_nat_nat_nat @ I @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I ) @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1008_SUP__upper,axiom,
! [I: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ I @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I ) @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1009_SUP__upper,axiom,
! [I: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ I @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1010_SUP__upper,axiom,
! [I: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ I @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1011_SUP__upper,axiom,
! [I: nat > nat > nat,A2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat] :
( ( member_nat_nat_nat2 @ I @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1012_SUP__upper,axiom,
! [I: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat] :
( ( member_nat_nat_nat @ I @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1013_SUP__upper,axiom,
! [I: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ( member952132173341509300at_nat @ I @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1014_SUP__mono_H,axiom,
! [F: nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ! [X: nat] : ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) )
=> ( 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_1015_SUP__least,axiom,
! [A2: set_nat,F: nat > set_nat_nat,U2: set_nat_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1016_SUP__least,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat_nat,U2: set_nat_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1017_SUP__least,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat_nat,U2: set_nat_nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1018_SUP__least,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat_nat,U2: set_nat_nat] :
( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1019_SUP__least,axiom,
! [A2: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat_nat,U2: set_nat_nat] :
( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1020_SUP__least,axiom,
! [A2: set_nat,F: nat > set_nat,U2: set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1021_SUP__least,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat,U2: set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1022_SUP__least,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat,U2: set_nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1023_SUP__least,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat,U2: set_nat] :
( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1024_SUP__least,axiom,
! [A2: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat,U2: set_nat] :
( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ A2 ) ) @ U2 ) ) ).
% SUP_least
thf(fact_1025_SUP__mono,axiom,
! [A2: set_nat,B3: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [N2: nat] :
( ( member_nat @ N2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1026_SUP__mono,axiom,
! [A2: set_nat_nat,B3: set_nat,F: ( nat > nat ) > set_nat,G: nat > set_nat] :
( ! [N2: nat > nat] :
( ( member_nat_nat @ N2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1027_SUP__mono,axiom,
! [A2: set_nat_nat_nat,B3: set_nat,F: ( nat > nat > nat ) > set_nat,G: nat > set_nat] :
( ! [N2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ N2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1028_SUP__mono,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat,F: ( ( nat > nat ) > nat ) > set_nat,G: nat > set_nat] :
( ! [N2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ N2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1029_SUP__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B3: set_nat,F: ( ( nat > nat ) > nat > nat ) > set_nat,G: nat > set_nat] :
( ! [N2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ N2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).
% SUP_mono
thf(fact_1030_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_nat_nat,X2: set_nat_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ X2 ) )
=> ( ! [Y4: set_nat_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I4 ) @ Y4 ) )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y4 ) )
=> ( ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1031_SUP__eqI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat_nat,X2: set_nat_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ X2 ) )
=> ( ! [Y4: set_nat_nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I4 ) @ Y4 ) )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y4 ) )
=> ( ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1032_SUP__eqI,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat_nat,X2: set_nat_nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ X2 ) )
=> ( ! [Y4: set_nat_nat] :
( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I4 ) @ Y4 ) )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y4 ) )
=> ( ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1033_SUP__eqI,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat_nat,X2: set_nat_nat] :
( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ X2 ) )
=> ( ! [Y4: set_nat_nat] :
( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I4 ) @ Y4 ) )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y4 ) )
=> ( ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1034_SUP__eqI,axiom,
! [A2: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat_nat,X2: set_nat_nat] :
( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ X2 ) )
=> ( ! [Y4: set_nat_nat] :
( ! [I4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I4 ) @ Y4 ) )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y4 ) )
=> ( ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1035_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_nat,X2: set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ X2 ) )
=> ( ! [Y4: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y4 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1036_SUP__eqI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat,X2: set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ X2 ) )
=> ( ! [Y4: set_nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y4 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1037_SUP__eqI,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat,X2: set_nat] :
( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ X2 ) )
=> ( ! [Y4: set_nat] :
( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y4 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1038_SUP__eqI,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat,X2: set_nat] :
( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ X2 ) )
=> ( ! [Y4: set_nat] :
( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y4 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1039_SUP__eqI,axiom,
! [A2: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat,X2: set_nat] :
( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ X2 ) )
=> ( ! [Y4: set_nat] :
( ! [I4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y4 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1040_SUP__lessD,axiom,
! [F: nat > set_nat,A2: set_nat,Y: set_nat,I: nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ Y )
=> ( ( member_nat @ I @ A2 )
=> ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_1041_SUP__lessD,axiom,
! [F: ( nat > nat ) > set_nat,A2: set_nat_nat,Y: set_nat,I: nat > nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) @ Y )
=> ( ( member_nat_nat @ I @ A2 )
=> ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_1042_SUP__lessD,axiom,
! [F: ( nat > nat > nat ) > set_nat,A2: set_nat_nat_nat,Y: set_nat,I: nat > nat > nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) @ Y )
=> ( ( member_nat_nat_nat2 @ I @ A2 )
=> ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_1043_SUP__lessD,axiom,
! [F: ( ( nat > nat ) > nat ) > set_nat,A2: set_nat_nat_nat2,Y: set_nat,I: ( nat > nat ) > nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) @ Y )
=> ( ( member_nat_nat_nat @ I @ A2 )
=> ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_1044_SUP__lessD,axiom,
! [F: ( ( nat > nat ) > nat > nat ) > set_nat,A2: set_nat_nat_nat_nat3,Y: set_nat,I: ( nat > nat ) > nat > nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ A2 ) ) @ Y )
=> ( ( member952132173341509300at_nat @ I @ A2 )
=> ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_1045_image__UN,axiom,
! [F: nat > nat > nat,B3: nat > set_nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( comple5448282615319421384at_nat
@ ( image_7301343469591561292at_nat
@ ^ [X3: nat] : ( image_nat_nat_nat2 @ F @ ( B3 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1046_image__UN,axiom,
! [F: nat > set_nat,B3: nat > set_nat,A2: set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X3: nat] : ( image_nat_set_nat @ F @ ( B3 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1047_image__UN,axiom,
! [F: nat > nat,B3: nat > set_nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( image_nat_nat @ F @ ( B3 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1048_UN__extend__simps_I10_J,axiom,
! [B3: ( nat > nat ) > set_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [A3: nat > nat] : ( B3 @ ( F @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B3 @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1049_UN__extend__simps_I10_J,axiom,
! [B3: ( nat > nat ) > set_nat,F: nat > nat > nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( B3 @ ( F @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B3 @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1050_UN__extend__simps_I10_J,axiom,
! [B3: set_nat > set_nat,F: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( B3 @ ( F @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B3 @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1051_UN__extend__simps_I10_J,axiom,
! [B3: nat > set_nat,F: nat > nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( B3 @ ( F @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1052_UN__empty2,axiom,
! [A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : bot_bot_set_nat
@ A2 ) )
= bot_bot_set_nat ) ).
