TPTP Problem File: SLH0447^1.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_02080_095844__6148566_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1522 ( 498 unt; 249 typ; 0 def)
% Number of atoms : 4072 (1216 equ; 0 cnn)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 12637 ( 354 ~; 49 |; 255 &;10125 @)
% ( 0 <=>;1854 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 3208 (3208 >; 0 *; 0 +; 0 <<)
% Number of symbols : 228 ( 225 usr; 26 con; 0-6 aty)
% Number of variables : 4170 ( 387 ^;3651 !; 132 ?;4170 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:47:20.250
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
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_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_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_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_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 (225)
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_Finite__Set_Ocard_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite_card_nat_nat: set_nat_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Fun_Oinj__on_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inj_on2461717442902640625at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
inj_on_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inj_on_nat_nat_nat2: ( nat > nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_nat_set_nat: ( nat > set_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_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,
inj_on290067182627368541at_nat: ( set_nat_nat > set_nat_nat ) > set_set_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on1908564424730267886et_nat: ( set_nat_nat > set_nat ) > set_set_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inj_on2031908246727262830at_nat: ( set_nat > set_nat_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inj_on2776966659131765557et_nat: ( set_nat > set_set_nat ) > set_set_nat > $o ).
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_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,
minus_7158188067284919257_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > $o ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
minus_2851842960567056136_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > ( ( ( nat > nat ) > nat ) > $o ) > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
minus_7240682219522218504_nat_o: ( ( nat > nat > nat ) > $o ) > ( ( nat > nat > nat ) > $o ) > ( nat > nat > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
minus_167519014754328503_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__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,
minus_4646100876039749548at_nat: set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 > set_nat_nat_nat_nat3 ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
minus_1221035652888719293at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > set_nat_nat_nat2 ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
minus_7721066311745899709at_nat: set_nat_nat_nat > set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_8121590178497047118at_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_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_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_If_001t__Nat__Onat,type,
if_nat: $o > nat > 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_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_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_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_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_I_Eo_J,type,
bot_bot_set_o: set_o ).
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_001_Eo,type,
ord_less_o: $o > $o > $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_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
ord_le2284792974450617250at_nat: set_na7938001796681673538at_nat > set_na7938001796681673538at_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le5849559942836194483at_nat: set_nat_nat_nat_nat5 > set_nat_nat_nat_nat5 > $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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
ord_le4355234527790737843at_nat: set_nat_nat_nat_nat4 > set_nat_nat_nat_nat4 > $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_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
image_8690456353314504180_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat3 > set_o ).
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_Mt__Nat__Onat_J_001_Eo,type,
image_nat_nat_nat_o: ( ( ( nat > nat ) > nat ) > $o ) > set_nat_nat_nat2 > set_o ).
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_001_Eo,type,
image_nat_nat_nat_o2: ( ( nat > nat > nat ) > $o ) > set_nat_nat_nat > set_o ).
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_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_001_Eo,type,
image_nat_nat_o: ( ( nat > nat ) > $o ) > set_nat_nat > set_o ).
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_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
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_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_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__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_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_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_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_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_6930934588239670658et_nat: ( set_nat_nat > set_nat ) > set_set_nat_nat > set_set_nat ).
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__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__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_001_Eo,type,
set_ord_atMost_o: $o > set_o ).
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_It__Nat__Onat_Mt__Nat__Onat_J,type,
fChoice_nat_nat: ( ( nat > nat ) > $o ) > nat > 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_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__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,
member7681264892014656106at_nat: set_nat_nat_nat_nat3 > set_se3022870823424313865at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member1694410638372364155at_nat: set_nat_nat_nat2 > set_set_nat_nat_nat2 > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8194441297229544571at_nat: set_nat_nat_nat > set_set_nat_nat_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_S,type,
s: ( nat > nat ) > nat > nat ).
thf(sy_v__092_060chi_062,type,
chi: ( nat > nat ) > nat ).
thf(sy_v_a____,type,
a: nat ).
thf(sy_v_ax____,type,
ax: nat > nat ).
thf(sy_v_b____,type,
b2: nat ).
thf(sy_v_bx____,type,
bx: nat > nat ).
thf(sy_v_f____,type,
f: nat > nat ).
thf(sy_v_j____,type,
j: nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_s____,type,
s2: nat ).
thf(sy_v_t,type,
t: nat ).
thf(sy_v_u____,type,
u: nat ).
thf(sy_v_v____,type,
v: nat ).
thf(sy_v_x____,type,
x: nat > nat > nat ).
thf(sy_v_y____,type,
y: nat > nat > nat ).
% Relevant facts (1268)
thf(fact_0_assms_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ t ).
% assms(2)
thf(fact_1__092_060open_0620_A_060_Ak_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ k ).
% \<open>0 < k\<close>
thf(fact_2__C1_C,axiom,
member_nat @ j @ ( b @ k ) ).
% "1"
thf(fact_3_j__prop,axiom,
ord_less_nat @ j @ n ).
% j_prop
thf(fact_4_f__classes__u,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ t )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [I: nat] : ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ v ) ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ u ) ) @ J @ t ) )
@ ( set_ord_lessThan_nat @ k ) )
@ ( hales_classes @ k @ t @ u ) ) ) ).
% f_classes_u
thf(fact_5_f__classes__v,axiom,
! [J: nat] :
( ( J = t )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [I: nat] : ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ v ) ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ u ) ) @ J @ t ) )
@ ( set_ord_lessThan_nat @ k ) )
@ ( hales_classes @ k @ t @ v ) ) ) ).
% f_classes_v
thf(fact_6_calculation,axiom,
( ( y @ a @ j )
= ( s
@ ( restrict_nat_nat
@ ^ [I: nat] : ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ v ) ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ u ) ) @ a @ t ) )
@ ( set_ord_lessThan_nat @ k ) )
@ j ) ) ).
% calculation
thf(fact_7_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_8__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062j_A_092_060in_062_AB_Ak_A_092_060Longrightarrow_062_Athesis_059_A_092_060exists_062ii_060k_O_Aj_A_092_060in_062_AB_Aii_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
( ~ ( member_nat @ j @ ( b @ k ) )
=> ? [Ii: nat] :
( ( ord_less_nat @ Ii @ k )
& ( member_nat @ j @ ( b @ Ii ) ) ) ) ).
% \<open>\<And>thesis. \<lbrakk>j \<in> B k \<Longrightarrow> thesis; \<exists>ii<k. j \<in> B ii \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_9_lessThan__iff,axiom,
! [I2: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I2 @ ( set_or1140352010380016476at_nat @ K ) )
= ( ord_less_nat_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_10_lessThan__iff,axiom,
! [I2: nat > nat > nat,K: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I2 @ ( set_or3808701207811398603at_nat @ K ) )
= ( ord_less_nat_nat_nat2 @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_11_lessThan__iff,axiom,
! [I2: ( nat > nat ) > nat,K: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I2 @ ( set_or2699333443382148811at_nat @ K ) )
= ( ord_less_nat_nat_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_12_lessThan__iff,axiom,
! [I2: ( nat > nat ) > nat > nat,K: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I2 @ ( set_or7562748684798938298at_nat @ K ) )
= ( ord_le4629963735342356977at_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_13_lessThan__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_14_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_15_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_16_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_17_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_18_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_19_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_20_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_21_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_22_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_23_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_24_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_25_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_26_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_27_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_28_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_29_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_30_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_31_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_32_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_33_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_34_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_35_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_36_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_37_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_38_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_39_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_40_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_41_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_42_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_43_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A2: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat] :
( ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat2] :
( ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_50_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_51_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_52_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_53_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_54_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_55_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_56_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_57_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_58_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_59_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_60_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_61_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_62_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ U ) ) ) ) ).
% lessThan_def
thf(fact_63_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_64_y__def,axiom,
( y
= ( restrict_nat_nat_nat2
@ ^ [S2: nat] :
( s
@ ( restrict_nat_nat
@ ^ [I: nat] : ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ v ) ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ u ) ) @ S2 @ t ) )
@ ( set_ord_lessThan_nat @ k ) ) )
@ ( set_ord_atMost_nat @ t ) ) ) ).
% y_def
thf(fact_65_assms_I1_J,axiom,
hales_4261547300027266985ce_nat @ s @ k @ n @ t @ k @ chi ).
% assms(1)
thf(fact_66_that_I1_J,axiom,
ord_less_nat @ a @ ( plus_plus_nat @ t @ one_one_nat ) ).
% that(1)
thf(fact_67_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_68_Bf__props_I6_J,axiom,
! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ k @ ( plus_plus_nat @ t @ one_one_nat ) ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( b @ k ) )
=> ( ( s @ X4 @ Xa )
= ( f @ Xa ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ k )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( b @ J2 ) )
=> ( ( s @ X4 @ Xa )
= ( X4 @ J2 ) ) ) ) ) ) ).
% Bf_props(6)
thf(fact_69_Bf__props_I3_J,axiom,
~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ b @ ( set_ord_lessThan_nat @ k ) ) ) ).
% Bf_props(3)
thf(fact_70_f__cube,axiom,
! [J: nat] :
( ( ord_less_nat @ J @ ( plus_plus_nat @ t @ one_one_nat ) )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [I: nat] : ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ v ) ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ u ) ) @ J @ t ) )
@ ( set_ord_lessThan_nat @ k ) )
@ ( hales_cube @ k @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% f_cube
thf(fact_71_Bf__props_I4_J,axiom,
( member_nat_nat @ f
@ ( piE_nat_nat @ ( b @ k )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% Bf_props(4)
thf(fact_72_restrict__apply_H,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( restrict_nat_nat @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_73_restrict__apply_H,axiom,
! [X: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_74_restrict__apply_H,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( restri4446420529079022766at_nat @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_75_restrict__ext,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat @ F @ A2 )
= ( restrict_nat_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_76_restrict__ext,axiom,
! [A2: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 )
= ( restrict_nat_nat_nat2 @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_77_restrict__ext,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( restri4446420529079022766at_nat @ F @ A2 )
= ( restri4446420529079022766at_nat @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_78_Bf__props_I1_J,axiom,
disjoi6798895846410478970at_nat @ b @ ( set_ord_atMost_nat @ k ) ).
% Bf_props(1)
thf(fact_79_that_I3_J,axiom,
ord_less_nat @ s2 @ ( plus_plus_nat @ t @ one_one_nat ) ).
% that(3)
thf(fact_80_that_I2_J,axiom,
ord_less_nat @ b2 @ ( plus_plus_nat @ t @ one_one_nat ) ).
% that(2)
thf(fact_81_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_82_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_83__092_060open_062k_A_060_Ak_A_L_A1_092_060close_062,axiom,
ord_less_nat @ k @ ( plus_plus_nat @ k @ one_one_nat ) ).
% \<open>k < k + 1\<close>
thf(fact_84_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_85_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_86_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_87_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_88_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_89_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_90_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_91_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_92_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_93_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_94_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_95_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_96_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_97_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_98_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_99_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_100_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_101_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_102_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_103_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_104_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_105_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_106_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_107_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_108_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ A2 @ B2 ) )
=> ( ( restrict_nat_nat_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_109_PiE__restrict,axiom,
! [F: nat > nat,A2: set_nat,B2: nat > set_nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ A2 @ B2 ) )
=> ( ( restrict_nat_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_110_PiE__restrict,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ A2 @ B2 ) )
=> ( ( restrict_nat_nat_nat2 @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_111_PiE__restrict,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ A2 @ B2 ) )
=> ( ( restri4446420529079022766at_nat @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_112_PiE__empty__range,axiom,
! [I2: nat > nat,I3: set_nat_nat,F2: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ I2 @ I3 )
=> ( ( ( F2 @ I2 )
= bot_bot_set_nat_nat )
=> ( ( piE_nat_nat_nat_nat3 @ I3 @ F2 )
= bot_bo3919185967433191911at_nat ) ) ) ).
% PiE_empty_range
thf(fact_113_PiE__empty__range,axiom,
! [I2: nat,I3: set_nat,F2: nat > set_nat_nat] :
( ( member_nat @ I2 @ I3 )
=> ( ( ( F2 @ I2 )
= bot_bot_set_nat_nat )
=> ( ( piE_nat_nat_nat2 @ I3 @ F2 )
= bot_bo7445843802507891576at_nat ) ) ) ).
% PiE_empty_range
thf(fact_114_PiE__empty__range,axiom,
! [I2: nat > nat > nat,I3: set_nat_nat_nat,F2: ( nat > nat > nat ) > set_nat] :
( ( member_nat_nat_nat2 @ I2 @ I3 )
=> ( ( ( F2 @ I2 )
= bot_bot_set_nat )
=> ( ( piE_nat_nat_nat_nat2 @ I3 @ F2 )
= bot_bo3013702615682746855at_nat ) ) ) ).
% PiE_empty_range
thf(fact_115_PiE__empty__range,axiom,
! [I2: ( nat > nat ) > nat,I3: set_nat_nat_nat2,F2: ( ( nat > nat ) > nat ) > set_nat] :
( ( member_nat_nat_nat @ I2 @ I3 )
=> ( ( ( F2 @ I2 )
= bot_bot_set_nat )
=> ( ( piE_nat_nat_nat_nat @ I3 @ F2 )
= bot_bo4508028030728203495at_nat ) ) ) ).
% PiE_empty_range
thf(fact_116_PiE__empty__range,axiom,
! [I2: ( nat > nat ) > nat > nat,I3: set_nat_nat_nat_nat3,F2: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ( member952132173341509300at_nat @ I2 @ I3 )
=> ( ( ( F2 @ I2 )
= bot_bot_set_nat )
=> ( ( piE_na4548495224246695081at_nat @ I3 @ F2 )
= bot_bo3386126977483763158at_nat ) ) ) ).
% PiE_empty_range
thf(fact_117_PiE__empty__range,axiom,
! [I2: nat,I3: set_nat,F2: nat > set_nat] :
( ( member_nat @ I2 @ I3 )
=> ( ( ( F2 @ I2 )
= bot_bot_set_nat )
=> ( ( piE_nat_nat @ I3 @ F2 )
= bot_bot_set_nat_nat ) ) ) ).
% PiE_empty_range
thf(fact_118_PiE__empty__range,axiom,
! [I2: nat > nat,I3: set_nat_nat,F2: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ I2 @ I3 )
=> ( ( ( F2 @ I2 )
= bot_bot_set_nat )
=> ( ( piE_nat_nat_nat @ I3 @ F2 )
= bot_bo945813143650711160at_nat ) ) ) ).
% PiE_empty_range
thf(fact_119_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_120_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_121_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_122_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_123_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_124_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_125_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_126_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_127_Bf__props_I2_J,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ b @ ( set_ord_atMost_nat @ k ) ) )
= ( set_ord_lessThan_nat @ n ) ) ).