% UN_empty2
thf(fact_1053_UN__empty,axiom,
! [B3: nat > set_nat_nat] :
( ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B3 @ bot_bot_set_nat ) )
= bot_bot_set_nat_nat ) ).
% UN_empty
thf(fact_1054_UN__empty,axiom,
! [B3: ( nat > nat ) > set_nat_nat] :
( ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ B3 @ bot_bot_set_nat_nat ) )
= bot_bot_set_nat_nat ) ).
% UN_empty
thf(fact_1055_UN__empty,axiom,
! [B3: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_1056_UN__empty,axiom,
! [B3: ( nat > nat ) > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B3 @ bot_bot_set_nat_nat ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_1057_UNION__empty__conv_I1_J,axiom,
! [B3: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B3 @ X3 )
= bot_bot_set_nat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_1058_UNION__empty__conv_I2_J,axiom,
! [B3: nat > set_nat,A2: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B3 @ X3 )
= bot_bot_set_nat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_1059_UN__mono,axiom,
! [A2: set_nat,B3: set_nat,F: nat > set_nat_nat,G: nat > set_nat_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1060_UN__mono,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat_nat,G: ( nat > nat > nat ) > set_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B3 )
=> ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1061_UN__mono,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat_nat,G: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B3 )
=> ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1062_UN__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat_nat,G: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B3 )
=> ( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1063_UN__mono,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,F: ( nat > nat ) > set_nat_nat,G: ( nat > nat ) > set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1064_UN__mono,axiom,
! [A2: set_nat,B3: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1065_UN__mono,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat,G: ( nat > nat > nat ) > set_nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B3 )
=> ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1066_UN__mono,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat,G: ( ( nat > nat ) > nat ) > set_nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B3 )
=> ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1067_UN__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat,G: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B3 )
=> ( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1068_UN__mono,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,F: ( nat > nat ) > set_nat,G: ( nat > nat ) > set_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_1069_UN__least,axiom,
! [A2: set_nat,B3: nat > set_nat_nat,C4: set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X ) @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B3 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_1070_UN__least,axiom,
! [A2: set_nat_nat,B3: ( nat > nat ) > set_nat_nat,C4: set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X ) @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ B3 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_1071_UN__least,axiom,
! [A2: set_nat_nat_nat,B3: ( nat > nat > nat ) > set_nat_nat,C4: set_nat_nat] :
( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X ) @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ B3 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_1072_UN__least,axiom,
! [A2: set_nat_nat_nat2,B3: ( ( nat > nat ) > nat ) > set_nat_nat,C4: set_nat_nat] :
( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X ) @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ B3 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_1073_UN__least,axiom,
! [A2: set_nat_nat_nat_nat3,B3: ( ( nat > nat ) > nat > nat ) > set_nat_nat,C4: set_nat_nat] :
( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ X ) @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ B3 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_1074_UN__least,axiom,
! [A2: set_nat,B3: nat > set_nat,C4: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_1075_UN__least,axiom,
! [A2: set_nat_nat,B3: ( nat > nat ) > set_nat,C4: set_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B3 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_1076_UN__least,axiom,
! [A2: set_nat_nat_nat,B3: ( nat > nat > nat ) > set_nat,C4: set_nat] :
( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ B3 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_1077_UN__least,axiom,
! [A2: set_nat_nat_nat2,B3: ( ( nat > nat ) > nat ) > set_nat,C4: set_nat] :
( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ B3 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_1078_UN__least,axiom,
! [A2: set_nat_nat_nat_nat3,B3: ( ( nat > nat ) > nat > nat ) > set_nat,C4: set_nat] :
( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ X ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ B3 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_1079_UN__upper,axiom,
! [A: nat,A2: set_nat,B3: nat > set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ A ) @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1080_UN__upper,axiom,
! [A: nat > nat,A2: set_nat_nat,B3: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ A ) @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1081_UN__upper,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B3: ( nat > nat > nat ) > set_nat_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ A ) @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1082_UN__upper,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B3: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ A ) @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1083_UN__upper,axiom,
! [A: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B3: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B3 @ A ) @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1084_UN__upper,axiom,
! [A: nat,A2: set_nat,B3: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1085_UN__upper,axiom,
! [A: nat > nat,A2: set_nat_nat,B3: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ A ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1086_UN__upper,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B3: ( nat > nat > nat ) > set_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ A ) @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1087_UN__upper,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B3: ( ( nat > nat ) > nat ) > set_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ A ) @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1088_UN__upper,axiom,
! [A: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B3: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ( member952132173341509300at_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B3 @ A ) @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ B3 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_1089_UN__subset__iff,axiom,
! [A2: nat > set_nat,I5: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I5 ) ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ord_less_eq_set_nat @ ( A2 @ X3 ) @ B3 ) ) ) ) ).
% UN_subset_iff
thf(fact_1090_SUP__eq__iff,axiom,
! [I5: set_nat_nat_nat,C: set_nat_nat,F: ( nat > nat > nat ) > set_nat_nat] :
( ( I5 != bot_bo7445843802507891576at_nat )
=> ( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ord_le9059583361652607317at_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ F @ I5 ) )
= C )
= ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1091_SUP__eq__iff,axiom,
! [I5: set_nat_nat_nat2,C: set_nat_nat,F: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ( I5 != bot_bo945813143650711160at_nat )
=> ( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ord_le9059583361652607317at_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ F @ I5 ) )
= C )
= ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1092_SUP__eq__iff,axiom,
! [I5: set_nat_nat_nat_nat3,C: set_nat_nat,F: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ( I5 != bot_bo3919185967433191911at_nat )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ord_le9059583361652607317at_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ F @ I5 ) )
= C )
= ( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1093_SUP__eq__iff,axiom,
! [I5: set_nat,C: set_nat_nat,F: nat > set_nat_nat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_le9059583361652607317at_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ I5 ) )
= C )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1094_SUP__eq__iff,axiom,
! [I5: set_nat_nat,C: set_nat_nat,F: ( nat > nat ) > set_nat_nat] :
( ( I5 != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ord_le9059583361652607317at_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ I5 ) )
= C )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1095_SUP__eq__iff,axiom,
! [I5: set_nat_nat_nat,C: set_nat,F: ( nat > nat > nat ) > set_nat] :
( ( I5 != bot_bo7445843802507891576at_nat )
=> ( ! [I3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I3 @ I5 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ I5 ) )
= C )
= ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1096_SUP__eq__iff,axiom,
! [I5: set_nat_nat_nat2,C: set_nat,F: ( ( nat > nat ) > nat ) > set_nat] :
( ( I5 != bot_bo945813143650711160at_nat )
=> ( ! [I3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I3 @ I5 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ I5 ) )
= C )
= ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1097_SUP__eq__iff,axiom,
! [I5: set_nat_nat_nat_nat3,C: set_nat,F: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ( I5 != bot_bo3919185967433191911at_nat )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I5 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ I5 ) )
= C )
= ( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1098_SUP__eq__iff,axiom,
! [I5: set_nat,C: set_nat,F: nat > set_nat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I5 ) )
= C )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1099_SUP__eq__iff,axiom,
! [I5: set_nat_nat,C: set_nat,F: ( nat > nat ) > set_nat] :
( ( I5 != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I5 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ I5 ) )
= C )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I5 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1100_SUP__subset__mono,axiom,
! [A2: set_nat,B3: set_nat,F: nat > set_nat_nat,G: nat > set_nat_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1101_SUP__subset__mono,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat_nat,G: ( nat > nat > nat ) > set_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B3 )
=> ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1102_SUP__subset__mono,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat_nat,G: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B3 )
=> ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1103_SUP__subset__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat_nat,G: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B3 )
=> ( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1104_SUP__subset__mono,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,F: ( nat > nat ) > set_nat_nat,G: ( nat > nat ) > set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1105_SUP__subset__mono,axiom,
! [A2: set_nat,B3: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1106_SUP__subset__mono,axiom,
! [A2: set_nat_nat_nat,B3: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat,G: ( nat > nat > nat ) > set_nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B3 )
=> ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1107_SUP__subset__mono,axiom,
! [A2: set_nat_nat_nat2,B3: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat,G: ( ( nat > nat ) > nat ) > set_nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B3 )
=> ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1108_SUP__subset__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat,G: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B3 )
=> ( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1109_SUP__subset__mono,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,F: ( nat > nat ) > set_nat,G: ( nat > nat ) > set_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ G @ B3 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1110_SUP__constant,axiom,
! [A2: set_nat,C: set_nat_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple5448282615319421384at_nat
@ ( image_7301343469591561292at_nat
@ ^ [Y2: nat] : C
@ A2 ) )
= bot_bot_set_nat_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple5448282615319421384at_nat
@ ( image_7301343469591561292at_nat
@ ^ [Y2: nat] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1111_SUP__constant,axiom,
! [A2: set_nat_nat,C: set_nat_nat] :
( ( ( A2 = bot_bot_set_nat_nat )
=> ( ( comple5448282615319421384at_nat
@ ( image_6905811865970898491at_nat
@ ^ [Y2: nat > nat] : C
@ A2 ) )
= bot_bot_set_nat_nat ) )
& ( ( A2 != bot_bot_set_nat_nat )
=> ( ( comple5448282615319421384at_nat
@ ( image_6905811865970898491at_nat
@ ^ [Y2: nat > nat] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1112_SUP__constant,axiom,
! [A2: set_nat,C: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1113_SUP__constant,axiom,
! [A2: set_nat_nat,C: set_nat] :
( ( ( A2 = bot_bot_set_nat_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [Y2: nat > nat] : C
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [Y2: nat > nat] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1114_SUP__empty,axiom,
! [F: nat > set_nat_nat] :
( ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ bot_bot_set_nat ) )
= bot_bot_set_nat_nat ) ).