% Bf_props(2)
thf(fact_128_Bf__props_I5_J,axiom,
( member952132173341509300at_nat @ s
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ k @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] : ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% Bf_props(5)
thf(fact_129_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_130_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_131_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_132_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_133_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_134_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_135_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_136_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_137_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_138_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_139_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_140_PiE__eq__iff__not__empty,axiom,
! [I3: set_nat_nat,F2: ( nat > nat ) > set_nat_nat,F3: ( nat > nat ) > set_nat_nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat_nat3 @ I3 @ F2 )
= ( piE_nat_nat_nat_nat3 @ I3 @ F3 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_141_PiE__eq__iff__not__empty,axiom,
! [I3: set_nat,F2: nat > set_nat_nat,F3: nat > set_nat_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat2 @ I3 @ F2 )
= ( piE_nat_nat_nat2 @ I3 @ F3 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_142_PiE__eq__iff__not__empty,axiom,
! [I3: set_nat_nat_nat,F2: ( nat > nat > nat ) > set_nat,F3: ( nat > nat > nat ) > set_nat] :
( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat ) )
=> ( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat_nat2 @ I3 @ F2 )
= ( piE_nat_nat_nat_nat2 @ I3 @ F3 ) )
= ( ! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ I3 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_143_PiE__eq__iff__not__empty,axiom,
! [I3: set_nat_nat_nat2,F2: ( ( nat > nat ) > nat ) > set_nat,F3: ( ( nat > nat ) > nat ) > set_nat] :
( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat ) )
=> ( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat_nat @ I3 @ F2 )
= ( piE_nat_nat_nat_nat @ I3 @ F3 ) )
= ( ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ I3 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_144_PiE__eq__iff__not__empty,axiom,
! [I3: set_nat_nat_nat_nat3,F2: ( ( nat > nat ) > nat > nat ) > set_nat,F3: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ! [I4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat ) )
=> ( ! [I4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_na4548495224246695081at_nat @ I3 @ F2 )
= ( piE_na4548495224246695081at_nat @ I3 @ F3 ) )
= ( ! [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ I3 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_145_PiE__eq__iff__not__empty,axiom,
! [I3: set_nat,F2: nat > set_nat,F3: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat @ I3 @ F2 )
= ( piE_nat_nat @ I3 @ F3 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_146_PiE__eq__iff__not__empty,axiom,
! [I3: set_nat_nat,F2: ( nat > nat ) > set_nat,F3: ( nat > nat ) > set_nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat @ I3 @ F2 )
= ( piE_nat_nat_nat @ I3 @ F3 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_147_PiE__eq__empty__iff,axiom,
! [I3: set_nat_nat,F2: ( nat > nat ) > set_nat_nat] :
( ( ( piE_nat_nat_nat_nat3 @ I3 @ F2 )
= bot_bo3919185967433191911at_nat )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
& ( ( F2 @ X2 )
= bot_bot_set_nat_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_148_PiE__eq__empty__iff,axiom,
! [I3: set_nat,F2: nat > set_nat_nat] :
( ( ( piE_nat_nat_nat2 @ I3 @ F2 )
= bot_bo7445843802507891576at_nat )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ I3 )
& ( ( F2 @ X2 )
= bot_bot_set_nat_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_149_PiE__eq__empty__iff,axiom,
! [I3: set_nat,F2: nat > set_nat] :
( ( ( piE_nat_nat @ I3 @ F2 )
= bot_bot_set_nat_nat )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ I3 )
& ( ( F2 @ X2 )
= bot_bot_set_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_150_PiE__eq__empty__iff,axiom,
! [I3: set_nat_nat,F2: ( nat > nat ) > set_nat] :
( ( ( piE_nat_nat_nat @ I3 @ F2 )
= bot_bo945813143650711160at_nat )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
& ( ( F2 @ X2 )
= bot_bot_set_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_151_PiE__eq__iff,axiom,
! [I3: set_nat_nat,F2: ( nat > nat ) > set_nat_nat,F3: ( nat > nat ) > set_nat_nat] :
( ( ( piE_nat_nat_nat_nat3 @ I3 @ F2 )
= ( piE_nat_nat_nat_nat3 @ I3 @ F3 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) )
| ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
& ( ( F2 @ X2 )
= bot_bot_set_nat_nat ) )
& ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
& ( ( F3 @ X2 )
= bot_bot_set_nat_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_152_PiE__eq__iff,axiom,
! [I3: set_nat,F2: nat > set_nat_nat,F3: nat > set_nat_nat] :
( ( ( piE_nat_nat_nat2 @ I3 @ F2 )
= ( piE_nat_nat_nat2 @ I3 @ F3 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) )
| ( ? [X2: nat] :
( ( member_nat @ X2 @ I3 )
& ( ( F2 @ X2 )
= bot_bot_set_nat_nat ) )
& ? [X2: nat] :
( ( member_nat @ X2 @ I3 )
& ( ( F3 @ X2 )
= bot_bot_set_nat_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_153_PiE__eq__iff,axiom,
! [I3: set_nat,F2: nat > set_nat,F3: nat > set_nat] :
( ( ( piE_nat_nat @ I3 @ F2 )
= ( piE_nat_nat @ I3 @ F3 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) )
| ( ? [X2: nat] :
( ( member_nat @ X2 @ I3 )
& ( ( F2 @ X2 )
= bot_bot_set_nat ) )
& ? [X2: nat] :
( ( member_nat @ X2 @ I3 )
& ( ( F3 @ X2 )
= bot_bot_set_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_154_PiE__eq__iff,axiom,
! [I3: set_nat_nat,F2: ( nat > nat ) > set_nat,F3: ( nat > nat ) > set_nat] :
( ( ( piE_nat_nat_nat @ I3 @ F2 )
= ( piE_nat_nat_nat @ I3 @ F3 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( ( F2 @ X2 )
= ( F3 @ X2 ) ) )
| ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
& ( ( F2 @ X2 )
= bot_bot_set_nat ) )
& ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
& ( ( F3 @ X2 )
= bot_bot_set_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_155_PiE__cong,axiom,
! [I3: set_nat,A2: nat > set_nat,B2: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_nat_nat @ I3 @ A2 )
= ( piE_nat_nat @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_156_PiE__cong,axiom,
! [I3: set_nat_nat,A2: ( nat > nat ) > set_nat_nat,B2: ( nat > nat ) > set_nat_nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_nat_nat_nat_nat3 @ I3 @ A2 )
= ( piE_nat_nat_nat_nat3 @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_157_PiE__cong,axiom,
! [I3: set_nat,A2: nat > set_nat_nat,B2: nat > set_nat_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_nat_nat_nat2 @ I3 @ A2 )
= ( piE_nat_nat_nat2 @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_158_PiE__cong,axiom,
! [I3: set_nat_nat,A2: ( nat > nat ) > set_nat,B2: ( nat > nat ) > set_nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( A2 @ I4 )
= ( B2 @ I4 ) ) )
=> ( ( piE_nat_nat_nat @ I3 @ A2 )
= ( piE_nat_nat_nat @ I3 @ B2 ) ) ) ).
% PiE_cong
thf(fact_159_PiE__mem,axiom,
! [F: nat > nat,S3: set_nat,T2: nat > set_nat,X: nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ S3 @ T2 ) )
=> ( ( member_nat @ X @ S3 )
=> ( member_nat @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_160_PiE__mem,axiom,
! [F: nat > nat > nat,S3: set_nat,T2: nat > set_nat_nat,X: nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ S3 @ T2 ) )
=> ( ( member_nat @ X @ S3 )
=> ( member_nat_nat @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_161_PiE__mem,axiom,
! [F: ( nat > nat ) > nat,S3: set_nat_nat,T2: ( nat > nat ) > set_nat,X: nat > nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ S3 @ T2 ) )
=> ( ( member_nat_nat @ X @ S3 )
=> ( member_nat @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_162_PiE__mem,axiom,
! [F: nat > nat > nat > nat,S3: set_nat,T2: nat > set_nat_nat_nat,X: nat] :
( ( member17114558718834868at_nat @ F @ ( piE_nat_nat_nat_nat5 @ S3 @ T2 ) )
=> ( ( member_nat @ X @ S3 )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_163_PiE__mem,axiom,
! [F: nat > ( nat > nat ) > nat,S3: set_nat,T2: nat > set_nat_nat_nat2,X: nat] :
( ( member2740455936716430260at_nat @ F @ ( piE_nat_nat_nat_nat4 @ S3 @ T2 ) )
=> ( ( member_nat @ X @ S3 )
=> ( member_nat_nat_nat @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_164_PiE__mem,axiom,
! [F: ( nat > nat > nat ) > nat,S3: set_nat_nat_nat,T2: ( nat > nat > nat ) > set_nat,X: nat > nat > nat] :
( ( member5318315686745620148at_nat @ F @ ( piE_nat_nat_nat_nat2 @ S3 @ T2 ) )
=> ( ( member_nat_nat_nat2 @ X @ S3 )
=> ( member_nat @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_165_PiE__mem,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,S3: set_nat_nat_nat2,T2: ( ( nat > nat ) > nat ) > set_nat,X: ( nat > nat ) > nat] :
( ( member2991261302380110260at_nat @ F @ ( piE_nat_nat_nat_nat @ S3 @ T2 ) )
=> ( ( member_nat_nat_nat @ X @ S3 )
=> ( member_nat @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_166_PiE__mem,axiom,
! [F: ( nat > nat ) > nat > nat,S3: set_nat_nat,T2: ( nat > nat ) > set_nat_nat,X: nat > nat] :
( ( member952132173341509300at_nat @ F @ ( piE_nat_nat_nat_nat3 @ S3 @ T2 ) )
=> ( ( member_nat_nat @ X @ S3 )
=> ( member_nat_nat @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_167_PiE__mem,axiom,
! [F: nat > ( nat > nat ) > nat > nat,S3: set_nat,T2: nat > set_nat_nat_nat_nat3,X: nat] :
( ( member8743709692935548195at_nat @ F @ ( piE_na2748089427378204713at_nat @ S3 @ T2 ) )
=> ( ( member_nat @ X @ S3 )
=> ( member952132173341509300at_nat @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_168_PiE__mem,axiom,
! [F: ( nat > nat ) > nat > nat > nat,S3: set_nat_nat,T2: ( nat > nat ) > set_nat_nat_nat,X: nat > nat] :
( ( member1679187572556404771at_nat @ F @ ( piE_na8678869062391380393at_nat @ S3 @ T2 ) )
=> ( ( member_nat_nat @ X @ S3 )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_169_PiE__ext,axiom,
! [X: nat > nat,K: set_nat,S: nat > set_nat,Y: nat > nat] :
( ( member_nat_nat @ X @ ( piE_nat_nat @ K @ S ) )
=> ( ( member_nat_nat @ Y @ ( piE_nat_nat @ K @ S ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ K )
=> ( ( X @ I4 )
= ( Y @ I4 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_170_PiE__ext,axiom,
! [X: ( nat > nat ) > nat > nat,K: set_nat_nat,S: ( nat > nat ) > set_nat_nat,Y: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ ( piE_nat_nat_nat_nat3 @ K @ S ) )
=> ( ( member952132173341509300at_nat @ Y @ ( piE_nat_nat_nat_nat3 @ K @ S ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ K )
=> ( ( X @ I4 )
= ( Y @ I4 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_171_PiE__ext,axiom,
! [X: nat > nat > nat,K: set_nat,S: nat > set_nat_nat,Y: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ ( piE_nat_nat_nat2 @ K @ S ) )
=> ( ( member_nat_nat_nat2 @ Y @ ( piE_nat_nat_nat2 @ K @ S ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ K )
=> ( ( X @ I4 )
= ( Y @ I4 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_172_PiE__ext,axiom,
! [X: ( nat > nat ) > nat,K: set_nat_nat,S: ( nat > nat ) > set_nat,Y: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ ( piE_nat_nat_nat @ K @ S ) )
=> ( ( member_nat_nat_nat @ Y @ ( piE_nat_nat_nat @ K @ S ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ K )
=> ( ( X @ I4 )
= ( Y @ I4 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_173_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_174_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat,B2: set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_175_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat,B2: set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3
@ ( piE_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_176_fun__ex,axiom,
! [A: nat,A2: set_nat,B: nat > nat > nat,B2: set_nat_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B2 )
=> ? [X3: nat > nat > nat > nat] :
( ( member17114558718834868at_nat @ X3
@ ( piE_nat_nat_nat_nat5 @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_177_fun__ex,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat @ B @ B2 )
=> ? [X3: nat > ( nat > nat ) > nat] :
( ( member2740455936716430260at_nat @ X3
@ ( piE_nat_nat_nat_nat4 @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_178_fun__ex,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B: nat,B2: set_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X3: ( nat > nat > nat ) > nat] :
( ( member5318315686745620148at_nat @ X3
@ ( piE_nat_nat_nat_nat2 @ A2
@ ^ [I: nat > nat > nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_179_fun__ex,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,B2: set_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ B2 )
=> ? [X3: ( ( nat > nat ) > nat ) > nat] :
( ( member2991261302380110260at_nat @ X3
@ ( piE_nat_nat_nat_nat @ A2
@ ^ [I: ( nat > nat ) > nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_180_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat,B2: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ B2 )
=> ? [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_181_fun__ex,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat > nat,B2: set_nat_nat_nat_nat3] :
( ( member_nat @ A @ A2 )
=> ( ( member952132173341509300at_nat @ B @ B2 )
=> ? [X3: nat > ( nat > nat ) > nat > nat] :
( ( member8743709692935548195at_nat @ X3
@ ( piE_na2748089427378204713at_nat @ A2
@ ^ [I: nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_182_fun__ex,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat > nat,B2: set_nat_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ B2 )
=> ? [X3: ( nat > nat ) > nat > nat > nat] :
( ( member1679187572556404771at_nat @ X3
@ ( piE_na8678869062391380393at_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ( ( X3 @ A )
= B ) ) ) ) ).
% fun_ex
thf(fact_183_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_184_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_185_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat,I3: set_nat_nat,X5: ( nat > nat ) > set_nat] :
( ( member_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ I3 ) @ ( piE_nat_nat_nat @ I3 @ X5 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( member_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_186_restrict__PiE__iff,axiom,
! [F: nat > nat,I3: set_nat,X5: nat > set_nat] :
( ( member_nat_nat @ ( restrict_nat_nat @ F @ I3 ) @ ( piE_nat_nat @ I3 @ X5 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( member_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_187_restrict__PiE__iff,axiom,
! [F: nat > nat > nat,I3: set_nat,X5: nat > set_nat_nat] :
( ( member_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ I3 ) @ ( piE_nat_nat_nat2 @ I3 @ X5 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_188_restrict__PiE__iff,axiom,
! [F: ( nat > nat ) > nat > nat,I3: set_nat_nat,X5: ( nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ ( restri4446420529079022766at_nat @ F @ I3 ) @ ( piE_nat_nat_nat_nat3 @ I3 @ X5 ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_189_cube__def,axiom,
( hales_cube
= ( ^ [N3: nat,T3: nat] :
( piE_nat_nat @ ( set_ord_lessThan_nat @ N3 )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ T3 ) ) ) ) ).
% cube_def
thf(fact_190_split__cube_I2_J,axiom,
! [X: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [Y2: nat] : ( X @ ( plus_plus_nat @ Y2 @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K ) )
@ ( hales_cube @ K @ T ) ) ) ).
% split_cube(2)
thf(fact_191_split__cube_I1_J,axiom,
! [X: nat > nat,K: nat,T: nat] :
( ( member_nat_nat @ X @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ X @ ( set_ord_lessThan_nat @ one_one_nat ) ) @ ( hales_cube @ one_one_nat @ T ) ) ) ).
% split_cube(1)
thf(fact_192_cube__props_I1_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( X3 @ zero_zero_nat )
= S ) ) ) ).
% cube_props(1)
thf(fact_193_layered__eq__classes,axiom,
! [S3: ( nat > nat ) > nat > nat,K: nat,N: nat,T: nat,R: nat,Chi: ( nat > nat ) > nat] :
( ( hales_4261547300027266985ce_nat @ S3 @ K @ N @ T @ R @ Chi )
=> ! [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 ) )
=> ( ( Chi @ ( S3 @ Xa ) )
= ( Chi @ ( S3 @ Xb ) ) ) ) ) ) ) ).
% layered_eq_classes
thf(fact_194_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_195_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_196_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_197_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_198_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_199_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_200_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_201_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_202_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_203_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_204_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_205_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_206_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_207_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_208_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_209_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_210_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_211_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_212_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_213_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_214_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_215_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_216_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_217_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_218_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_219_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_220_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_221_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_222_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_223_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_224_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_225_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_226_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_227_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_228_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_229_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_230_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 ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_231_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 ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_232_cube__restrict,axiom,
! [J: nat,N: nat,Y: nat > nat,T: nat] :
( ( ord_less_nat @ J @ N )
=> ( ( member_nat_nat @ Y @ ( hales_cube @ N @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ Y @ ( set_ord_lessThan_nat @ J ) ) @ ( hales_cube @ J @ T ) ) ) ) ).
% cube_restrict
thf(fact_233_x__def,axiom,
( x
= ( restrict_nat_nat_nat2
@ ^ [I: nat] :
( restrict_nat_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ ( minus_minus_nat @ k @ I ) ) @ zero_zero_nat @ t )
@ ( set_ord_lessThan_nat @ k ) )
@ ( set_ord_atMost_nat @ k ) ) ) ).
% x_def
thf(fact_234_uv__props,axiom,
( ( member_nat @ u @ ( set_ord_atMost_nat @ k ) )
& ( member_nat @ v @ ( set_ord_atMost_nat @ k ) )
& ( ord_less_nat @ u @ v )
& ( ( chi @ ( s @ ( x @ u ) ) )
= ( chi @ ( s @ ( x @ v ) ) ) ) ) ).
% uv_props
thf(fact_235_line1,axiom,
! [S: nat] :
( ( ord_less_eq_nat @ S @ t )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [I: nat] : ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ v ) ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ u ) ) @ S @ t ) )
@ ( set_ord_lessThan_nat @ k ) )
@ ( hales_cube @ k @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% line1
thf(fact_236_image__add__0,axiom,
! [S3: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S3 )
= S3 ) ).
% image_add_0
thf(fact_237__092_060open_062y_A_092_060in_062_A_123_O_O_060t_A_L_A1_125_A_092_060rightarrow_062_092_060_094sub_062E_Acube_An_A_It_A_L_A1_J_092_060close_062,axiom,
( member_nat_nat_nat2 @ y
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat] : ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% \<open>y \<in> {..<t + 1} \<rightarrow>\<^sub>E cube n (t + 1)\<close>
thf(fact_238__092_060open_062_092_060chi_062_A_092_060in_062_Acube_An_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060k_125_092_060close_062,axiom,
( member_nat_nat_nat @ chi
@ ( piE_nat_nat_nat @ ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ k ) ) ) ).
% \<open>\<chi> \<in> cube n (t + 1) \<rightarrow>\<^sub>E {..<k}\<close>
thf(fact_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_254_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_255_image__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_256_image__eqI,axiom,
! [B: nat > nat,F: nat > nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_257_image__eqI,axiom,
! [B: nat,F: ( nat > nat ) > nat,X: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_258_image__eqI,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_259_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_260_image__eqI,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,X: nat > nat,A2: set_nat_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_261_image__eqI,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,X: nat > nat > nat,A2: set_nat_nat_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_262_image__eqI,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,X: ( nat > nat ) > nat,A2: set_nat_nat_nat2] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_263_image__eqI,axiom,
! [B: ( nat > nat ) > nat > nat,F: nat > ( nat > nat ) > nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member952132173341509300at_nat @ B @ ( image_6393715451659844596at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_264_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_265_empty__iff,axiom,
! [C: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ C @ bot_bo7445843802507891576at_nat ) ).
% empty_iff
thf(fact_266_empty__iff,axiom,
! [C: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ C @ bot_bo945813143650711160at_nat ) ).
% empty_iff
thf(fact_267_empty__iff,axiom,
! [C: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ C @ bot_bo3919185967433191911at_nat ) ).
% empty_iff
thf(fact_268_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_269_all__not__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ! [X2: nat > nat] :
~ ( member_nat_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_270_all__not__in__conv,axiom,
! [A2: set_nat_nat_nat] :
( ( ! [X2: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ X2 @ A2 ) )
= ( A2 = bot_bo7445843802507891576at_nat ) ) ).