% SUP_empty
thf(fact_1115_SUP__empty,axiom,
! [F: ( nat > nat ) > set_nat_nat] :
( ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ bot_bot_set_nat_nat ) )
= bot_bot_set_nat_nat ) ).
% SUP_empty
thf(fact_1116_SUP__empty,axiom,
! [F: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_1117_SUP__empty,axiom,
! [F: ( nat > nat ) > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ bot_bot_set_nat_nat ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_1118_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_1119_dim1__subspace__elims_I2_J,axiom,
! [B3: nat > set_nat,N: nat,F: nat > nat,T: nat,S5: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B3 @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S5
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ T )
@ ^ [I2: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ one_one_nat ) )
=> ( ( S5 @ X @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ J ) )
=> ( ( S5 @ X @ Xa2 )
= ( X @ J ) ) ) ) ) )
=> ( ( inf_inf_set_nat @ ( B3 @ zero_zero_nat ) @ ( B3 @ one_one_nat ) )
= bot_bot_set_nat ) ) ) ) ) ) ) ).
% dim1_subspace_elims(2)
thf(fact_1120_dim1__subspace__elims_I1_J,axiom,
! [B3: nat > set_nat,N: nat,F: nat > nat,T: nat,S5: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B3 @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S5
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ T )
@ ^ [I2: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ one_one_nat ) )
=> ( ( S5 @ X @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ J ) )
=> ( ( S5 @ X @ Xa2 )
= ( X @ J ) ) ) ) ) )
=> ( ( sup_sup_set_nat @ ( B3 @ zero_zero_nat ) @ ( B3 @ one_one_nat ) )
= ( set_ord_lessThan_nat @ N ) ) ) ) ) ) ) ) ).
% dim1_subspace_elims(1)
thf(fact_1121__092_060open_062_123_O_O1_125_A_061_A_1230_M_A1_125_092_060close_062,axiom,
( ( set_ord_atMost_nat @ one_one_nat )
= ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).
% \<open>{..1} = {0, 1}\<close>
thf(fact_1122_insertCI,axiom,
! [A: nat,B3: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B3 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B3 ) ) ) ).
% insertCI
thf(fact_1123_insertCI,axiom,
! [A: nat > nat,B3: set_nat_nat,B: nat > nat] :
( ( ~ ( member_nat_nat @ A @ B3 )
=> ( A = B ) )
=> ( member_nat_nat @ A @ ( insert_nat_nat @ B @ B3 ) ) ) ).
% insertCI
thf(fact_1124_insertCI,axiom,
! [A: nat > nat > nat,B3: set_nat_nat_nat,B: nat > nat > nat] :
( ( ~ ( member_nat_nat_nat2 @ A @ B3 )
=> ( A = B ) )
=> ( member_nat_nat_nat2 @ A @ ( insert_nat_nat_nat2 @ B @ B3 ) ) ) ).
% insertCI
thf(fact_1125_insertCI,axiom,
! [A: ( nat > nat ) > nat,B3: set_nat_nat_nat2,B: ( nat > nat ) > nat] :
( ( ~ ( member_nat_nat_nat @ A @ B3 )
=> ( A = B ) )
=> ( member_nat_nat_nat @ A @ ( insert_nat_nat_nat @ B @ B3 ) ) ) ).
% insertCI
thf(fact_1126_insertCI,axiom,
! [A: ( nat > nat ) > nat > nat,B3: set_nat_nat_nat_nat3,B: ( nat > nat ) > nat > nat] :
( ( ~ ( member952132173341509300at_nat @ A @ B3 )
=> ( A = B ) )
=> ( member952132173341509300at_nat @ A @ ( insert1655528249163143117at_nat @ B @ B3 ) ) ) ).
% insertCI
thf(fact_1127_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1128_insert__iff,axiom,
! [A: nat > nat,B: nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ A @ ( insert_nat_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1129_insert__iff,axiom,
! [A: nat > nat > nat,B: nat > nat > nat,A2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ A @ ( insert_nat_nat_nat2 @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_nat_nat2 @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1130_insert__iff,axiom,
! [A: ( nat > nat ) > nat,B: ( nat > nat ) > nat,A2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ A @ ( insert_nat_nat_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_nat_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1131_insert__iff,axiom,
! [A: ( nat > nat ) > nat > nat,B: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ A @ ( insert1655528249163143117at_nat @ B @ A2 ) )
= ( ( A = B )
| ( member952132173341509300at_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1132_insert__absorb2,axiom,
! [X2: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A2 ) )
= ( insert_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_1133_insert__absorb2,axiom,
! [X2: nat > nat,A2: set_nat_nat] :
( ( insert_nat_nat @ X2 @ ( insert_nat_nat @ X2 @ A2 ) )
= ( insert_nat_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_1134_IntI,axiom,
! [C: nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( member_nat_nat @ C @ A2 )
=> ( ( member_nat_nat @ C @ B3 )
=> ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_1135_IntI,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ A2 )
=> ( ( member_nat_nat_nat2 @ C @ B3 )
=> ( member_nat_nat_nat2 @ C @ ( inf_in5274420515160781174at_nat @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_1136_IntI,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ A2 )
=> ( ( member_nat_nat_nat @ C @ B3 )
=> ( member_nat_nat_nat @ C @ ( inf_in7997761893158376566at_nat @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_1137_IntI,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ A2 )
=> ( ( member952132173341509300at_nat @ C @ B3 )
=> ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_1138_IntI,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_1139_Int__iff,axiom,
! [C: nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B3 ) )
= ( ( member_nat_nat @ C @ A2 )
& ( member_nat_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_1140_Int__iff,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( inf_in5274420515160781174at_nat @ A2 @ B3 ) )
= ( ( member_nat_nat_nat2 @ C @ A2 )
& ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_1141_Int__iff,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( inf_in7997761893158376566at_nat @ A2 @ B3 ) )
= ( ( member_nat_nat_nat @ C @ A2 )
& ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_1142_Int__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A2 @ B3 ) )
= ( ( member952132173341509300at_nat @ C @ A2 )
& ( member952132173341509300at_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_1143_Int__iff,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B3 ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_1144_UnCI,axiom,
! [C: nat > nat,B3: set_nat_nat,A2: set_nat_nat] :
( ( ~ ( member_nat_nat @ C @ B3 )
=> ( member_nat_nat @ C @ A2 ) )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B3 ) ) ) ).
% UnCI
thf(fact_1145_UnCI,axiom,
! [C: nat > nat > nat,B3: set_nat_nat_nat,A2: set_nat_nat_nat] :
( ( ~ ( member_nat_nat_nat2 @ C @ B3 )
=> ( member_nat_nat_nat2 @ C @ A2 ) )
=> ( member_nat_nat_nat2 @ C @ ( sup_su3334021163961628176at_nat @ A2 @ B3 ) ) ) ).
% UnCI
thf(fact_1146_UnCI,axiom,
! [C: ( nat > nat ) > nat,B3: set_nat_nat_nat2,A2: set_nat_nat_nat2] :
( ( ~ ( member_nat_nat_nat @ C @ B3 )
=> ( member_nat_nat_nat @ C @ A2 ) )
=> ( member_nat_nat_nat @ C @ ( sup_su6057362541959223568at_nat @ A2 @ B3 ) ) ) ).
% UnCI
thf(fact_1147_UnCI,axiom,
! [C: ( nat > nat ) > nat > nat,B3: set_nat_nat_nat_nat3,A2: set_nat_nat_nat_nat3] :
( ( ~ ( member952132173341509300at_nat @ C @ B3 )
=> ( member952132173341509300at_nat @ C @ A2 ) )
=> ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A2 @ B3 ) ) ) ).
% UnCI
thf(fact_1148_UnCI,axiom,
! [C: nat,B3: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% UnCI
thf(fact_1149_Un__iff,axiom,
! [C: nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B3 ) )
= ( ( member_nat_nat @ C @ A2 )
| ( member_nat_nat @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_1150_Un__iff,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( sup_su3334021163961628176at_nat @ A2 @ B3 ) )
= ( ( member_nat_nat_nat2 @ C @ A2 )
| ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_1151_Un__iff,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( sup_su6057362541959223568at_nat @ A2 @ B3 ) )
= ( ( member_nat_nat_nat @ C @ A2 )
| ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_1152_Un__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A2 @ B3 ) )
= ( ( member952132173341509300at_nat @ C @ A2 )
| ( member952132173341509300at_nat @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_1153_Un__iff,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_1154_image__insert,axiom,
! [F: nat > set_nat,A: nat,B3: set_nat] :
( ( image_nat_set_nat @ F @ ( insert_nat @ A @ B3 ) )
= ( insert_set_nat @ ( F @ A ) @ ( image_nat_set_nat @ F @ B3 ) ) ) ).
% image_insert
thf(fact_1155_image__insert,axiom,
! [F: nat > nat,A: nat,B3: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A @ B3 ) )
= ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B3 ) ) ) ).
% image_insert
thf(fact_1156_image__insert,axiom,
! [F: nat > nat > nat,A: nat,B3: set_nat] :
( ( image_nat_nat_nat2 @ F @ ( insert_nat @ A @ B3 ) )
= ( insert_nat_nat @ ( F @ A ) @ ( image_nat_nat_nat2 @ F @ B3 ) ) ) ).
% image_insert
thf(fact_1157_image__insert,axiom,
! [F: ( nat > nat ) > nat,A: nat > nat,B3: set_nat_nat] :
( ( image_nat_nat_nat @ F @ ( insert_nat_nat @ A @ B3 ) )
= ( insert_nat @ ( F @ A ) @ ( image_nat_nat_nat @ F @ B3 ) ) ) ).
% image_insert
thf(fact_1158_image__insert,axiom,
! [F: ( nat > nat ) > nat > nat,A: nat > nat,B3: set_nat_nat] :
( ( image_3205354838064109189at_nat @ F @ ( insert_nat_nat @ A @ B3 ) )
= ( insert_nat_nat @ ( F @ A ) @ ( image_3205354838064109189at_nat @ F @ B3 ) ) ) ).