% all_not_in_conv
thf(fact_271_all__not__in__conv,axiom,
! [A2: set_nat_nat_nat2] :
( ( ! [X2: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ X2 @ A2 ) )
= ( A2 = bot_bo945813143650711160at_nat ) ) ).
% all_not_in_conv
thf(fact_272_all__not__in__conv,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ( ! [X2: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ X2 @ A2 ) )
= ( A2 = bot_bo3919185967433191911at_nat ) ) ).
% all_not_in_conv
thf(fact_273_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_274_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_275_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_276_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_277_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_278_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_279_image__ident,axiom,
! [Y3: set_nat_nat] :
( ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : X2
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_280_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_281_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_282_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_283__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062ax_Abx_O_Aax_A_092_060in_062_AS_A_096_Ax_A_096_A_123_O_Ok_125_A_092_060and_062_Abx_A_092_060in_062_AS_A_096_Ax_A_096_A_123_O_Ok_125_A_092_060and_062_Aax_A_092_060noteq_062_Abx_A_092_060and_062_A_092_060chi_062_Aax_A_061_A_092_060chi_062_Abx_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Ax: nat > nat,Bx: nat > nat] :
~ ( ( member_nat_nat @ Ax @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) )
& ( member_nat_nat @ Bx @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) )
& ( Ax != Bx )
& ( ( chi @ Ax )
= ( chi @ Bx ) ) ) ).
% \<open>\<And>thesis. (\<And>ax bx. ax \<in> S ` x ` {..k} \<and> bx \<in> S ` x ` {..k} \<and> ax \<noteq> bx \<and> \<chi> ax = \<chi> bx \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_284__092_060open_062_092_060exists_062a_Ab_O_Aa_A_092_060in_062_AS_A_096_Ax_A_096_A_123_O_Ok_125_A_092_060and_062_Ab_A_092_060in_062_AS_A_096_Ax_A_096_A_123_O_Ok_125_A_092_060and_062_Aa_A_092_060noteq_062_Ab_A_092_060and_062_A_092_060chi_062_Aa_A_061_A_092_060chi_062_Ab_092_060close_062,axiom,
? [A4: nat > nat,B4: nat > nat] :
( ( member_nat_nat @ A4 @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) )
& ( member_nat_nat @ B4 @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) )
& ( A4 != B4 )
& ( ( chi @ A4 )
= ( chi @ B4 ) ) ) ).
% \<open>\<exists>a b. a \<in> S ` x ` {..k} \<and> b \<in> S ` x ` {..k} \<and> a \<noteq> b \<and> \<chi> a = \<chi> b\<close>
thf(fact_285__092_060open_062_092_060exists_062u_Av_O_Au_A_092_060in_062_A_123_O_Ok_125_A_092_060and_062_Av_A_092_060in_062_A_123_O_Ok_125_A_092_060and_062_Au_A_092_060noteq_062_Av_A_092_060and_062_A_092_060chi_062_A_IS_A_Ix_Au_J_J_A_061_A_092_060chi_062_A_IS_A_Ix_Av_J_J_092_060close_062,axiom,
? [U2: nat,V: nat] :
( ( member_nat @ U2 @ ( set_ord_atMost_nat @ k ) )
& ( member_nat @ V @ ( set_ord_atMost_nat @ k ) )
& ( U2 != V )
& ( ( chi @ ( s @ ( x @ U2 ) ) )
= ( chi @ ( s @ ( x @ V ) ) ) ) ) ).
% \<open>\<exists>u v. u \<in> {..k} \<and> v \<in> {..k} \<and> u \<noteq> v \<and> \<chi> (S (x u)) = \<chi> (S (x v))\<close>
thf(fact_286_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_287_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_288_atMost__subset__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_289_atMost__subset__iff,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ ( set_or250740698829186286at_nat @ X ) @ ( set_or250740698829186286at_nat @ Y ) )
= ( ord_le9059583361652607317at_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_290_atMost__subset__iff,axiom,
! [X: nat > nat,Y: nat > nat] :
( ( ord_le9059583361652607317at_nat @ ( set_or9140604705432621368at_nat @ X ) @ ( set_or9140604705432621368at_nat @ Y ) )
= ( ord_less_eq_nat_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_291_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_292_atMost__iff,axiom,
! [I2: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I2 @ ( set_or9140604705432621368at_nat @ K ) )
= ( ord_less_eq_nat_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_293_atMost__iff,axiom,
! [I2: nat > nat > nat,K: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I2 @ ( set_or6142498856979658663at_nat @ K ) )
= ( ord_le3127000006974329230at_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_294_atMost__iff,axiom,
! [I2: ( nat > nat ) > nat,K: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I2 @ ( set_or5033131092550408871at_nat @ K ) )
= ( ord_le2017632242545079438at_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_295_atMost__iff,axiom,
! [I2: ( nat > nat ) > nat > nat,K: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I2 @ ( set_or3591701359631937174at_nat @ K ) )
= ( ord_le747776305331315197at_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_296_atMost__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_297_atMost__iff,axiom,
! [I2: set_nat_nat,K: set_nat_nat] :
( ( member_set_nat_nat @ I2 @ ( set_or250740698829186286at_nat @ K ) )
= ( ord_le9059583361652607317at_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_298_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_299_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_300_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_301_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_302_Sup__atMost,axiom,
! [Y: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_303_Sup__atMost,axiom,
! [Y: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_304__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062u_Av_O_Au_A_092_060in_062_A_123_O_Ok_125_A_092_060and_062_Av_A_092_060in_062_A_123_O_Ok_125_A_092_060and_062_Au_A_060_Av_A_092_060and_062_A_092_060chi_062_A_IS_A_Ix_Au_J_J_A_061_A_092_060chi_062_A_IS_A_Ix_Av_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [U2: nat,V: nat] :
~ ( ( member_nat @ U2 @ ( set_ord_atMost_nat @ k ) )
& ( member_nat @ V @ ( set_ord_atMost_nat @ k ) )
& ( ord_less_nat @ U2 @ V )
& ( ( chi @ ( s @ ( x @ U2 ) ) )
= ( chi @ ( s @ ( x @ V ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>u v. u \<in> {..k} \<and> v \<in> {..k} \<and> u < v \<and> \<chi> (S (x u)) = \<chi> (S (x v)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_305_A,axiom,
! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ k )
=> ( member_nat_nat @ ( x @ I2 ) @ ( hales_cube @ k @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% A
thf(fact_306_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_307_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_308_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_309_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_310_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_311_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_312_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_313_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_314_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_315__092_060open_062_092_060And_062i_O_Ai_A_092_060le_062_Ak_A_092_060Longrightarrow_062_AS_A_Ix_Ai_J_A_092_060in_062_Acube_An_A_It_A_L_A1_J_092_060close_062,axiom,
! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ k )
=> ( member_nat_nat @ ( s @ ( x @ I2 ) ) @ ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% \<open>\<And>i. i \<le> k \<Longrightarrow> S (x i) \<in> cube n (t + 1)\<close>
thf(fact_316__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062B_Af_O_A_092_060lbrakk_062disjoint__family__on_AB_A_123_O_Ok_125_059_A_092_060Union_062_A_IB_A_096_A_123_O_Ok_125_J_A_061_A_123_O_O_060n_125_059_A_123_125_A_092_060notin_062_AB_A_096_A_123_O_O_060k_125_059_Af_A_092_060in_062_AB_Ak_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060t_A_L_A1_125_059_AS_A_092_060in_062_Acube_Ak_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_Acube_An_A_It_A_L_A1_J_059_A_092_060forall_062y_092_060in_062cube_Ak_A_It_A_L_A1_J_O_A_I_092_060forall_062i_092_060in_062B_Ak_O_AS_Ay_Ai_A_061_Af_Ai_J_A_092_060and_062_A_I_092_060forall_062j_060k_O_A_092_060forall_062i_092_060in_062B_Aj_O_AS_Ay_Ai_A_061_Ay_Aj_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [B5: nat > set_nat] :
( ( disjoi6798895846410478970at_nat @ B5 @ ( set_ord_atMost_nat @ k ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ ( set_ord_atMost_nat @ k ) ) )
= ( set_ord_lessThan_nat @ n ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B5 @ ( set_ord_lessThan_nat @ k ) ) )
=> ! [F4: nat > nat] :
( ( member_nat_nat @ F4
@ ( piE_nat_nat @ ( B5 @ k )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) ) )
=> ( ( member952132173341509300at_nat @ s
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ k @ ( plus_plus_nat @ t @ one_one_nat ) )
@ ^ [I: nat > nat] : ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ) )
=> ~ ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ k @ ( plus_plus_nat @ t @ one_one_nat ) ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( B5 @ k ) )
=> ( ( s @ X4 @ Xa )
= ( F4 @ Xa ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ k )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( B5 @ J2 ) )
=> ( ( s @ X4 @ Xa )
= ( X4 @ J2 ) ) ) ) ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>B f. \<lbrakk>disjoint_family_on B {..k}; \<Union> (B ` {..k}) = {..<n}; {} \<notin> B ` {..<k}; f \<in> B k \<rightarrow>\<^sub>E {..<t + 1}; S \<in> cube k (t + 1) \<rightarrow>\<^sub>E cube n (t + 1); \<forall>y\<in>cube k (t + 1). (\<forall>i\<in>B k. S y i = f i) \<and> (\<forall>j<k. \<forall>i\<in>B j. S y i = y j)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_317_ab__props,axiom,
( ( member_nat_nat @ ax @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) )
& ( member_nat_nat @ bx @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) )
& ( ax != bx )
& ( ( chi @ ax )
= ( chi @ bx ) ) ) ).
% ab_props
thf(fact_318_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M3: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_319_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 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_320_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_321_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_322_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_323_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_324_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_325_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_326_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_327_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_328_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_329_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_330_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_331_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_332_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_333_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_334_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_335_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_336_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_337_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_338_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_339_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_340_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_341_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_342_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_343_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_344_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I4: nat,J4: nat] :
( ( ord_less_nat @ I4 @ J4 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J4 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_345_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_346_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_347_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_348_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_349_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_350_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_351_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_352_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_353_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_354_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_355_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_356_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_357_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_358_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_359_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_360_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_361_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_362_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_363_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_364_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_365_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_366_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_367_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_368_PiE__eq,axiom,
! [I3: set_nat,S3: nat > set_nat,T2: nat > set_nat] :
( ( ( piE_nat_nat @ I3 @ S3 )
= ( piE_nat_nat @ I3 @ T2 ) )
= ( ( ( ( piE_nat_nat @ I3 @ S3 )
= bot_bot_set_nat_nat )
& ( ( piE_nat_nat @ I3 @ T2 )
= bot_bot_set_nat_nat ) )
| ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( ( S3 @ X2 )
= ( T2 @ X2 ) ) ) ) ) ).
% PiE_eq
thf(fact_369_PiE__eq,axiom,
! [I3: set_nat_nat,S3: ( nat > nat ) > set_nat_nat,T2: ( nat > nat ) > set_nat_nat] :
( ( ( piE_nat_nat_nat_nat3 @ I3 @ S3 )
= ( piE_nat_nat_nat_nat3 @ I3 @ T2 ) )
= ( ( ( ( piE_nat_nat_nat_nat3 @ I3 @ S3 )
= bot_bo3919185967433191911at_nat )
& ( ( piE_nat_nat_nat_nat3 @ I3 @ T2 )
= bot_bo3919185967433191911at_nat ) )
| ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( ( S3 @ X2 )
= ( T2 @ X2 ) ) ) ) ) ).
% PiE_eq
thf(fact_370_PiE__eq,axiom,
! [I3: set_nat,S3: nat > set_nat_nat,T2: nat > set_nat_nat] :
( ( ( piE_nat_nat_nat2 @ I3 @ S3 )
= ( piE_nat_nat_nat2 @ I3 @ T2 ) )
= ( ( ( ( piE_nat_nat_nat2 @ I3 @ S3 )
= bot_bo7445843802507891576at_nat )
& ( ( piE_nat_nat_nat2 @ I3 @ T2 )
= bot_bo7445843802507891576at_nat ) )
| ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( ( S3 @ X2 )
= ( T2 @ X2 ) ) ) ) ) ).
% PiE_eq
thf(fact_371_PiE__eq,axiom,
! [I3: set_nat_nat,S3: ( nat > nat ) > set_nat,T2: ( nat > nat ) > set_nat] :
( ( ( piE_nat_nat_nat @ I3 @ S3 )
= ( piE_nat_nat_nat @ I3 @ T2 ) )
= ( ( ( ( piE_nat_nat_nat @ I3 @ S3 )
= bot_bo945813143650711160at_nat )
& ( ( piE_nat_nat_nat @ I3 @ T2 )
= bot_bo945813143650711160at_nat ) )
| ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( ( S3 @ X2 )
= ( T2 @ X2 ) ) ) ) ) ).
% PiE_eq
thf(fact_372_all__PiE__elements,axiom,
! [I3: set_nat,S3: nat > set_nat,P: nat > nat > $o] :
( ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( piE_nat_nat @ I3 @ S3 ) )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ I3 )
=> ( P @ Y2 @ ( X2 @ Y2 ) ) ) ) )
= ( ( ( piE_nat_nat @ I3 @ S3 )
= bot_bot_set_nat_nat )
| ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( S3 @ X2 ) )
=> ( P @ X2 @ Y2 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_373_all__PiE__elements,axiom,
! [I3: set_nat_nat,S3: ( nat > nat ) > set_nat_nat,P: ( nat > nat ) > ( nat > nat ) > $o] :
( ( ! [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ ( piE_nat_nat_nat_nat3 @ I3 @ S3 ) )
=> ! [Y2: nat > nat] :
( ( member_nat_nat @ Y2 @ I3 )
=> ( P @ Y2 @ ( X2 @ Y2 ) ) ) ) )
= ( ( ( piE_nat_nat_nat_nat3 @ I3 @ S3 )
= bot_bo3919185967433191911at_nat )
| ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ! [Y2: nat > nat] :
( ( member_nat_nat @ Y2 @ ( S3 @ X2 ) )
=> ( P @ X2 @ Y2 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_374_all__PiE__elements,axiom,
! [I3: set_nat,S3: nat > set_nat_nat,P: nat > ( nat > nat ) > $o] :
( ( ! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ ( piE_nat_nat_nat2 @ I3 @ S3 ) )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ I3 )
=> ( P @ Y2 @ ( X2 @ Y2 ) ) ) ) )
= ( ( ( piE_nat_nat_nat2 @ I3 @ S3 )
= bot_bo7445843802507891576at_nat )
| ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ! [Y2: nat > nat] :
( ( member_nat_nat @ Y2 @ ( S3 @ X2 ) )
=> ( P @ X2 @ Y2 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_375_all__PiE__elements,axiom,
! [I3: set_nat_nat,S3: ( nat > nat ) > set_nat,P: ( nat > nat ) > nat > $o] :
( ( ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ ( piE_nat_nat_nat @ I3 @ S3 ) )
=> ! [Y2: nat > nat] :
( ( member_nat_nat @ Y2 @ I3 )
=> ( P @ Y2 @ ( X2 @ Y2 ) ) ) ) )
= ( ( ( piE_nat_nat_nat @ I3 @ S3 )
= bot_bo945813143650711160at_nat )
| ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( S3 @ X2 ) )
=> ( P @ X2 @ Y2 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_376_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U: set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ U ) ) ) ) ).
% atMost_def
thf(fact_377_atMost__def,axiom,
( set_or250740698829186286at_nat
= ( ^ [U: set_nat_nat] :
( collect_set_nat_nat
@ ^ [X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ U ) ) ) ) ).
% atMost_def
thf(fact_378_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_eq_nat @ X2 @ U ) ) ) ) ).
% atMost_def
thf(fact_379_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_380_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_381_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_382_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_383_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_384_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_385_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_386_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_387_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_388_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_389_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_390_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_391_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_392_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_393_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_394_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_395_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_396_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_397_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_398_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_399_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_400_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_401_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_402_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_403_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_404_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_405_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_406_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_407_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_408_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_409_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_410_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_411_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_412_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_413_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_414_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_415_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_416_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_417_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_418_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_419_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ).
% imageI
thf(fact_420_imageI,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_421_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X ) @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_422_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat_nat @ ( F @ X ) @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_423_imageI,axiom,
! [X: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_424_imageI,axiom,
! [X: nat > nat > nat,A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_425_imageI,axiom,
! [X: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_426_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > ( nat > nat ) > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member952132173341509300at_nat @ ( F @ X ) @ ( image_6393715451659844596at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_427_image__iff,axiom,
! [Z: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_428_image__iff,axiom,
! [Z: nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( member_nat @ Z @ ( image_nat_nat_nat @ F @ A2 ) )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_429_image__iff,axiom,
! [Z: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ Z @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_430_image__iff,axiom,
! [Z: nat > nat,F: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ Z @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_431_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 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_432_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 ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_433_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 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_434_bex__imageD,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_435_image__cong,axiom,
! [M3: set_nat,N4: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M3 = N4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_set_nat @ F @ M3 )
= ( image_nat_set_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_436_image__cong,axiom,
! [M3: set_nat,N4: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ( M3 = N4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat_nat2 @ F @ M3 )
= ( image_nat_nat_nat2 @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_437_image__cong,axiom,
! [M3: set_nat_nat,N4: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ( M3 = N4 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ N4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_3205354838064109189at_nat @ F @ M3 )
= ( image_3205354838064109189at_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_438_image__cong,axiom,
! [M3: set_nat_nat,N4: set_nat_nat,F: ( nat > nat ) > nat,G: ( nat > nat ) > nat] :
( ( M3 = N4 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ N4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat_nat @ F @ M3 )
= ( image_nat_nat_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_439_ball__imageD,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_440_ball__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_441_ball__imageD,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_442_ball__imageD,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_443_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_444_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_445_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_446_rev__image__eqI,axiom,
! [X: nat > nat,A2: set_nat_nat,B: nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_447_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat > nat > nat,F: nat > nat > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_448_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_449_rev__image__eqI,axiom,
! [X: nat > nat,A2: set_nat_nat,B: nat > nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_450_rev__image__eqI,axiom,
! [X: nat > nat > nat,A2: set_nat_nat_nat,B: nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_451_rev__image__eqI,axiom,
! [X: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_452_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: ( nat > nat ) > nat > nat,F: nat > ( nat > nat ) > nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member952132173341509300at_nat @ B @ ( image_6393715451659844596at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_453_emptyE,axiom,
! [A: nat > nat] :
~ ( member_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_454_emptyE,axiom,
! [A: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ A @ bot_bo7445843802507891576at_nat ) ).
% emptyE
thf(fact_455_emptyE,axiom,
! [A: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ A @ bot_bo945813143650711160at_nat ) ).
% emptyE
thf(fact_456_emptyE,axiom,
! [A: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ A @ bot_bo3919185967433191911at_nat ) ).
% emptyE
thf(fact_457_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_458_equals0D,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( A2 = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_459_equals0D,axiom,
! [A2: set_nat_nat_nat,A: nat > nat > nat] :
( ( A2 = bot_bo7445843802507891576at_nat )
=> ~ ( member_nat_nat_nat2 @ A @ A2 ) ) ).
% equals0D
thf(fact_460_equals0D,axiom,
! [A2: set_nat_nat_nat2,A: ( nat > nat ) > nat] :
( ( A2 = bot_bo945813143650711160at_nat )
=> ~ ( member_nat_nat_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_461_equals0D,axiom,
! [A2: set_nat_nat_nat_nat3,A: ( nat > nat ) > nat > nat] :
( ( A2 = bot_bo3919185967433191911at_nat )
=> ~ ( member952132173341509300at_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_462_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_463_equals0I,axiom,
! [A2: set_nat_nat] :
( ! [Y4: nat > nat] :
~ ( member_nat_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_464_equals0I,axiom,
! [A2: set_nat_nat_nat] :
( ! [Y4: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ Y4 @ A2 )
=> ( A2 = bot_bo7445843802507891576at_nat ) ) ).
% equals0I
thf(fact_465_equals0I,axiom,
! [A2: set_nat_nat_nat2] :
( ! [Y4: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ Y4 @ A2 )
=> ( A2 = bot_bo945813143650711160at_nat ) ) ).
% equals0I
thf(fact_466_equals0I,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ! [Y4: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ Y4 @ A2 )
=> ( A2 = bot_bo3919185967433191911at_nat ) ) ).
% equals0I
thf(fact_467_equals0I,axiom,
! [A2: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_468_ex__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ? [X2: nat > nat] : ( member_nat_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_469_ex__in__conv,axiom,
! [A2: set_nat_nat_nat] :
( ( ? [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A2 ) )
= ( A2 != bot_bo7445843802507891576at_nat ) ) ).
% ex_in_conv
thf(fact_470_ex__in__conv,axiom,
! [A2: set_nat_nat_nat2] :
( ( ? [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A2 ) )
= ( A2 != bot_bo945813143650711160at_nat ) ) ).
% ex_in_conv
thf(fact_471_ex__in__conv,axiom,
! [A2: set_nat_nat_nat_nat3] :
( ( ? [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A2 ) )
= ( A2 != bot_bo3919185967433191911at_nat ) ) ).
% ex_in_conv
thf(fact_472_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_473_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_474_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ? [B4: nat > nat] : ( member_nat_nat @ B4 @ ( minus_8121590178497047118at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_475_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( ord_le6871433888996735800at_nat @ A2 @ B2 )
=> ? [B4: nat > nat > nat] : ( member_nat_nat_nat2 @ B4 @ ( minus_7721066311745899709at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_476_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( ord_le371403230139555384at_nat @ A2 @ B2 )
=> ? [B4: ( nat > nat ) > nat] : ( member_nat_nat_nat @ B4 @ ( minus_1221035652888719293at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_477_psubset__imp__ex__mem,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( ord_le6177938698872215975at_nat @ A2 @ B2 )
=> ? [B4: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ B4 @ ( minus_4646100876039749548at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_478_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_479_psubsetD,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: nat > nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_480_psubsetD,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le6871433888996735800at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_481_psubsetD,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le371403230139555384at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_482_psubsetD,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,C: ( nat > nat ) > nat > nat] :
( ( ord_le6177938698872215975at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ C @ A2 )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_483_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_484_imageE,axiom,
! [B: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_485_imageE,axiom,
! [B: nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( member_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) )
=> ~ ! [X3: nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_486_imageE,axiom,
! [B: nat > nat,F: nat > nat > nat,A2: set_nat] :
( ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_487_imageE,axiom,
! [B: nat,F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat] :
( ( member_nat @ B @ ( image_913610194320715013at_nat @ F @ A2 ) )
=> ~ ! [X3: nat > nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat2 @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_488_imageE,axiom,
! [B: nat,F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2] :
( ( member_nat @ B @ ( image_7809927846809980933at_nat @ F @ A2 ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_489_imageE,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ~ ! [X3: nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_490_imageE,axiom,
! [B: nat > nat > nat,F: nat > nat > nat > nat,A2: set_nat] :
( ( member_nat_nat_nat2 @ B @ ( image_6919068903512877573at_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_491_imageE,axiom,
! [B: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,A2: set_nat] :
( ( member_nat_nat_nat @ B @ ( image_5809701139083627781at_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_492_imageE,axiom,
! [B: nat,F: ( ( nat > nat ) > nat > nat ) > nat,A2: set_nat_nat_nat_nat3] :
( ( member_nat @ B @ ( image_8194121248528334964at_nat @ F @ A2 ) )
=> ~ ! [X3: ( nat > nat ) > nat > nat] :
( ( B
= ( F @ X3 ) )
=> ~ ( member952132173341509300at_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_493_image__image,axiom,
! [F: nat > set_nat,G: nat > nat,A2: set_nat] :
( ( image_nat_set_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_494_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
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_495_image__image,axiom,
! [F: nat > nat,G: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat_nat @ G @ A2 ) )
= ( image_nat_nat_nat
@ ^ [X2: nat > nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_496_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
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_497_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
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_498_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
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_499_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
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_500_image__image,axiom,
! [F: nat > set_nat,G: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( image_nat_set_nat @ F @ ( image_nat_nat_nat @ G @ A2 ) )
= ( image_7432509271690132940et_nat
@ ^ [X2: nat > nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_501_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
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_502_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
@ ^ [X2: nat > nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_503_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_504_Compr__image__eq,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_set_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_505_Compr__image__eq,axiom,
! [F: nat > nat > nat,A2: set_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat_nat2 @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_506_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat_nat @ F
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_507_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_3205354838064109189at_nat @ F
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_508_Compr__image__eq,axiom,
! [F: nat > nat > nat > nat,A2: set_nat,P: ( nat > nat > nat ) > $o] :
( ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ ( image_6919068903512877573at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_6919068903512877573at_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_509_Compr__image__eq,axiom,
! [F: nat > ( nat > nat ) > nat,A2: set_nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ ( image_5809701139083627781at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_5809701139083627781at_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_510_Compr__image__eq,axiom,
! [F: ( nat > nat > nat ) > nat,A2: set_nat_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_913610194320715013at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_913610194320715013at_nat @ F
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_511_Compr__image__eq,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A2: set_nat_nat_nat2,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_7809927846809980933at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_7809927846809980933at_nat @ F
@ ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_512_Compr__image__eq,axiom,
! [F: ( nat > nat > nat ) > nat > nat,A2: set_nat_nat_nat,P: ( nat > nat ) > $o] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_1545173636400105204at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_1545173636400105204at_nat @ F
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_513_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_514_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_515_less__set__def,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
( ord_less_nat_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A5 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_516_less__set__def,axiom,
( ord_le6871433888996735800at_nat
= ( ^ [A5: set_nat_nat_nat,B6: set_nat_nat_nat] :
( ord_le3977685358511927117_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A5 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_517_less__set__def,axiom,
( ord_le371403230139555384at_nat
= ( ^ [A5: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
( ord_le8812218136411540557_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A5 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_518_less__set__def,axiom,
( ord_le6177938698872215975at_nat
= ( ^ [A5: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
( ord_le4961065272816086430_nat_o
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A5 )
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_519_dim1__subspace__elims_I3_J,axiom,
! [B2: nat > set_nat,N: nat,F: nat > nat,T: nat,S3: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B2 @ one_one_nat )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S3
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ T )
@ ^ [I: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ one_one_nat ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J4: nat] :
( ( ord_less_nat @ J4 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ J4 ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( X3 @ J4 ) ) ) ) ) )
=> ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( B2 @ one_one_nat ) )
=> ( ( S3 @ X4 @ Xa )
= ( F @ Xa ) ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( B2 @ zero_zero_nat ) )
=> ( ( S3 @ X4 @ Xa )
= ( X4 @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ).
% dim1_subspace_elims(3)
thf(fact_520_dim1__subspace__elims_I4_J,axiom,
! [B2: nat > set_nat,N: nat,F: nat > nat,T: nat,S3: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B2 @ one_one_nat )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S3
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ T )
@ ^ [I: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ one_one_nat ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J4: nat] :
( ( ord_less_nat @ J4 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ J4 ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( X3 @ J4 ) ) ) ) ) )
=> ( ( B2 @ zero_zero_nat )
!= bot_bot_set_nat ) ) ) ) ) ) ) ).
% dim1_subspace_elims(4)
thf(fact_521_dim0__layered__subspace__ex,axiom,
! [Chi: ( nat > nat ) > nat,N: nat,T: nat,R: nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ R ) ) )
=> ? [S4: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S4 @ zero_zero_nat @ N @ T @ R @ Chi ) ) ).
% dim0_layered_subspace_ex
thf(fact_522_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_523__092_060open_062_092_060chi_062_A_096_AS_A_096_Ax_A_096_A_123_O_Ok_125_A_092_060subseteq_062_A_092_060chi_062_A_096_Acube_An_A_It_A_L_A1_J_092_060close_062,axiom,
ord_less_eq_set_nat @ ( image_nat_nat_nat @ chi @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) ) @ ( image_nat_nat_nat @ chi @ ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ).
% \<open>\<chi> ` S ` x ` {..k} \<subseteq> \<chi> ` cube n (t + 1)\<close>
thf(fact_524__092_060open_062card_A_I_092_060chi_062_A_096_AS_A_096_Ax_A_096_A_123_O_Ok_125_J_A_092_060le_062_Acard_A_I_092_060chi_062_A_096_Acube_An_A_It_A_L_A1_J_J_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat_nat @ chi @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) ) ) @ ( finite_card_nat @ ( image_nat_nat_nat @ chi @ ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% \<open>card (\<chi> ` S ` x ` {..k}) \<le> card (\<chi> ` cube n (t + 1))\<close>
thf(fact_525__092_060open_062S_A_096_Ax_A_096_A_123_O_Ok_125_A_092_060subseteq_062_Acube_An_A_It_A_L_A1_J_092_060close_062,axiom,
ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) @ ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ).
% \<open>S ` x ` {..k} \<subseteq> cube n (t + 1)\<close>
thf(fact_526__092_060open_062_092_060chi_062_A_096_Acube_An_A_It_A_L_A1_J_A_092_060subseteq_062_A_123_O_O_060k_125_092_060close_062,axiom,
ord_less_eq_set_nat @ ( image_nat_nat_nat @ chi @ ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ) @ ( set_ord_lessThan_nat @ k ) ).
% \<open>\<chi> ` cube n (t + 1) \<subseteq> {..<k}\<close>
thf(fact_527_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_528_SUP__const,axiom,
! [A2: set_nat,F: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_529_SUP__const,axiom,
! [A2: set_nat,F: $o] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_530_cSUP__const,axiom,
! [A2: set_nat,C: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_531_cSUP__const,axiom,
! [A2: set_nat_nat,C: nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat_nat
@ ^ [X2: nat > nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_532_cSUP__const,axiom,
! [A2: set_nat,C: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_533_cSUP__const,axiom,
! [A2: set_nat,C: $o] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X2: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_534_subsetI,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( member_nat_nat_nat2 @ X3 @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_535_subsetI,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( member_nat_nat_nat @ X3 @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_536_subsetI,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
=> ( member952132173341509300at_nat @ X3 @ B2 ) )
=> ( ord_le5260717879541182899at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_537_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_538_subsetI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat @ X3 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_539_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_540_subset__antisym,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_541_Union__iff,axiom,
! [A2: nat > nat,C4: set_set_nat_nat] :
( ( member_nat_nat @ A2 @ ( comple5448282615319421384at_nat @ C4 ) )
= ( ? [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ C4 )
& ( member_nat_nat @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_542_Union__iff,axiom,
! [A2: nat > nat > nat,C4: set_set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ A2 @ ( comple8167887107183641911at_nat @ C4 ) )
= ( ? [X2: set_nat_nat_nat] :
( ( member8194441297229544571at_nat @ X2 @ C4 )
& ( member_nat_nat_nat2 @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_543_Union__iff,axiom,
! [A2: ( nat > nat ) > nat,C4: set_set_nat_nat_nat2] :
( ( member_nat_nat_nat @ A2 @ ( comple1667856448326461495at_nat @ C4 ) )
= ( ? [X2: set_nat_nat_nat2] :
( ( member1694410638372364155at_nat @ X2 @ C4 )
& ( member_nat_nat_nat @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_544_Union__iff,axiom,
! [A2: ( nat > nat ) > nat > nat,C4: set_se3022870823424313865at_nat] :
( ( member952132173341509300at_nat @ A2 @ ( comple2605510978757769510at_nat @ C4 ) )
= ( ? [X2: set_nat_nat_nat_nat3] :
( ( member7681264892014656106at_nat @ X2 @ C4 )
& ( member952132173341509300at_nat @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_545_Union__iff,axiom,
! [A2: nat,C4: set_set_nat] :
( ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C4 ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ C4 )
& ( member_nat @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_546_UnionI,axiom,
! [X5: set_nat_nat,C4: set_set_nat_nat,A2: nat > nat] :
( ( member_set_nat_nat @ X5 @ C4 )
=> ( ( member_nat_nat @ A2 @ X5 )
=> ( member_nat_nat @ A2 @ ( comple5448282615319421384at_nat @ C4 ) ) ) ) ).
% UnionI
thf(fact_547_UnionI,axiom,
! [X5: set_nat_nat_nat,C4: set_set_nat_nat_nat,A2: nat > nat > nat] :
( ( member8194441297229544571at_nat @ X5 @ C4 )
=> ( ( member_nat_nat_nat2 @ A2 @ X5 )
=> ( member_nat_nat_nat2 @ A2 @ ( comple8167887107183641911at_nat @ C4 ) ) ) ) ).
% UnionI
thf(fact_548_UnionI,axiom,
! [X5: set_nat_nat_nat2,C4: set_set_nat_nat_nat2,A2: ( nat > nat ) > nat] :
( ( member1694410638372364155at_nat @ X5 @ C4 )
=> ( ( member_nat_nat_nat @ A2 @ X5 )
=> ( member_nat_nat_nat @ A2 @ ( comple1667856448326461495at_nat @ C4 ) ) ) ) ).
% UnionI
thf(fact_549_UnionI,axiom,
! [X5: set_nat_nat_nat_nat3,C4: set_se3022870823424313865at_nat,A2: ( nat > nat ) > nat > nat] :
( ( member7681264892014656106at_nat @ X5 @ C4 )
=> ( ( member952132173341509300at_nat @ A2 @ X5 )
=> ( member952132173341509300at_nat @ A2 @ ( comple2605510978757769510at_nat @ C4 ) ) ) ) ).
% UnionI
thf(fact_550_UnionI,axiom,
! [X5: set_nat,C4: set_set_nat,A2: nat] :
( ( member_set_nat @ X5 @ C4 )
=> ( ( member_nat @ A2 @ X5 )
=> ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C4 ) ) ) ) ).
% UnionI
thf(fact_551_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ X2 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_552_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ X2 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_553_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_554_DiffI,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ A2 )
=> ( ~ ( member_nat_nat @ C @ B2 )
=> ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_555_DiffI,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ A2 )
=> ( ~ ( member_nat_nat_nat2 @ C @ B2 )
=> ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_556_DiffI,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ A2 )
=> ( ~ ( member_nat_nat_nat @ C @ B2 )
=> ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_557_DiffI,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ A2 )
=> ( ~ ( member952132173341509300at_nat @ C @ B2 )
=> ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_558_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_559_Diff__iff,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
= ( ( member_nat_nat @ C @ A2 )
& ~ ( member_nat_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_560_Diff__iff,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) )
= ( ( member_nat_nat_nat2 @ C @ A2 )
& ~ ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_561_Diff__iff,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
= ( ( member_nat_nat_nat @ C @ A2 )
& ~ ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_562_Diff__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) )
= ( ( member952132173341509300at_nat @ C @ A2 )
& ~ ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_563__092_060open_062k_A_L_A1_A_061_Acard_A_123_O_Ok_125_092_060close_062,axiom,
( ( plus_plus_nat @ k @ one_one_nat )
= ( finite_card_nat @ ( set_ord_atMost_nat @ k ) ) ) ).