% image_insert
thf(fact_1159_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1160_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ A2 ) )
= ( image_nat_set_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1161_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_nat_nat @ ( F @ X2 ) @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( image_nat_nat_nat2 @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1162_insert__image,axiom,
! [X2: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_nat_nat_nat @ F @ A2 ) )
= ( image_nat_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1163_insert__image,axiom,
! [X2: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( ( insert_nat_nat @ ( F @ X2 ) @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1164_insert__image,axiom,
! [X2: nat > nat > nat,A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_913610194320715013at_nat @ F @ A2 ) )
= ( image_913610194320715013at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1165_insert__image,axiom,
! [X2: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_7809927846809980933at_nat @ F @ A2 ) )
= ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1166_insert__image,axiom,
! [X2: nat > nat > nat,A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
=> ( ( insert_nat_nat @ ( F @ X2 ) @ ( image_1545173636400105204at_nat @ F @ A2 ) )
= ( image_1545173636400105204at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1167_insert__image,axiom,
! [X2: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
=> ( ( insert_nat_nat @ ( F @ X2 ) @ ( image_1262493855416953332at_nat @ F @ A2 ) )
= ( image_1262493855416953332at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1168_insert__image,axiom,
! [X2: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > nat] :
( ( member952132173341509300at_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_8194121248528334964at_nat @ F @ A2 ) )
= ( image_8194121248528334964at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1169_singletonI,axiom,
! [A: nat > nat > nat] : ( member_nat_nat_nat2 @ A @ ( insert_nat_nat_nat2 @ A @ bot_bo7445843802507891576at_nat ) ) ).
% singletonI
thf(fact_1170_singletonI,axiom,
! [A: ( nat > nat ) > nat] : ( member_nat_nat_nat @ A @ ( insert_nat_nat_nat @ A @ bot_bo945813143650711160at_nat ) ) ).
% singletonI
thf(fact_1171_singletonI,axiom,
! [A: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ A @ ( insert1655528249163143117at_nat @ A @ bot_bo3919185967433191911at_nat ) ) ).
% singletonI
thf(fact_1172_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1173_singletonI,axiom,
! [A: nat > nat] : ( member_nat_nat @ A @ ( insert_nat_nat @ A @ bot_bot_set_nat_nat ) ) ).
% singletonI
thf(fact_1174_insert__subset,axiom,
! [X2: nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B3 )
= ( ( member_nat @ X2 @ B3 )
& ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_1175_insert__subset,axiom,
! [X2: nat > nat > nat,A2: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ ( insert_nat_nat_nat2 @ X2 @ A2 ) @ B3 )
= ( ( member_nat_nat_nat2 @ X2 @ B3 )
& ( ord_le3211623285424100676at_nat @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_1176_insert__subset,axiom,
! [X2: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ ( insert_nat_nat_nat @ X2 @ A2 ) @ B3 )
= ( ( member_nat_nat_nat @ X2 @ B3 )
& ( ord_le5934964663421696068at_nat @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_1177_insert__subset,axiom,
! [X2: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
( ( ord_le5260717879541182899at_nat @ ( insert1655528249163143117at_nat @ X2 @ A2 ) @ B3 )
= ( ( member952132173341509300at_nat @ X2 @ B3 )
& ( ord_le5260717879541182899at_nat @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_1178_insert__subset,axiom,
! [X2: nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( insert_nat_nat @ X2 @ A2 ) @ B3 )
= ( ( member_nat_nat @ X2 @ B3 )
& ( ord_le9059583361652607317at_nat @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_1179_Un__empty,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B3 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B3 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_1180_Un__empty,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A2 @ B3 )
= bot_bot_set_nat_nat )
= ( ( A2 = bot_bot_set_nat_nat )
& ( B3 = bot_bot_set_nat_nat ) ) ) ).
% Un_empty
thf(fact_1181_Int__subset__iff,axiom,
! [C4: set_nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ ( inf_inf_set_nat @ A2 @ B3 ) )
= ( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( ord_less_eq_set_nat @ C4 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_1182_Int__subset__iff,axiom,
! [C4: set_nat_nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C4 @ ( inf_inf_set_nat_nat @ A2 @ B3 ) )
= ( ( ord_le9059583361652607317at_nat @ C4 @ A2 )
& ( ord_le9059583361652607317at_nat @ C4 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_1183_Int__insert__left__if0,axiom,
! [A: nat > nat,C4: set_nat_nat,B3: set_nat_nat] :
( ~ ( member_nat_nat @ A @ C4 )
=> ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A @ B3 ) @ C4 )
= ( inf_inf_set_nat_nat @ B3 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_1184_Int__insert__left__if0,axiom,
! [A: nat > nat > nat,C4: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ~ ( member_nat_nat_nat2 @ A @ C4 )
=> ( ( inf_in5274420515160781174at_nat @ ( insert_nat_nat_nat2 @ A @ B3 ) @ C4 )
= ( inf_in5274420515160781174at_nat @ B3 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_1185_Int__insert__left__if0,axiom,
! [A: ( nat > nat ) > nat,C4: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ~ ( member_nat_nat_nat @ A @ C4 )
=> ( ( inf_in7997761893158376566at_nat @ ( insert_nat_nat_nat @ A @ B3 ) @ C4 )
= ( inf_in7997761893158376566at_nat @ B3 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_1186_Int__insert__left__if0,axiom,
! [A: ( nat > nat ) > nat > nat,C4: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
( ~ ( member952132173341509300at_nat @ A @ C4 )
=> ( ( inf_in2949407623404935909at_nat @ ( insert1655528249163143117at_nat @ A @ B3 ) @ C4 )
= ( inf_in2949407623404935909at_nat @ B3 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_1187_Int__insert__left__if0,axiom,
! [A: nat,C4: set_nat,B3: set_nat] :
( ~ ( member_nat @ A @ C4 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B3 ) @ C4 )