% \<open>k + 1 = card {..k}\<close>
thf(fact_564_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_565_empty__subsetI,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% empty_subsetI
thf(fact_566_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_567_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_568_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_569_Sup__bot__conv_I1_J,axiom,
! [A2: set_o] :
( ( ( complete_Sup_Sup_o @ A2 )
= bot_bot_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_570_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_571_Sup__bot__conv_I2_J,axiom,
! [A2: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A2 ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_572_ball__UN,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_573_bex__UN,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_574_UN__ball__bex__simps_I2_J,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_575_UN__ball__bex__simps_I4_J,axiom,
! [B2: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X2 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_576_psubsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_577_psubsetI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_578_cSup__atMost,axiom,
! [X: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ X ) )
= X ) ).
% cSup_atMost
thf(fact_579_cSup__atMost,axiom,
! [X: nat] :
( ( complete_Sup_Sup_nat @ ( set_ord_atMost_nat @ X ) )
= X ) ).
% cSup_atMost
thf(fact_580_cSup__atMost,axiom,
! [X: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ X ) )
= X ) ).
% cSup_atMost
thf(fact_581_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_582_card__lessThan,axiom,
! [U3: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U3 ) )
= U3 ) ).
% card_lessThan
thf(fact_583_SUP__identity__eq,axiom,
! [A2: set_nat_nat] :
( ( comple2450677804321093138at_nat
@ ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : X2
@ A2 ) )
= ( comple2450677804321093138at_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_584_SUP__identity__eq,axiom,
! [A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X2: set_nat] : X2
@ A2 ) )
= ( comple7399068483239264473et_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_585_SUP__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_586_SUP__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A2 ) )
= ( complete_Sup_Sup_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_587_UN__iff,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( member_nat @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_588_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat,B2: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_589_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat > nat,B2: nat > set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ ( B2 @ A ) )
=> ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_590_UN__I,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat,B2: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_591_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat > nat > nat,B2: nat > set_nat_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ ( B2 @ A ) )
=> ( member_nat_nat_nat2 @ B @ ( comple8167887107183641911at_nat @ ( image_6130888460295934395at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_592_UN__I,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat,B2: nat > set_nat_nat_nat2] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat_nat_nat @ B @ ( B2 @ A ) )
=> ( member_nat_nat_nat @ B @ ( comple1667856448326461495at_nat @ ( image_8854229838293529787at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_593_UN__I,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat,B2: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat @ B @ ( B2 @ A ) )
=> ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_594_UN__I,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B: nat,B2: ( nat > nat > nat ) > set_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_595_UN__I,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B: nat,B2: ( ( nat > nat ) > nat ) > set_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_596_UN__I,axiom,
! [A: nat,A2: set_nat,B: ( nat > nat ) > nat > nat,B2: nat > set_nat_nat_nat_nat3] :
( ( member_nat @ A @ A2 )
=> ( ( member952132173341509300at_nat @ B @ ( B2 @ A ) )
=> ( member952132173341509300at_nat @ B @ ( comple2605510978757769510at_nat @ ( image_3332361743537024938at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_597_UN__I,axiom,
! [A: nat > nat,A2: set_nat_nat,B: nat > nat > nat,B2: ( nat > nat ) > set_nat_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( member_nat_nat_nat2 @ B @ ( B2 @ A ) )
=> ( member_nat_nat_nat2 @ B @ ( comple8167887107183641911at_nat @ ( image_470123710477037866at_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_598_Sup__apply,axiom,
( comple2450677804321093138at_nat
= ( ^ [A5: set_nat_nat,X2: nat] :
( complete_Sup_Sup_nat
@ ( image_nat_nat_nat
@ ^ [F5: nat > nat] : ( F5 @ X2 )
@ A5 ) ) ) ) ).
% Sup_apply
thf(fact_599_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_600_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_601_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_602_Diff__eq__empty__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ( minus_8121590178497047118at_nat @ A2 @ B2 )
= bot_bot_set_nat_nat )
= ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_603__092_060open_062x_A_096_A_123_O_Ok_125_A_092_060subseteq_062_Acube_Ak_A_It_A_L_A1_J_092_060close_062,axiom,
ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) @ ( hales_cube @ k @ ( plus_plus_nat @ t @ one_one_nat ) ) ).
% \<open>x ` {..k} \<subseteq> cube k (t + 1)\<close>
thf(fact_604__092_060open_062S_A_096_Acube_Ak_A_It_A_L_A1_J_A_092_060subseteq_062_Acube_An_A_It_A_L_A1_J_092_060close_062,axiom,
ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ s @ ( hales_cube @ k @ ( plus_plus_nat @ t @ one_one_nat ) ) ) @ ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ).
% \<open>S ` cube k (t + 1) \<subseteq> cube n (t + 1)\<close>
thf(fact_605_SUP__bot,axiom,
! [A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : bot_bot_set_nat
@ A2 ) )
= bot_bot_set_nat ) ).
% SUP_bot
thf(fact_606_SUP__bot__conv_I1_J,axiom,
! [B2: nat > set_nat,A2: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_607_SUP__bot__conv_I2_J,axiom,
! [B2: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_608_SUP__apply,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,X: nat] :
( ( comple2450677804321093138at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ X )
= ( complete_Sup_Sup_nat
@ ( image_nat_nat_nat
@ ^ [Y2: nat > nat] : ( F @ Y2 @ X )
@ A2 ) ) ) ).
% SUP_apply
thf(fact_609_SUP__apply,axiom,
! [F: nat > nat > nat,A2: set_nat,X: nat] :
( ( comple2450677804321093138at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ X )
= ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [Y2: nat] : ( F @ Y2 @ X )
@ A2 ) ) ) ).
% SUP_apply
thf(fact_610__C_K_C,axiom,
ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat_nat @ chi @ ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) @ k ).
% "*"
thf(fact_611__092_060open_062card_A_I_092_060chi_062_A_096_Acube_An_A_It_A_L_A1_J_J_A_092_060le_062_Acard_A_123_O_O_060k_125_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat_nat @ chi @ ( hales_cube @ n @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) @ ( finite_card_nat @ ( set_ord_lessThan_nat @ k ) ) ).
% \<open>card (\<chi> ` cube n (t + 1)) \<le> card {..<k}\<close>
thf(fact_612__092_060open_062S_A_096_Ax_A_096_A_123_O_Ok_125_A_092_060subseteq_062_AS_A_096_Acube_Ak_A_It_A_L_A1_J_092_060close_062,axiom,
ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) @ ( image_3205354838064109189at_nat @ s @ ( hales_cube @ k @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ).
% \<open>S ` x ` {..k} \<subseteq> S ` cube k (t + 1)\<close>
thf(fact_613_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_614_DiffE,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_615_DiffE,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% DiffE
thf(fact_616_DiffE,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_617_DiffE,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) )
=> ~ ( ( member952132173341509300at_nat @ C @ A2 )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_618_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_619_DiffD1,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
=> ( member_nat_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_620_DiffD1,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) )
=> ( member_nat_nat_nat2 @ C @ A2 ) ) ).
% DiffD1
thf(fact_621_DiffD1,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
=> ( member_nat_nat_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_622_DiffD1,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) )
=> ( member952132173341509300at_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_623_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_624_DiffD2,axiom,
! [C: nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) )
=> ~ ( member_nat_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_625_DiffD2,axiom,
! [C: nat > nat > nat,A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A2 @ B2 ) )
=> ~ ( member_nat_nat_nat2 @ C @ B2 ) ) ).
% DiffD2
thf(fact_626_DiffD2,axiom,
! [C: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A2 @ B2 ) )
=> ~ ( member_nat_nat_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_627_DiffD2,axiom,
! [C: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A2 @ B2 ) )
=> ~ ( member952132173341509300at_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_628_Diff__mono,axiom,
! [A2: set_nat,C4: set_nat,D3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( ord_less_eq_set_nat @ D3 @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_629_Diff__mono,axiom,
! [A2: set_nat_nat,C4: set_nat_nat,D3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C4 )
=> ( ( ord_le9059583361652607317at_nat @ D3 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) @ ( minus_8121590178497047118at_nat @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_630_Diff__subset,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_631_Diff__subset,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_632_double__diff,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C4 )
=> ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_633_double__diff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C4 )
=> ( ( minus_8121590178497047118at_nat @ B2 @ ( minus_8121590178497047118at_nat @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_634_set__diff__eq,axiom,
( minus_8121590178497047118at_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A5 )
& ~ ( member_nat_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_635_set__diff__eq,axiom,
( minus_7721066311745899709at_nat
= ( ^ [A5: set_nat_nat_nat,B6: set_nat_nat_nat] :
( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A5 )
& ~ ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_636_set__diff__eq,axiom,
( minus_1221035652888719293at_nat
= ( ^ [A5: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A5 )
& ~ ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_637_set__diff__eq,axiom,
( minus_4646100876039749548at_nat
= ( ^ [A5: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A5 )
& ~ ( member952132173341509300at_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_638_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A5 )
& ~ ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_639_minus__set__def,axiom,
( minus_8121590178497047118at_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
( collect_nat_nat
@ ( minus_167519014754328503_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A5 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_640_minus__set__def,axiom,
( minus_7721066311745899709at_nat
= ( ^ [A5: set_nat_nat_nat,B6: set_nat_nat_nat] :
( collect_nat_nat_nat2
@ ( minus_7240682219522218504_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A5 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_641_minus__set__def,axiom,
( minus_1221035652888719293at_nat
= ( ^ [A5: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
( collect_nat_nat_nat
@ ( minus_2851842960567056136_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A5 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_642_minus__set__def,axiom,
( minus_4646100876039749548at_nat
= ( ^ [A5: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
( collec3567154360959927026at_nat
@ ( minus_7158188067284919257_nat_o
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A5 )
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_643_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_644_in__mono,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,X: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ X @ A2 )
=> ( member_nat_nat_nat2 @ X @ B2 ) ) ) ).
% in_mono
thf(fact_645_in__mono,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,X: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ X @ A2 )
=> ( member_nat_nat_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_646_in__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,X: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ X @ A2 )
=> ( member952132173341509300at_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_647_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_648_in__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,X: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_649_subsetD,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat2 @ C @ A2 )
=> ( member_nat_nat_nat2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_650_subsetD,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ( member_nat_nat_nat @ C @ A2 )
=> ( member_nat_nat_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_651_subsetD,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,C: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B2 )
=> ( ( member952132173341509300at_nat @ C @ A2 )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_652_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_653_subsetD,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_654_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_655_equalityE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_656_subset__eq,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A5: set_nat_nat_nat,B6: set_nat_nat_nat] :
! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A5 )
=> ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_657_subset__eq,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A5: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A5 )
=> ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_658_subset__eq,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A5: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
! [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A5 )
=> ( member952132173341509300at_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_659_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( member_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_660_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A5 )
=> ( member_nat_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_661_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_662_equalityD1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_663_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_664_equalityD2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 = B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_665_subset__iff,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A5: set_nat_nat_nat,B6: set_nat_nat_nat] :
! [T3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ T3 @ A5 )
=> ( member_nat_nat_nat2 @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_666_subset__iff,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A5: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
! [T3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ T3 @ A5 )
=> ( member_nat_nat_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_667_subset__iff,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A5: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
! [T3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ T3 @ A5 )
=> ( member952132173341509300at_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_668_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A5 )
=> ( member_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_669_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
! [T3: nat > nat] :
( ( member_nat_nat @ T3 @ A5 )
=> ( member_nat_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_670_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_671_subset__refl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_672_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_673_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_674_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C4 )
=> ( ord_less_eq_set_nat @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_675_subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C4 )
=> ( ord_le9059583361652607317at_nat @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_676_set__eq__subset,axiom,
( ( ^ [Y6: set_nat,Z2: set_nat] : ( Y6 = Z2 ) )
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_677_set__eq__subset,axiom,
( ( ^ [Y6: set_nat_nat,Z2: set_nat_nat] : ( Y6 = Z2 ) )
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_678_Collect__subset,axiom,
! [A2: set_nat_nat_nat,P: ( nat > nat > nat ) > $o] :
( ord_le3211623285424100676at_nat
@ ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_679_Collect__subset,axiom,
! [A2: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o] :
( ord_le5934964663421696068at_nat
@ ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_680_Collect__subset,axiom,
! [A2: set_nat_nat_nat_nat3,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ord_le5260717879541182899at_nat
@ ( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_681_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_682_Collect__subset,axiom,
! [A2: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_683_less__eq__set__def,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A5: set_nat_nat_nat,B6: set_nat_nat_nat] :
( ord_le5384859702510996545_nat_o
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A5 )
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_684_less__eq__set__def,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A5: set_nat_nat_nat2,B6: set_nat_nat_nat2] :
( ord_le996020443555834177_nat_o
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A5 )
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_685_less__eq__set__def,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A5: set_nat_nat_nat_nat3,B6: set_nat_nat_nat_nat3] :
( ord_le5430825838364970130_nat_o
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A5 )
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_686_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_687_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A5 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_688_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_689_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X2: nat > nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_690_Union__mono,axiom,
! [A2: set_set_nat_nat,B2: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_691_Union__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_692_Union__least,axiom,
! [A2: set_set_nat_nat,C4: set_nat_nat] :
( ! [X6: set_nat_nat] :
( ( member_set_nat_nat @ X6 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X6 @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ C4 ) ) ).
% Union_least
thf(fact_693_Union__least,axiom,
! [A2: set_set_nat,C4: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A2 )
=> ( ord_less_eq_set_nat @ X6 @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ C4 ) ) ).
% Union_least
thf(fact_694_Union__upper,axiom,
! [B2: set_nat_nat,A2: set_set_nat_nat] :
( ( member_set_nat_nat @ B2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ ( comple5448282615319421384at_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_695_Union__upper,axiom,
! [B2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_696_Union__subsetI,axiom,
! [A2: set_set_nat_nat,B2: set_set_nat_nat] :
( ! [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ A2 )
=> ? [Y5: set_nat_nat] :
( ( member_set_nat_nat @ Y5 @ B2 )
& ( ord_le9059583361652607317at_nat @ X3 @ Y5 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_697_Union__subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ? [Y5: set_nat] :
( ( member_set_nat @ Y5 @ B2 )
& ( ord_less_eq_set_nat @ X3 @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_698_PiE__mono,axiom,
! [A2: set_nat_nat_nat,B2: ( nat > nat > nat ) > set_nat,C4: ( nat > nat > nat ) > set_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le4355234527790737843at_nat @ ( piE_nat_nat_nat_nat2 @ A2 @ B2 ) @ ( piE_nat_nat_nat_nat2 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_699_PiE__mono,axiom,
! [A2: set_nat_nat_nat2,B2: ( ( nat > nat ) > nat ) > set_nat,C4: ( ( nat > nat ) > nat ) > set_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le5849559942836194483at_nat @ ( piE_nat_nat_nat_nat @ A2 @ B2 ) @ ( piE_nat_nat_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_700_PiE__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B2: ( ( nat > nat ) > nat > nat ) > set_nat,C4: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le2284792974450617250at_nat @ ( piE_na4548495224246695081at_nat @ A2 @ B2 ) @ ( piE_na4548495224246695081at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_701_PiE__mono,axiom,
! [A2: set_nat_nat,B2: ( nat > nat ) > set_nat,C4: ( nat > nat ) > set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le5934964663421696068at_nat @ ( piE_nat_nat_nat @ A2 @ B2 ) @ ( piE_nat_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_702_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_nat,C4: nat > set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ A2 @ B2 ) @ ( piE_nat_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_703_PiE__mono,axiom,
! [A2: set_nat_nat_nat,B2: ( nat > nat > nat ) > set_nat_nat,C4: ( nat > nat > nat ) > set_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le3125778081881428130at_nat @ ( piE_na7122919648973241129at_nat @ A2 @ B2 ) @ ( piE_na7122919648973241129at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_704_PiE__mono,axiom,
! [A2: set_nat_nat_nat2,B2: ( ( nat > nat ) > nat ) > set_nat_nat,C4: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le3190276326201062306at_nat @ ( piE_na6840239867990089257at_nat @ A2 @ B2 ) @ ( piE_na6840239867990089257at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_705_PiE__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B2: ( ( nat > nat ) > nat > nat ) > set_nat_nat,C4: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le9041126503034175505at_nat @ ( piE_na6564615839001774232at_nat @ A2 @ B2 ) @ ( piE_na6564615839001774232at_nat @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_706_PiE__mono,axiom,
! [A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat,C4: ( nat > nat ) > set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le5260717879541182899at_nat @ ( piE_nat_nat_nat_nat3 @ A2 @ B2 ) @ ( piE_nat_nat_nat_nat3 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_707_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_nat_nat,C4: nat > set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ ( C4 @ X3 ) ) )
=> ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ A2 @ B2 ) @ ( piE_nat_nat_nat2 @ A2 @ C4 ) ) ) ).
% PiE_mono
thf(fact_708_Sup__set__def,axiom,
( comple5448282615319421384at_nat
= ( ^ [A5: set_set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] : ( complete_Sup_Sup_o @ ( image_set_nat_nat_o @ ( member_nat_nat @ X2 ) @ A5 ) ) ) ) ) ).
% Sup_set_def
thf(fact_709_Sup__set__def,axiom,
( comple8167887107183641911at_nat
= ( ^ [A5: set_set_nat_nat_nat] :
( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] : ( complete_Sup_Sup_o @ ( image_5198217506544545261_nat_o @ ( member_nat_nat_nat2 @ X2 ) @ A5 ) ) ) ) ) ).
% Sup_set_def
thf(fact_710_Sup__set__def,axiom,
( comple1667856448326461495at_nat
= ( ^ [A5: set_set_nat_nat_nat2] :
( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] : ( complete_Sup_Sup_o @ ( image_8774134582277556973_nat_o @ ( member_nat_nat_nat @ X2 ) @ A5 ) ) ) ) ) ).
% Sup_set_def
thf(fact_711_Sup__set__def,axiom,
( comple2605510978757769510at_nat
= ( ^ [A5: set_se3022870823424313865at_nat] :
( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] : ( complete_Sup_Sup_o @ ( image_7580978635682194622_nat_o @ ( member952132173341509300at_nat @ X2 ) @ A5 ) ) ) ) ) ).
% Sup_set_def
thf(fact_712_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A5: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X2 ) @ A5 ) ) ) ) ) ).
% Sup_set_def
thf(fact_713_subset__PiE,axiom,
! [I3: set_nat_nat,S3: ( nat > nat ) > set_nat,T2: ( nat > nat ) > set_nat] :
( ( ord_le5934964663421696068at_nat @ ( piE_nat_nat_nat @ I3 @ S3 ) @ ( piE_nat_nat_nat @ I3 @ T2 ) )
= ( ( ( piE_nat_nat_nat @ I3 @ S3 )
= bot_bo945813143650711160at_nat )
| ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( ord_less_eq_set_nat @ ( S3 @ X2 ) @ ( T2 @ X2 ) ) ) ) ) ).
% subset_PiE
thf(fact_714_subset__PiE,axiom,
! [I3: set_nat_nat,S3: ( nat > nat ) > set_nat_nat,T2: ( nat > nat ) > set_nat_nat] :
( ( ord_le5260717879541182899at_nat @ ( piE_nat_nat_nat_nat3 @ I3 @ S3 ) @ ( piE_nat_nat_nat_nat3 @ I3 @ T2 ) )
= ( ( ( piE_nat_nat_nat_nat3 @ I3 @ S3 )
= bot_bo3919185967433191911at_nat )
| ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( ord_le9059583361652607317at_nat @ ( S3 @ X2 ) @ ( T2 @ X2 ) ) ) ) ) ).
% subset_PiE
thf(fact_715_subset__PiE,axiom,
! [I3: set_nat,S3: nat > set_nat_nat,T2: nat > set_nat_nat] :
( ( ord_le3211623285424100676at_nat @ ( piE_nat_nat_nat2 @ I3 @ S3 ) @ ( piE_nat_nat_nat2 @ I3 @ T2 ) )
= ( ( ( piE_nat_nat_nat2 @ I3 @ S3 )
= bot_bo7445843802507891576at_nat )
| ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( ord_le9059583361652607317at_nat @ ( S3 @ X2 ) @ ( T2 @ X2 ) ) ) ) ) ).
% subset_PiE
thf(fact_716_subset__PiE,axiom,
! [I3: set_nat,S3: nat > set_nat,T2: nat > set_nat] :
( ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ I3 @ S3 ) @ ( piE_nat_nat @ I3 @ T2 ) )
= ( ( ( piE_nat_nat @ I3 @ S3 )
= bot_bot_set_nat_nat )
| ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( ord_less_eq_set_nat @ ( S3 @ X2 ) @ ( T2 @ X2 ) ) ) ) ) ).
% subset_PiE
thf(fact_717_image__diff__subset,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_718_image__diff__subset,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ ( image_nat_nat_nat @ F @ B2 ) ) @ ( image_nat_nat_nat @ F @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_719_image__diff__subset,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ ( image_3205354838064109189at_nat @ F @ B2 ) ) @ ( image_3205354838064109189at_nat @ F @ ( minus_8121590178497047118at_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_720_image__diff__subset,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ ( image_nat_nat_nat2 @ F @ B2 ) ) @ ( image_nat_nat_nat2 @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_721_subset__image__iff,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_722_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_723_subset__image__iff,axiom,
! [B2: set_nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_724_subset__image__iff,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_725_subset__image__iff,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A2 )
& ( B2
= ( image_3205354838064109189at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_726_image__subset__iff,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_727_image__subset__iff,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_728_image__subset__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_729_image__subset__iff,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_730_subset__imageE,axiom,
! [B2: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_set_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_731_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_732_subset__imageE,axiom,
! [B2: set_nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_733_subset__imageE,axiom,
! [B2: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_nat_nat2 @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_734_subset__imageE,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A2 )
=> ( B2
!= ( image_3205354838064109189at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_735_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_736_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_nat,B2: set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_737_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,B2: set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_738_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat,B2: set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_739_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat > nat,B2: set_nat_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat_nat2 @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_740_image__subsetI,axiom,
! [A2: set_nat,F: nat > ( nat > nat ) > nat,B2: set_nat_nat_nat2] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_741_image__subsetI,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat,B2: set_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_913610194320715013at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_742_image__subsetI,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,B2: set_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_743_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat > nat,B2: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_nat_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_744_image__subsetI,axiom,
! [A2: set_nat,F: nat > ( nat > nat ) > nat > nat,B2: set_nat_nat_nat_nat3] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member952132173341509300at_nat @ ( F @ X3 ) @ B2 ) )
=> ( ord_le5260717879541182899at_nat @ ( image_6393715451659844596at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_745_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_746_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_747_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ ( image_nat_nat_nat2 @ F @ B2 ) ) ) ).
% image_mono
thf(fact_748_image__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ ( image_nat_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_749_image__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ ( image_3205354838064109189at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_750_psubsetE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_751_psubsetE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_752_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_753_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_754_psubset__imp__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_755_psubset__imp__subset,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_756_psubset__subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C4 )
=> ( ord_less_set_nat @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_757_psubset__subset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_758_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B6 )
& ~ ( ord_less_eq_set_nat @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_759_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B6 )
& ~ ( ord_le9059583361652607317at_nat @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_760_subset__psubset__trans,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C4 )
=> ( ord_less_set_nat @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_761_subset__psubset__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat_nat @ B2 @ C4 )
=> ( ord_less_set_nat_nat @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_762_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_set_nat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_763_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B6: set_nat_nat] :
( ( ord_less_set_nat_nat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_764_UN__mono,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat_nat,G: ( nat > nat > nat ) > set_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_765_UN__mono,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat_nat,G: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_766_UN__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B2: 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 @ B2 )
=> ( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_767_UN__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat_nat,G: nat > set_nat_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_768_UN__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > set_nat_nat,G: ( nat > nat ) > set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_769_UN__mono,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat,G: ( nat > nat > nat ) > set_nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_770_UN__mono,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat,G: ( ( nat > nat ) > nat ) > set_nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_771_UN__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B2: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat,G: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ B2 )
=> ( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_772_UN__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_773_UN__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > set_nat,G: ( nat > nat ) > set_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_774_UN__least,axiom,
! [A2: set_nat,B2: nat > set_nat_nat,C4: set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B2 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_775_UN__least,axiom,
! [A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat,C4: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ B2 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_776_UN__least,axiom,
! [A2: set_nat_nat_nat,B2: ( nat > nat > nat ) > set_nat_nat,C4: set_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ B2 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_777_UN__least,axiom,
! [A2: set_nat_nat_nat2,B2: ( ( nat > nat ) > nat ) > set_nat_nat,C4: set_nat_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ B2 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_778_UN__least,axiom,
! [A2: set_nat_nat_nat_nat3,B2: ( ( nat > nat ) > nat > nat ) > set_nat_nat,C4: set_nat_nat] :
( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ X3 ) @ C4 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ B2 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_779_UN__least,axiom,
! [A2: set_nat,B2: nat > set_nat,C4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_780_UN__least,axiom,
! [A2: set_nat_nat,B2: ( nat > nat ) > set_nat,C4: set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B2 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_781_UN__least,axiom,
! [A2: set_nat_nat_nat,B2: ( nat > nat > nat ) > set_nat,C4: set_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ B2 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_782_UN__least,axiom,
! [A2: set_nat_nat_nat2,B2: ( ( nat > nat ) > nat ) > set_nat,C4: set_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ B2 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_783_UN__least,axiom,
! [A2: set_nat_nat_nat_nat3,B2: ( ( nat > nat ) > nat > nat ) > set_nat,C4: set_nat] :
( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X3 ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ B2 @ A2 ) ) @ C4 ) ) ).
% UN_least
thf(fact_784_UN__upper,axiom,
! [A: nat,A2: set_nat,B2: nat > set_nat_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ A ) @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_785_UN__upper,axiom,
! [A: nat > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ A ) @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_786_UN__upper,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B2: ( nat > nat > nat ) > set_nat_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ A ) @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_787_UN__upper,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: ( ( nat > nat ) > nat ) > set_nat_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ A ) @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_788_UN__upper,axiom,
! [A: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: ( ( nat > nat ) > nat > nat ) > set_nat_nat] :
( ( member952132173341509300at_nat @ A @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( B2 @ A ) @ ( comple5448282615319421384at_nat @ ( image_6952571752803954585at_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_789_UN__upper,axiom,
! [A: nat,A2: set_nat,B2: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_790_UN__upper,axiom,
! [A: nat > nat,A2: set_nat_nat,B2: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_791_UN__upper,axiom,
! [A: nat > nat > nat,A2: set_nat_nat_nat,B2: ( nat > nat > nat ) > set_nat] :
( ( member_nat_nat_nat2 @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_792_UN__upper,axiom,
! [A: ( nat > nat ) > nat,A2: set_nat_nat_nat2,B2: ( ( nat > nat ) > nat ) > set_nat] :
( ( member_nat_nat_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_793_UN__upper,axiom,
! [A: ( nat > nat ) > nat > nat,A2: set_nat_nat_nat_nat3,B2: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ( member952132173341509300at_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_794_UN__subset__iff,axiom,
! [A2: nat > set_nat,I3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I3 ) ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( ord_less_eq_set_nat @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_795_PiE__eq__subset,axiom,
! [I3: set_nat_nat_nat,F2: ( nat > nat > nat ) > set_nat,F3: ( nat > nat > nat ) > set_nat,I2: nat > nat > nat] :
( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat ) )
=> ( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat_nat2 @ I3 @ F2 )
= ( piE_nat_nat_nat_nat2 @ I3 @ F3 ) )
=> ( ( member_nat_nat_nat2 @ I2 @ I3 )
=> ( ord_less_eq_set_nat @ ( F2 @ I2 ) @ ( F3 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_796_PiE__eq__subset,axiom,
! [I3: set_nat_nat_nat2,F2: ( ( nat > nat ) > nat ) > set_nat,F3: ( ( nat > nat ) > nat ) > set_nat,I2: ( nat > nat ) > nat] :
( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat ) )
=> ( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat_nat @ I3 @ F2 )
= ( piE_nat_nat_nat_nat @ I3 @ F3 ) )
=> ( ( member_nat_nat_nat @ I2 @ I3 )
=> ( ord_less_eq_set_nat @ ( F2 @ I2 ) @ ( F3 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_797_PiE__eq__subset,axiom,
! [I3: set_nat_nat_nat_nat3,F2: ( ( nat > nat ) > nat > nat ) > set_nat,F3: ( ( nat > nat ) > nat > nat ) > set_nat,I2: ( nat > nat ) > nat > nat] :
( ! [I4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat ) )
=> ( ! [I4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_na4548495224246695081at_nat @ I3 @ F2 )
= ( piE_na4548495224246695081at_nat @ I3 @ F3 ) )
=> ( ( member952132173341509300at_nat @ I2 @ I3 )
=> ( ord_less_eq_set_nat @ ( F2 @ I2 ) @ ( F3 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_798_PiE__eq__subset,axiom,
! [I3: set_nat,F2: nat > set_nat,F3: nat > set_nat,I2: nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat @ I3 @ F2 )
= ( piE_nat_nat @ I3 @ F3 ) )
=> ( ( member_nat @ I2 @ I3 )
=> ( ord_less_eq_set_nat @ ( F2 @ I2 ) @ ( F3 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_799_PiE__eq__subset,axiom,
! [I3: set_nat_nat,F2: ( nat > nat ) > set_nat,F3: ( nat > nat ) > set_nat,I2: nat > nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat @ I3 @ F2 )
= ( piE_nat_nat_nat @ I3 @ F3 ) )
=> ( ( member_nat_nat @ I2 @ I3 )
=> ( ord_less_eq_set_nat @ ( F2 @ I2 ) @ ( F3 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_800_PiE__eq__subset,axiom,
! [I3: set_nat_nat_nat,F2: ( nat > nat > nat ) > set_nat_nat,F3: ( nat > nat > nat ) > set_nat_nat,I2: nat > nat > nat] :
( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na7122919648973241129at_nat @ I3 @ F2 )
= ( piE_na7122919648973241129at_nat @ I3 @ F3 ) )
=> ( ( member_nat_nat_nat2 @ I2 @ I3 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ I2 ) @ ( F3 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_801_PiE__eq__subset,axiom,
! [I3: set_nat_nat_nat2,F2: ( ( nat > nat ) > nat ) > set_nat_nat,F3: ( ( nat > nat ) > nat ) > set_nat_nat,I2: ( nat > nat ) > nat] :
( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na6840239867990089257at_nat @ I3 @ F2 )
= ( piE_na6840239867990089257at_nat @ I3 @ F3 ) )
=> ( ( member_nat_nat_nat @ I2 @ I3 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ I2 ) @ ( F3 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_802_PiE__eq__subset,axiom,
! [I3: set_nat_nat_nat_nat3,F2: ( ( nat > nat ) > nat > nat ) > set_nat_nat,F3: ( ( nat > nat ) > nat > nat ) > set_nat_nat,I2: ( nat > nat ) > nat > nat] :
( ! [I4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_na6564615839001774232at_nat @ I3 @ F2 )
= ( piE_na6564615839001774232at_nat @ I3 @ F3 ) )
=> ( ( member952132173341509300at_nat @ I2 @ I3 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ I2 ) @ ( F3 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_803_PiE__eq__subset,axiom,
! [I3: set_nat_nat,F2: ( nat > nat ) > set_nat_nat,F3: ( nat > nat ) > set_nat_nat,I2: nat > nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat_nat3 @ I3 @ F2 )
= ( piE_nat_nat_nat_nat3 @ I3 @ F3 ) )
=> ( ( member_nat_nat @ I2 @ I3 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ I2 ) @ ( F3 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_804_PiE__eq__subset,axiom,
! [I3: set_nat,F2: nat > set_nat_nat,F3: nat > set_nat_nat,I2: nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F3 @ I4 )
!= bot_bot_set_nat_nat ) )
=> ( ( ( piE_nat_nat_nat2 @ I3 @ F2 )
= ( piE_nat_nat_nat2 @ I3 @ F3 ) )
=> ( ( member_nat @ I2 @ I3 )
=> ( ord_le9059583361652607317at_nat @ ( F2 @ I2 ) @ ( F3 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_805_PiE__uniqueness,axiom,
! [F: nat > set_nat,A2: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B2 )
=> ? [X3: nat > set_nat] :
( ( member_nat_set_nat @ X3
@ ( piE_nat_set_nat @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: nat > set_nat] :
( ( ( member_nat_set_nat @ Y5
@ ( piE_nat_set_nat @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_806_PiE__uniqueness,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: nat > nat] :
( ( ( member_nat_nat @ Y5
@ ( piE_nat_nat @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_807_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B2 )
=> ? [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3
@ ( piE_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( nat > nat ) > nat] :
( ( ( member_nat_nat_nat @ Y5
@ ( piE_nat_nat_nat @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_808_PiE__uniqueness,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) @ B2 )
=> ? [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: ( nat > nat ) > nat > nat] :
( ( ( member952132173341509300at_nat @ Y5
@ ( piE_nat_nat_nat_nat3 @ A2
@ ^ [I: nat > nat] : B2 ) )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_809_PiE__uniqueness,axiom,
! [F: nat > nat > nat,A2: set_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B2 )
=> ? [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( X3 @ Xa )
= ( F @ Xa ) ) )
& ! [Y5: nat > nat > nat] :
( ( ( member_nat_nat_nat2 @ Y5
@ ( piE_nat_nat_nat2 @ A2
@ ^ [I: nat] : B2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( Y5 @ Xa2 )
= ( F @ Xa2 ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% PiE_uniqueness
thf(fact_810_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_811_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > set_nat,D3: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C4 @ A2 ) )
= ( Sup @ ( image_nat_set_nat @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_812_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > nat > nat,D3: nat > nat > nat,Sup: set_nat_nat > nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat_nat2 @ C4 @ A2 ) )
= ( Sup @ ( image_nat_nat_nat2 @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_813_Sup_OSUP__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > nat > nat,D3: ( nat > nat ) > nat > nat,Sup: set_nat_nat > nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image_3205354838064109189at_nat @ C4 @ A2 ) )
= ( Sup @ ( image_3205354838064109189at_nat @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_814_Sup_OSUP__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > nat,D3: ( nat > nat ) > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat_nat @ C4 @ A2 ) )
= ( Sup @ ( image_nat_nat_nat @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_815_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > set_nat,D3: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C4 @ A2 ) )
= ( Inf @ ( image_nat_set_nat @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_816_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > nat > nat,D3: nat > nat > nat,Inf: set_nat_nat > nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat_nat2 @ C4 @ A2 ) )
= ( Inf @ ( image_nat_nat_nat2 @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_817_Inf_OINF__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > nat > nat,D3: ( nat > nat ) > nat > nat,Inf: set_nat_nat > nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image_3205354838064109189at_nat @ C4 @ A2 ) )
= ( Inf @ ( image_3205354838064109189at_nat @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_818_Inf_OINF__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > nat,D3: ( nat > nat ) > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat_nat @ C4 @ A2 ) )
= ( Inf @ ( image_nat_nat_nat @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_819_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_820_UnionE,axiom,
! [A2: nat > nat,C4: set_set_nat_nat] :
( ( member_nat_nat @ A2 @ ( comple5448282615319421384at_nat @ C4 ) )
=> ~ ! [X6: set_nat_nat] :
( ( member_nat_nat @ A2 @ X6 )
=> ~ ( member_set_nat_nat @ X6 @ C4 ) ) ) ).