= ( inf_inf_set_nat @ B3 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_1188_Int__insert__left__if1,axiom,
! [A: nat > nat,C4: set_nat_nat,B3: set_nat_nat] :
( ( member_nat_nat @ A @ C4 )
=> ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A @ B3 ) @ C4 )
= ( insert_nat_nat @ A @ ( inf_inf_set_nat_nat @ B3 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1189_Int__insert__left__if1,axiom,
! [A: nat > nat > nat,C4: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ A @ C4 )
=> ( ( inf_in5274420515160781174at_nat @ ( insert_nat_nat_nat2 @ A @ B3 ) @ C4 )
= ( insert_nat_nat_nat2 @ A @ ( inf_in5274420515160781174at_nat @ B3 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1190_Int__insert__left__if1,axiom,
! [A: ( nat > nat ) > nat,C4: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ A @ C4 )
=> ( ( inf_in7997761893158376566at_nat @ ( insert_nat_nat_nat @ A @ B3 ) @ C4 )
= ( insert_nat_nat_nat @ A @ ( inf_in7997761893158376566at_nat @ B3 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1191_Int__insert__left__if1,axiom,
! [A: ( nat > nat ) > nat > nat,C4: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ A @ C4 )
=> ( ( inf_in2949407623404935909at_nat @ ( insert1655528249163143117at_nat @ A @ B3 ) @ C4 )
= ( insert1655528249163143117at_nat @ A @ ( inf_in2949407623404935909at_nat @ B3 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1192_Int__insert__left__if1,axiom,
! [A: nat,C4: set_nat,B3: set_nat] :
( ( member_nat @ A @ C4 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B3 ) @ C4 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B3 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1193_insert__inter__insert,axiom,
! [A: nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A @ A2 ) @ ( insert_nat_nat @ A @ B3 ) )
= ( insert_nat_nat @ A @ ( inf_inf_set_nat_nat @ A2 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_1194_insert__inter__insert,axiom,
! [A: nat,A2: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ ( insert_nat @ A @ B3 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_1195_Int__insert__right__if0,axiom,
! [A: nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ~ ( member_nat_nat @ A @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ ( insert_nat_nat @ A @ B3 ) )
= ( inf_inf_set_nat_nat @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1196_Int__insert__right__if0,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ~ ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( inf_in5274420515160781174at_nat @ A2 @ ( insert_nat_nat_nat2 @ A @ B3 ) )
= ( inf_in5274420515160781174at_nat @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1197_Int__insert__right__if0,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ~ ( member_nat_nat_nat @ A @ A2 )
=> ( ( inf_in7997761893158376566at_nat @ A2 @ ( insert_nat_nat_nat @ A @ B3 ) )
= ( inf_in7997761893158376566at_nat @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1198_Int__insert__right__if0,axiom,
! [A: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
( ~ ( member952132173341509300at_nat @ A @ A2 )
=> ( ( inf_in2949407623404935909at_nat @ A2 @ ( insert1655528249163143117at_nat @ A @ B3 ) )
= ( inf_in2949407623404935909at_nat @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1199_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B3: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B3 ) )
= ( inf_inf_set_nat @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_1200_Int__insert__right__if1,axiom,
! [A: nat > nat,A2: set_nat_nat,B3: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ ( insert_nat_nat @ A @ B3 ) )
= ( insert_nat_nat @ A @ ( inf_inf_set_nat_nat @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1201_Int__insert__right__if1,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( inf_in5274420515160781174at_nat @ A2 @ ( insert_nat_nat_nat2 @ A @ B3 ) )
= ( insert_nat_nat_nat2 @ A @ ( inf_in5274420515160781174at_nat @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1202_Int__insert__right__if1,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( inf_in7997761893158376566at_nat @ A2 @ ( insert_nat_nat_nat @ A @ B3 ) )
= ( insert_nat_nat_nat @ A @ ( inf_in7997761893158376566at_nat @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1203_Int__insert__right__if1,axiom,
! [A: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B3: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ A @ A2 )
=> ( ( inf_in2949407623404935909at_nat @ A2 @ ( insert1655528249163143117at_nat @ A @ B3 ) )
= ( insert1655528249163143117at_nat @ A @ ( inf_in2949407623404935909at_nat @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1204_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B3 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1205_Un__subset__iff,axiom,
! [A2: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ C4 )
= ( ( ord_less_eq_set_nat @ A2 @ C4 )
& ( ord_less_eq_set_nat @ B3 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_1206_Un__subset__iff,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B3 ) @ C4 )
= ( ( ord_le9059583361652607317at_nat @ A2 @ C4 )
& ( ord_le9059583361652607317at_nat @ B3 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_1207_Un__insert__left,axiom,
! [A: nat > nat,B3: set_nat_nat,C4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( insert_nat_nat @ A @ B3 ) @ C4 )
= ( insert_nat_nat @ A @ ( sup_sup_set_nat_nat @ B3 @ C4 ) ) ) ).
% Un_insert_left
thf(fact_1208_Un__insert__left,axiom,
! [A: nat,B3: set_nat,C4: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B3 ) @ C4 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B3 @ C4 ) ) ) ).
% Un_insert_left
thf(fact_1209_Un__insert__right,axiom,
! [A2: set_nat_nat,A: nat > nat,B3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ ( insert_nat_nat @ A @ B3 ) )
= ( insert_nat_nat @ A @ ( sup_sup_set_nat_nat @ A2 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_1210_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B3: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B3 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_1211_Int__Un__eq_I4_J,axiom,
! [T3: set_nat,S5: set_nat] :
( ( sup_sup_set_nat @ T3 @ ( inf_inf_set_nat @ S5 @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_1212_Int__Un__eq_I3_J,axiom,
! [S5: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ S5 @ ( inf_inf_set_nat @ S5 @ T3 ) )
= S5 ) ).