% UnionE
thf(fact_821_UnionE,axiom,
! [A2: nat > nat > nat,C4: set_set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ A2 @ ( comple8167887107183641911at_nat @ C4 ) )
=> ~ ! [X6: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ A2 @ X6 )
=> ~ ( member8194441297229544571at_nat @ X6 @ C4 ) ) ) ).
% UnionE
thf(fact_822_UnionE,axiom,
! [A2: ( nat > nat ) > nat,C4: set_set_nat_nat_nat2] :
( ( member_nat_nat_nat @ A2 @ ( comple1667856448326461495at_nat @ C4 ) )
=> ~ ! [X6: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ A2 @ X6 )
=> ~ ( member1694410638372364155at_nat @ X6 @ C4 ) ) ) ).
% UnionE
thf(fact_823_UnionE,axiom,
! [A2: ( nat > nat ) > nat > nat,C4: set_se3022870823424313865at_nat] :
( ( member952132173341509300at_nat @ A2 @ ( comple2605510978757769510at_nat @ C4 ) )
=> ~ ! [X6: set_nat_nat_nat_nat3] :
( ( member952132173341509300at_nat @ A2 @ X6 )
=> ~ ( member7681264892014656106at_nat @ X6 @ C4 ) ) ) ).
% UnionE
thf(fact_824_UnionE,axiom,
! [A2: nat,C4: set_set_nat] :
( ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C4 ) )
=> ~ ! [X6: set_nat] :
( ( member_nat @ A2 @ X6 )
=> ~ ( member_set_nat @ X6 @ C4 ) ) ) ).
% UnionE
thf(fact_825_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat_nat > nat > nat,A2: set_nat_nat] :
( ( Sup
@ ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_826_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat_nat > nat > nat,A2: set_nat_nat] :
( ( Inf
@ ( image_3205354838064109189at_nat
@ ^ [X2: nat > nat] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_827_classes__subset__cube,axiom,
! [N: nat,T: nat,I2: nat] : ( ord_le9059583361652607317at_nat @ ( hales_classes @ N @ T @ I2 ) @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% classes_subset_cube
thf(fact_828_Sup__fun__def,axiom,
( comple2450677804321093138at_nat
= ( ^ [A5: set_nat_nat,X2: nat] :
( complete_Sup_Sup_nat
@ ( image_nat_nat_nat
@ ^ [F5: nat > nat] : ( F5 @ X2 )
@ A5 ) ) ) ) ).
% Sup_fun_def
thf(fact_829_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 ) )
& ! [D4: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D4 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_830_cSup__eq__maximum,axiom,
! [Z: set_nat_nat,X5: set_set_nat_nat] :
( ( member_set_nat_nat @ Z @ X5 )
=> ( ! [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ X5 )
=> ( ord_le9059583361652607317at_nat @ X3 @ Z ) )
=> ( ( comple5448282615319421384at_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_831_cSup__eq__maximum,axiom,
! [Z: set_nat,X5: set_set_nat] :
( ( member_set_nat @ Z @ X5 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
=> ( ord_less_eq_set_nat @ X3 @ Z ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_832_cSup__eq__maximum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat @ Z @ X5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_833_cSup__eq__maximum,axiom,
! [Z: $o,X5: set_o] :
( ( member_o @ Z @ X5 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ X3 @ Z ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_834_Sup__subset__mono,axiom,
! [A2: set_set_nat_nat,B2: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_835_Sup__subset__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_836_Sup__subset__mono,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_837_Sup__upper2,axiom,
! [U3: set_nat_nat,A2: set_set_nat_nat,V2: set_nat_nat] :
( ( member_set_nat_nat @ U3 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ V2 @ U3 )
=> ( ord_le9059583361652607317at_nat @ V2 @ ( comple5448282615319421384at_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_838_Sup__upper2,axiom,
! [U3: set_nat,A2: set_set_nat,V2: set_nat] :
( ( member_set_nat @ U3 @ A2 )
=> ( ( ord_less_eq_set_nat @ V2 @ U3 )
=> ( ord_less_eq_set_nat @ V2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_839_Sup__upper2,axiom,
! [U3: $o,A2: set_o,V2: $o] :
( ( member_o @ U3 @ A2 )
=> ( ( ord_less_eq_o @ V2 @ U3 )
=> ( ord_less_eq_o @ V2 @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_840_Sup__le__iff,axiom,
! [A2: set_set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ B )
= ( ! [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_841_Sup__le__iff,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ B )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_842_Sup__le__iff,axiom,
! [A2: set_o,B: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ B )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_o @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_843_Sup__upper,axiom,
! [X: set_nat_nat,A2: set_set_nat_nat] :
( ( member_set_nat_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ X @ ( comple5448282615319421384at_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_844_Sup__upper,axiom,
! [X: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_845_Sup__upper,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ X @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_846_Sup__least,axiom,
! [A2: set_set_nat_nat,Z: set_nat_nat] :
( ! [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X3 @ Z ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_847_Sup__least,axiom,
! [A2: set_set_nat,Z: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_848_Sup__least,axiom,
! [A2: set_o,Z: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ X3 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_849_Sup__mono,axiom,
! [A2: set_set_nat_nat,B2: set_set_nat_nat] :
( ! [A4: set_nat_nat] :
( ( member_set_nat_nat @ A4 @ A2 )
=> ? [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ B2 )
& ( ord_le9059583361652607317at_nat @ A4 @ X4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A2 ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_850_Sup__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ A2 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ A4 @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_851_Sup__mono,axiom,
! [A2: set_o,B2: set_o] :
( ! [A4: $o] :
( ( member_o @ A4 @ A2 )
=> ? [X4: $o] :
( ( member_o @ X4 @ B2 )
& ( ord_less_eq_o @ A4 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_852_Sup__eqI,axiom,
! [A2: set_set_nat_nat,X: set_nat_nat] :
( ! [Y4: set_nat_nat] :
( ( member_set_nat_nat @ Y4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ Y4 @ X ) )
=> ( ! [Y4: set_nat_nat] :
( ! [Z3: set_nat_nat] :
( ( member_set_nat_nat @ Z3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ Z3 @ Y4 ) )
=> ( ord_le9059583361652607317at_nat @ X @ Y4 ) )
=> ( ( comple5448282615319421384at_nat @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_853_Sup__eqI,axiom,
! [A2: set_set_nat,X: set_nat] :
( ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ A2 )
=> ( ord_less_eq_set_nat @ Y4 @ X ) )
=> ( ! [Y4: set_nat] :
( ! [Z3: set_nat] :
( ( member_set_nat @ Z3 @ A2 )
=> ( ord_less_eq_set_nat @ Z3 @ Y4 ) )
=> ( ord_less_eq_set_nat @ X @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_854_Sup__eqI,axiom,
! [A2: set_o,X: $o] :
( ! [Y4: $o] :
( ( member_o @ Y4 @ A2 )
=> ( ord_less_eq_o @ Y4 @ X ) )
=> ( ! [Y4: $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ A2 )
=> ( ord_less_eq_o @ Z3 @ Y4 ) )
=> ( ord_less_eq_o @ X @ Y4 ) )
=> ( ( complete_Sup_Sup_o @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_855_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > nat,D3: nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_856_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > $o,D3: nat > $o] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C4 @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_857_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > set_nat,D3: nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_858_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C4: nat > nat > nat,D3: nat > nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple2450677804321093138at_nat @ ( image_nat_nat_nat2 @ C4 @ A2 ) )
= ( comple2450677804321093138at_nat @ ( image_nat_nat_nat2 @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_859_SUP__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > nat,D3: ( nat > nat ) > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_860_SUP__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > $o,D3: ( nat > nat ) > $o] :
( ( A2 = B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ C4 @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_nat_o @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_861_SUP__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > set_nat,D3: ( nat > nat ) > set_nat] :
( ( A2 = B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ C4 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_862_SUP__cong,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C4: ( nat > nat ) > nat > nat,D3: ( nat > nat ) > nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( comple2450677804321093138at_nat @ ( image_3205354838064109189at_nat @ C4 @ A2 ) )
= ( comple2450677804321093138at_nat @ ( image_3205354838064109189at_nat @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_863_SUP__cong,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,C4: ( nat > nat > nat ) > nat,D3: ( nat > nat > nat ) > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_913610194320715013at_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_913610194320715013at_nat @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_864_SUP__cong,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,C4: ( ( nat > nat ) > nat ) > nat,D3: ( ( nat > nat ) > nat ) > nat] :
( ( A2 = B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ B2 )
=> ( ( C4 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_7809927846809980933at_nat @ C4 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_7809927846809980933at_nat @ D3 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_865_empty__Union__conv,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_866_Union__empty__conv,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_867_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_868_SUP__commute,axiom,
! [F: nat > nat > set_nat,B2: set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J3: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( F @ I @ J3 )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_869_image__Union,axiom,
! [F: ( nat > nat ) > nat > nat,S3: set_set_nat_nat] :
( ( image_3205354838064109189at_nat @ F @ ( comple5448282615319421384at_nat @ S3 ) )
= ( comple5448282615319421384at_nat @ ( image_3832368097948589297at_nat @ ( image_3205354838064109189at_nat @ F ) @ S3 ) ) ) ).
% image_Union
thf(fact_870_image__Union,axiom,
! [F: ( nat > nat ) > nat,S3: set_set_nat_nat] :
( ( image_nat_nat_nat @ F @ ( comple5448282615319421384at_nat @ S3 ) )
= ( comple7399068483239264473et_nat @ ( image_6930934588239670658et_nat @ ( image_nat_nat_nat @ F ) @ S3 ) ) ) ).
% image_Union
thf(fact_871_image__Union,axiom,
! [F: nat > set_nat,S3: set_set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ S3 ) )
= ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ S3 ) ) ) ).
% image_Union
thf(fact_872_image__Union,axiom,
! [F: nat > nat > nat,S3: set_set_nat] :
( ( image_nat_nat_nat2 @ F @ ( comple7399068483239264473et_nat @ S3 ) )
= ( comple5448282615319421384at_nat @ ( image_7054278410236665602at_nat @ ( image_nat_nat_nat2 @ F ) @ S3 ) ) ) ).
% image_Union
thf(fact_873_image__Union,axiom,
! [F: nat > nat,S3: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S3 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S3 ) ) ) ).
% image_Union
thf(fact_874_UN__extend__simps_I6_J,axiom,
! [A2: nat > set_nat,C4: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ C4 ) ) @ B2 )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( minus_minus_set_nat @ ( A2 @ X2 ) @ B2 )
@ C4 ) ) ) ).
% UN_extend_simps(6)
thf(fact_875_UN__UN__flatten,axiom,
! [C4: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_876_UN__E,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_877_UN__E,axiom,
! [B: nat > nat,B2: nat > set_nat_nat,A2: set_nat] :
( ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B2 @ A2 ) ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat_nat @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_878_UN__E,axiom,
! [B: nat,B2: ( nat > nat ) > set_nat,A2: set_nat_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B2 @ A2 ) ) )
=> ~ ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_879_UN__E,axiom,
! [B: nat > nat,B2: ( nat > nat ) > set_nat_nat,A2: set_nat_nat] :
( ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ B2 @ A2 ) ) )
=> ~ ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ~ ( member_nat_nat @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_880_UN__E,axiom,
! [B: nat > nat > nat,B2: nat > set_nat_nat_nat,A2: set_nat] :
( ( member_nat_nat_nat2 @ B @ ( comple8167887107183641911at_nat @ ( image_6130888460295934395at_nat @ B2 @ A2 ) ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat_nat_nat2 @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_881_UN__E,axiom,
! [B: ( nat > nat ) > nat,B2: nat > set_nat_nat_nat2,A2: set_nat] :
( ( member_nat_nat_nat @ B @ ( comple1667856448326461495at_nat @ ( image_8854229838293529787at_nat @ B2 @ A2 ) ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat_nat_nat @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_882_UN__E,axiom,
! [B: nat,B2: ( nat > nat > nat ) > set_nat,A2: set_nat_nat_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ B2 @ A2 ) ) )
=> ~ ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_883_UN__E,axiom,
! [B: nat,B2: ( ( nat > nat ) > nat ) > set_nat,A2: set_nat_nat_nat2] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ B2 @ A2 ) ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_884_UN__E,axiom,
! [B: nat > nat,B2: ( nat > nat > nat ) > set_nat_nat,A2: set_nat_nat_nat] :
( ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_7416711816588782250at_nat @ B2 @ A2 ) ) )
=> ~ ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ~ ( member_nat_nat @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_885_UN__E,axiom,
! [B: nat > nat,B2: ( ( nat > nat ) > nat ) > set_nat_nat,A2: set_nat_nat_nat2] :
( ( member_nat_nat @ B @ ( comple5448282615319421384at_nat @ ( image_2070201431993601450at_nat @ B2 @ A2 ) ) )
=> ~ ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ~ ( member_nat_nat @ B @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_886_UN__extend__simps_I8_J,axiom,
! [B2: nat > set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y2: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_887_UN__extend__simps_I9_J,axiom,
! [C4: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_888_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > $o,G: nat > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
=> ( ! [J4: nat] :
( ( member_nat @ J4 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J4 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_889_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
=> ( ! [J4: nat] :
( ( member_nat @ J4 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J4 ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_890_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat_nat,F: nat > $o,G: ( nat > nat ) > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
=> ( ! [J4: nat > nat] :
( ( member_nat_nat @ J4 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J4 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_891_SUP__eq,axiom,
! [A2: set_nat_nat,B2: set_nat,F: ( nat > nat ) > $o,G: nat > $o] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
=> ( ! [J4: nat] :
( ( member_nat @ J4 @ B2 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J4 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_892_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat_nat,G: nat > set_nat_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_le9059583361652607317at_nat @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
=> ( ! [J4: nat] :
( ( member_nat @ J4 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_le9059583361652607317at_nat @ ( G @ J4 ) @ ( F @ X4 ) ) ) )
=> ( ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) )
= ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_893_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat_nat,F: nat > set_nat,G: ( nat > nat ) > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
=> ( ! [J4: nat > nat] :
( ( member_nat_nat @ J4 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J4 ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_894_SUP__eq,axiom,
! [A2: set_nat_nat,B2: set_nat,F: ( nat > nat ) > set_nat,G: nat > set_nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_less_eq_set_nat @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
=> ( ! [J4: nat] :
( ( member_nat @ J4 @ B2 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J4 ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_895_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat_nat_nat,F: nat > $o,G: ( nat > nat > nat ) > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
=> ( ! [J4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ J4 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J4 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o2 @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_896_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat_nat_nat2,F: nat > $o,G: ( ( nat > nat ) > nat ) > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
=> ( ! [J4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ J4 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J4 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_897_SUP__eq,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > $o,G: ( nat > nat ) > $o] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ A2 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B2 )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
=> ( ! [J4: nat > nat] :
( ( member_nat_nat @ J4 @ B2 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
& ( ord_less_eq_o @ ( G @ J4 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_898_less__eq__Sup,axiom,
! [A2: set_set_nat_nat,U3: set_nat_nat] :
( ! [V: set_nat_nat] :
( ( member_set_nat_nat @ V @ A2 )
=> ( ord_le9059583361652607317at_nat @ U3 @ V ) )
=> ( ( A2 != bot_bo7376149671870096959at_nat )
=> ( ord_le9059583361652607317at_nat @ U3 @ ( comple5448282615319421384at_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_899_less__eq__Sup,axiom,
! [A2: set_set_nat,U3: set_nat] :
( ! [V: set_nat] :
( ( member_set_nat @ V @ A2 )
=> ( ord_less_eq_set_nat @ U3 @ V ) )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U3 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_900_less__eq__Sup,axiom,
! [A2: set_o,U3: $o] :
( ! [V: $o] :
( ( member_o @ V @ A2 )
=> ( ord_less_eq_o @ U3 @ V ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ U3 @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_901_cSup__least,axiom,
! [X5: set_set_nat_nat,Z: set_nat_nat] :
( ( X5 != bot_bo7376149671870096959at_nat )
=> ( ! [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ X5 )
=> ( ord_le9059583361652607317at_nat @ X3 @ Z ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_902_cSup__least,axiom,
! [X5: set_set_nat,Z: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
=> ( ord_less_eq_set_nat @ X3 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_903_cSup__least,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_904_cSup__least,axiom,
! [X5: set_o,Z: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ X3 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_905_cSup__eq__non__empty,axiom,
! [X5: set_set_nat_nat,A: set_nat_nat] :
( ( X5 != bot_bo7376149671870096959at_nat )
=> ( ! [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ X5 )
=> ( ord_le9059583361652607317at_nat @ X3 @ 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_906_cSup__eq__non__empty,axiom,
! [X5: set_set_nat,A: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
=> ( ord_less_eq_set_nat @ X3 @ 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_907_cSup__eq__non__empty,axiom,
! [X5: set_nat,A: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ 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_908_cSup__eq__non__empty,axiom,
! [X5: set_o,A: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ X3 @ A ) )
=> ( ! [Y4: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ Y4 ) )
=> ( ord_less_eq_o @ A @ Y4 ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_909_less__cSupD,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z @ ( complete_Sup_Sup_nat @ X5 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( ord_less_nat @ Z @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_910_less__cSupE,axiom,
! [Y: nat,X5: set_nat] :
( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X5 ) )
=> ( ( X5 != bot_bot_set_nat )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_911_SUP__eq__const,axiom,
! [I3: set_nat_nat,F: ( nat > nat ) > set_nat,X: set_nat] :
( ( I3 != bot_bot_set_nat_nat )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_912_SUP__eq__const,axiom,
! [I3: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat,X: set_nat] :
( ( I3 != bot_bo7445843802507891576at_nat )
=> ( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_913_SUP__eq__const,axiom,
! [I3: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat,X: set_nat] :
( ( I3 != bot_bo945813143650711160at_nat )
=> ( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_914_SUP__eq__const,axiom,
! [I3: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > set_nat,X: set_nat] :
( ( I3 != bot_bo3919185967433191911at_nat )
=> ( ! [I4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_915_SUP__eq__const,axiom,
! [I3: set_nat,F: nat > set_nat,X: set_nat] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_916_SUP__eq__const,axiom,
! [I3: set_nat_nat,F: ( nat > nat ) > $o,X: $o] :
( ( I3 != bot_bot_set_nat_nat )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_917_SUP__eq__const,axiom,
! [I3: set_nat_nat_nat,F: ( nat > nat > nat ) > $o,X: $o] :
( ( I3 != bot_bo7445843802507891576at_nat )
=> ( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o2 @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_918_SUP__eq__const,axiom,
! [I3: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > $o,X: $o] :
( ( I3 != bot_bo945813143650711160at_nat )
=> ( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_919_SUP__eq__const,axiom,
! [I3: set_nat_nat_nat_nat3,F: ( ( nat > nat ) > nat > nat ) > $o,X: $o] :
( ( I3 != bot_bo3919185967433191911at_nat )
=> ( ! [I4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_8690456353314504180_nat_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_920_SUP__eq__const,axiom,
! [I3: set_nat,F: nat > $o,X: $o] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_921_SUP__eqI,axiom,
! [A2: set_nat,F: nat > $o,X: $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ X ) )
=> ( ! [Y4: $o] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y4 ) )
=> ( ord_less_eq_o @ X @ Y4 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_922_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_nat,X: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ X ) )
=> ( ! [Y4: set_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y4 ) )
=> ( ord_less_eq_set_nat @ X @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_923_SUP__eqI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > $o,X: $o] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ X ) )
=> ( ! [Y4: $o] :
( ! [I5: nat > nat] :
( ( member_nat_nat @ I5 @ A2 )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y4 ) )
=> ( ord_less_eq_o @ X @ Y4 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_924_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_nat_nat,X: set_nat_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I4 ) @ X ) )
=> ( ! [Y4: set_nat_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I5 ) @ Y4 ) )
=> ( ord_le9059583361652607317at_nat @ X @ Y4 ) )
=> ( ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_925_SUP__eqI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat,X: set_nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ X ) )
=> ( ! [Y4: set_nat] :
( ! [I5: nat > nat] :
( ( member_nat_nat @ I5 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y4 ) )
=> ( ord_less_eq_set_nat @ X @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_926_SUP__eqI,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > $o,X: $o] :
( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ X ) )
=> ( ! [Y4: $o] :
( ! [I5: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I5 @ A2 )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y4 ) )
=> ( ord_less_eq_o @ X @ Y4 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o2 @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_927_SUP__eqI,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > $o,X: $o] :
( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ X ) )
=> ( ! [Y4: $o] :
( ! [I5: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I5 @ A2 )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y4 ) )
=> ( ord_less_eq_o @ X @ Y4 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_928_SUP__eqI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat_nat,X: set_nat_nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I4 ) @ X ) )
=> ( ! [Y4: set_nat_nat] :
( ! [I5: nat > nat] :
( ( member_nat_nat @ I5 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I5 ) @ Y4 ) )
=> ( ord_le9059583361652607317at_nat @ X @ Y4 ) )
=> ( ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_929_SUP__eqI,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat,X: set_nat] :
( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ X ) )
=> ( ! [Y4: set_nat] :
( ! [I5: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I5 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y4 ) )
=> ( ord_less_eq_set_nat @ X @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_930_SUP__eqI,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat,X: set_nat] :
( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ X ) )
=> ( ! [Y4: set_nat] :
( ! [I5: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I5 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y4 ) )
=> ( ord_less_eq_set_nat @ X @ Y4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) )
= X ) ) ) ).