% Int_Un_eq(3)
thf(fact_1213_Int__Un__eq_I2_J,axiom,
! [S5: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S5 @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_1214_Int__Un__eq_I1_J,axiom,
! [S5: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S5 @ T3 ) @ S5 )
= S5 ) ).
% Int_Un_eq(1)
thf(fact_1215_Un__Int__eq_I4_J,axiom,
! [T3: set_nat,S5: set_nat] :
( ( inf_inf_set_nat @ T3 @ ( sup_sup_set_nat @ S5 @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_1216_Un__Int__eq_I3_J,axiom,
! [S5: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ S5 @ ( sup_sup_set_nat @ S5 @ T3 ) )
= S5 ) ).
% Un_Int_eq(3)
thf(fact_1217_Un__Int__eq_I2_J,axiom,
! [S5: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S5 @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_1218_B__props,axiom,
( ( ( sup_sup_set_nat @ ( b @ zero_zero_nat ) @ ( b @ one_one_nat ) )
= ( set_ord_lessThan_nat @ n ) )
& ( ( inf_inf_set_nat @ ( b @ zero_zero_nat ) @ ( b @ one_one_nat ) )
= bot_bot_set_nat ) ) ).
% B_props
thf(fact_1219_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_1220_is__line__elim__t__1,axiom,
! [L2: nat > nat > nat,N: nat,T: nat] :
( ( hales_is_line @ L2 @ N @ T )
=> ( ( T = one_one_nat )
=> ~ ! [B_0: set_nat,B_1: set_nat] :
~ ( ( ( sup_sup_set_nat @ B_0 @ B_1 )
= ( set_ord_lessThan_nat @ N ) )
& ( ( inf_inf_set_nat @ B_0 @ B_1 )
= bot_bot_set_nat )
& ( B_0 != bot_bot_set_nat )
& ! [X4: nat] :
( ( member_nat @ X4 @ B_1 )
=> ! [Xa: nat] :
( ( ord_less_nat @ Xa @ T )
=> ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ T )
=> ( ( L2 @ Xa @ X4 )
= ( L2 @ Y3 @ X4 ) ) ) ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ B_0 )
=> ! [S2: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( ( L2 @ S2 @ X4 )
= S2 ) ) ) ) ) ) ).
% is_line_elim_t_1
thf(fact_1221_cube1__alt__def,axiom,
! [N: nat] :
( ( hales_cube @ N @ one_one_nat )
= ( insert_nat_nat
@ ( restrict_nat_nat
@ ^ [X3: nat] : zero_zero_nat
@ ( set_ord_lessThan_nat @ N ) )
@ bot_bot_set_nat_nat ) ) ).
% cube1_alt_def
thf(fact_1222_cube0__alt__def,axiom,
! [T: nat] :
( ( hales_cube @ zero_zero_nat @ T )
= ( insert_nat_nat
@ ^ [X3: nat] : undefined_nat
@ bot_bot_set_nat_nat ) ) ).
% cube0_alt_def
thf(fact_1223_L_H__def,axiom,
( l
= ( fun_upd_nat_nat_nat @ l2 @ t
@ ^ [J3: nat] : ( if_nat @ ( member_nat @ J3 @ ( b @ one_one_nat ) ) @ ( l2 @ ( minus_minus_nat @ t @ one_one_nat ) @ J3 ) @ ( if_nat @ ( member_nat @ J3 @ ( b @ zero_zero_nat ) ) @ t @ undefined_nat ) ) ) ) ).
% L'_def
thf(fact_1224_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1225_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1226_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1227_diff__diff__left,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_1228_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1229_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1230_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1231_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1232_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1233_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1234_set__incr__altdef,axiom,
( hales_set_incr
= ( ^ [N3: nat] : ( image_nat_nat @ ( plus_plus_nat @ N3 ) ) ) ) ).
% set_incr_altdef
thf(fact_1235_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1236_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1237_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1238_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1239_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1240_less__diff__conv,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1241_le__diff__conv,axiom,
! [J2: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1242_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1243_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1244_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1245_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I )
= K )
= ( J2
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1246_diff__commute,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_1247_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1248_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1249_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1250_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1251_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1252_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1253_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1254_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1255_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1256_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1257_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1258_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1259_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1260_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1261_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1262_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 ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1263_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 ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1264_less__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1265_set__incr__def,axiom,
( hales_set_incr
= ( ^ [N3: nat] :
( image_nat_nat
@ ^ [A3: nat] : ( plus_plus_nat @ A3 @ N3 ) ) ) ) ).
% set_incr_def
% Helper facts (8)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X7: nat] : ( P @ X7 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: ( nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat @ P ) )
= ( ? [X7: nat > nat] : ( P @ X7 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_T,axiom,
! [P: ( ( nat > nat ) > nat ) > $o] :
( ( P @ ( fChoice_nat_nat_nat @ P ) )
= ( ? [X7: ( nat > nat ) > nat] : ( P @ X7 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [P: ( nat > nat > nat ) > $o] :
( ( P @ ( fChoice_nat_nat_nat2 @ P ) )
= ( ? [X7: nat > nat > nat] : ( P @ X7 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( P @ ( fChoic52552927678224201at_nat @ P ) )
= ( ? [X7: ( nat > nat ) > nat > nat] : ( P @ X7 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( l2 @ i @ j )
= ( f @ j ) ) ).
%------------------------------------------------------------------------------