% SUP_eqI
thf(fact_931_SUP__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [N2: nat] :
( ( member_nat @ N2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( 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 @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_932_SUP__mono,axiom,
! [A2: set_nat_nat,B2: set_nat,F: ( nat > nat ) > set_nat,G: nat > set_nat] :
( ! [N2: nat > nat] :
( ( member_nat_nat @ N2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( 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 @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_933_SUP__mono,axiom,
! [A2: set_nat_nat_nat,B2: 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 @ B2 )
& ( 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 @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_934_SUP__mono,axiom,
! [A2: set_nat_nat_nat2,B2: 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 @ B2 )
& ( 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 @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_935_SUP__mono,axiom,
! [A2: set_nat_nat_nat_nat3,B2: 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 @ B2 )
& ( 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 @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_936_SUP__least,axiom,
! [A2: set_nat,F: nat > $o,U3: $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ U3 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ U3 ) ) ).
% SUP_least
thf(fact_937_SUP__least,axiom,
! [A2: set_nat,F: nat > set_nat,U3: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ U3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ U3 ) ) ).
% SUP_least
thf(fact_938_SUP__least,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > $o,U3: $o] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ U3 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A2 ) ) @ U3 ) ) ).
% SUP_least
thf(fact_939_SUP__least,axiom,
! [A2: set_nat,F: nat > set_nat_nat,U3: set_nat_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I4 ) @ U3 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) @ U3 ) ) ).
% SUP_least
thf(fact_940_SUP__least,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat,U3: set_nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ U3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) @ U3 ) ) ).
% SUP_least
thf(fact_941_SUP__least,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > $o,U3: $o] :
( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ U3 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o2 @ F @ A2 ) ) @ U3 ) ) ).
% SUP_least
thf(fact_942_SUP__least,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > $o,U3: $o] :
( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ U3 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o @ F @ A2 ) ) @ U3 ) ) ).
% SUP_least
thf(fact_943_SUP__least,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat_nat,U3: set_nat_nat] :
( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I4 ) @ U3 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) ) @ U3 ) ) ).
% SUP_least
thf(fact_944_SUP__least,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat,U3: set_nat] :
( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ U3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) @ U3 ) ) ).
% SUP_least
thf(fact_945_SUP__least,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat,U3: set_nat] :
( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ U3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) @ U3 ) ) ).
% SUP_least
thf(fact_946_SUP__mono_H,axiom,
! [F: nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ! [X3: nat] : ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( 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_947_SUP__upper,axiom,
! [I2: nat,A2: set_nat,F: nat > $o] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_948_SUP__upper,axiom,
! [I2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_949_SUP__upper,axiom,
! [I2: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > $o] :
( ( member_nat_nat @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_950_SUP__upper,axiom,
! [I2: nat,A2: set_nat,F: nat > set_nat_nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I2 ) @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_951_SUP__upper,axiom,
! [I2: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_952_SUP__upper,axiom,
! [I2: nat > nat > nat,A2: set_nat_nat_nat,F: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o2 @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_953_SUP__upper,axiom,
! [I2: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_954_SUP__upper,axiom,
! [I2: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ I2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ I2 ) @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_955_SUP__upper,axiom,
! [I2: nat > nat > nat,A2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat] :
( ( member_nat_nat_nat2 @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_956_SUP__upper,axiom,
! [I2: ( nat > nat ) > nat,A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat] :
( ( member_nat_nat_nat @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_957_SUP__le__iff,axiom,
! [F: nat > set_nat,A2: set_nat,U3: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ U3 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ U3 ) ) ) ) ).
% SUP_le_iff
thf(fact_958_SUP__upper2,axiom,
! [I2: nat,A2: set_nat,U3: $o,F: nat > $o] :
( ( member_nat @ I2 @ A2 )
=> ( ( ord_less_eq_o @ U3 @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U3 @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_959_SUP__upper2,axiom,
! [I2: nat,A2: set_nat,U3: set_nat,F: nat > set_nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( ord_less_eq_set_nat @ U3 @ ( F @ I2 ) )
=> ( ord_less_eq_set_nat @ U3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_960_SUP__upper2,axiom,
! [I2: nat > nat,A2: set_nat_nat,U3: $o,F: ( nat > nat ) > $o] :
( ( member_nat_nat @ I2 @ A2 )
=> ( ( ord_less_eq_o @ U3 @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U3 @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_961_SUP__upper2,axiom,
! [I2: nat,A2: set_nat,U3: set_nat_nat,F: nat > set_nat_nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ U3 @ ( F @ I2 ) )
=> ( ord_le9059583361652607317at_nat @ U3 @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_962_SUP__upper2,axiom,
! [I2: nat > nat,A2: set_nat_nat,U3: set_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ I2 @ A2 )
=> ( ( ord_less_eq_set_nat @ U3 @ ( F @ I2 ) )
=> ( ord_less_eq_set_nat @ U3 @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_963_SUP__upper2,axiom,
! [I2: nat > nat > nat,A2: set_nat_nat_nat,U3: $o,F: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ I2 @ A2 )
=> ( ( ord_less_eq_o @ U3 @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U3 @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o2 @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_964_SUP__upper2,axiom,
! [I2: ( nat > nat ) > nat,A2: set_nat_nat_nat2,U3: $o,F: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ I2 @ A2 )
=> ( ( ord_less_eq_o @ U3 @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U3 @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_965_SUP__upper2,axiom,
! [I2: nat > nat,A2: set_nat_nat,U3: set_nat_nat,F: ( nat > nat ) > set_nat_nat] :
( ( member_nat_nat @ I2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ U3 @ ( F @ I2 ) )
=> ( ord_le9059583361652607317at_nat @ U3 @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_966_SUP__upper2,axiom,
! [I2: nat > nat > nat,A2: set_nat_nat_nat,U3: set_nat,F: ( nat > nat > nat ) > set_nat] :
( ( member_nat_nat_nat2 @ I2 @ A2 )
=> ( ( ord_less_eq_set_nat @ U3 @ ( F @ I2 ) )
=> ( ord_less_eq_set_nat @ U3 @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_967_SUP__upper2,axiom,
! [I2: ( nat > nat ) > nat,A2: set_nat_nat_nat2,U3: set_nat,F: ( ( nat > nat ) > nat ) > set_nat] :
( ( member_nat_nat_nat @ I2 @ A2 )
=> ( ( ord_less_eq_set_nat @ U3 @ ( F @ I2 ) )
=> ( ord_less_eq_set_nat @ U3 @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_968_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > $o,G: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_969_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_970_SUP__subset__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > $o,G: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_971_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat_nat,G: nat > set_nat_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_972_SUP__subset__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > set_nat,G: ( nat > nat ) > set_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_973_SUP__subset__mono,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,F: ( nat > nat > nat ) > $o,G: ( nat > nat > nat ) > $o] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o2 @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o2 @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_974_SUP__subset__mono,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > $o,G: ( ( nat > nat ) > nat ) > $o] :
( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_975_SUP__subset__mono,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > set_nat_nat,G: ( nat > nat ) > set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ A2 ) ) @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_976_SUP__subset__mono,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat,F: ( nat > nat > nat ) > set_nat,G: ( nat > nat > nat ) > set_nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ B2 )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_977_SUP__subset__mono,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > set_nat,G: ( ( nat > nat ) > nat ) > set_nat] :
( ( ord_le5934964663421696068at_nat @ A2 @ B2 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_978_SUP__lessD,axiom,
! [F: nat > set_nat,A2: set_nat,Y: set_nat,I2: nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ Y )
=> ( ( member_nat @ I2 @ A2 )
=> ( ord_less_set_nat @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_979_SUP__lessD,axiom,
! [F: ( nat > nat ) > set_nat,A2: set_nat_nat,Y: set_nat,I2: nat > nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) @ Y )
=> ( ( member_nat_nat @ I2 @ A2 )
=> ( ord_less_set_nat @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_980_SUP__lessD,axiom,
! [F: ( nat > nat > nat ) > set_nat,A2: set_nat_nat_nat,Y: set_nat,I2: nat > nat > nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ A2 ) ) @ Y )
=> ( ( member_nat_nat_nat2 @ I2 @ A2 )
=> ( ord_less_set_nat @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_981_SUP__lessD,axiom,
! [F: ( ( nat > nat ) > nat ) > set_nat,A2: set_nat_nat_nat2,Y: set_nat,I2: ( nat > nat ) > nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ A2 ) ) @ Y )
=> ( ( member_nat_nat_nat @ I2 @ A2 )
=> ( ord_less_set_nat @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_982_SUP__lessD,axiom,
! [F: ( ( nat > nat ) > nat > nat ) > set_nat,A2: set_nat_nat_nat_nat3,Y: set_nat,I2: ( nat > nat ) > nat > nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ A2 ) ) @ Y )
=> ( ( member952132173341509300at_nat @ I2 @ A2 )
=> ( ord_less_set_nat @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_983_SUP__lessD,axiom,
! [F: nat > $o,A2: set_nat,Y: $o,I2: nat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ Y )
=> ( ( member_nat @ I2 @ A2 )
=> ( ord_less_o @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_984_SUP__lessD,axiom,
! [F: ( nat > nat ) > $o,A2: set_nat_nat,Y: $o,I2: nat > nat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A2 ) ) @ Y )
=> ( ( member_nat_nat @ I2 @ A2 )
=> ( ord_less_o @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_985_SUP__lessD,axiom,
! [F: ( nat > nat > nat ) > $o,A2: set_nat_nat_nat,Y: $o,I2: nat > nat > nat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o2 @ F @ A2 ) ) @ Y )
=> ( ( member_nat_nat_nat2 @ I2 @ A2 )
=> ( ord_less_o @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_986_SUP__lessD,axiom,
! [F: ( ( nat > nat ) > nat ) > $o,A2: set_nat_nat_nat2,Y: $o,I2: ( nat > nat ) > nat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o @ F @ A2 ) ) @ Y )
=> ( ( member_nat_nat_nat @ I2 @ A2 )
=> ( ord_less_o @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_987_SUP__lessD,axiom,
! [F: ( ( nat > nat ) > nat > nat ) > $o,A2: set_nat_nat_nat_nat3,Y: $o,I2: ( nat > nat ) > nat > nat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_8690456353314504180_nat_o @ F @ A2 ) ) @ Y )
=> ( ( member952132173341509300at_nat @ I2 @ A2 )
=> ( ord_less_o @ ( F @ I2 ) @ Y ) ) ) ).
% SUP_lessD
thf(fact_988_image__UN,axiom,
! [F: ( nat > nat ) > nat,B2: nat > set_nat_nat,A2: set_nat] :
( ( image_nat_nat_nat @ F @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_nat_nat_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_989_image__UN,axiom,
! [F: nat > set_nat,B2: nat > set_nat,A2: set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X2: nat] : ( image_nat_set_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_990_image__UN,axiom,
! [F: nat > nat > nat,B2: nat > set_nat,A2: set_nat] :
( ( image_nat_nat_nat2 @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple5448282615319421384at_nat
@ ( image_7301343469591561292at_nat
@ ^ [X2: nat] : ( image_nat_nat_nat2 @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_991_image__UN,axiom,
! [F: nat > nat,B2: nat > set_nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_nat_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_992_UN__extend__simps_I10_J,axiom,
! [B2: ( nat > nat ) > set_nat,F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [A3: nat > nat] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_993_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [A3: nat > nat] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_994_UN__extend__simps_I10_J,axiom,
! [B2: set_nat > set_nat,F: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_995_UN__extend__simps_I10_J,axiom,
! [B2: ( nat > nat ) > set_nat,F: nat > nat > nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_996_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_nat,F: nat > nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( B2 @ ( F @ A3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_997_UN__empty2,axiom,
! [A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : bot_bot_set_nat
@ A2 ) )
= bot_bot_set_nat ) ).
% UN_empty2
thf(fact_998_UN__empty,axiom,
! [B2: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_999_UNION__empty__conv_I1_J,axiom,
! [B2: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_nat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_1000_UNION__empty__conv_I2_J,axiom,
! [B2: nat > set_nat,A2: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2 @ X2 )
= bot_bot_set_nat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_1001_cSUP__least,axiom,
! [A2: set_nat,F: nat > nat,M3: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1002_cSUP__least,axiom,
! [A2: set_nat,F: nat > $o,M3: $o] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ M3 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1003_cSUP__least,axiom,
! [A2: set_nat,F: nat > set_nat,M3: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ M3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1004_cSUP__least,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,M3: nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1005_cSUP__least,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > $o,M3: $o] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ M3 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1006_cSUP__least,axiom,
! [A2: set_nat,F: nat > set_nat_nat,M3: set_nat_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ M3 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1007_cSUP__least,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat,M3: set_nat] :
( ( A2 != bot_bot_set_nat_nat )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ M3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1008_cSUP__least,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > nat,M3: nat] :
( ( A2 != bot_bo7445843802507891576at_nat )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_913610194320715013at_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1009_cSUP__least,axiom,
! [A2: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,M3: nat] :
( ( A2 != bot_bo945813143650711160at_nat )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_7809927846809980933at_nat @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1010_cSUP__least,axiom,
! [A2: set_nat_nat_nat,F: ( nat > nat > nat ) > $o,M3: $o] :
( ( A2 != bot_bo7445843802507891576at_nat )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ M3 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o2 @ F @ A2 ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_1011_SUP__eq__iff,axiom,
! [I3: set_nat,C: $o,F: nat > $o] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ord_less_eq_o @ C @ ( F @ I4 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I3 ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1012_SUP__eq__iff,axiom,
! [I3: set_nat,C: set_nat,F: nat > set_nat] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I3 ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1013_SUP__eq__iff,axiom,
! [I3: set_nat_nat,C: $o,F: ( nat > nat ) > $o] :
( ( I3 != bot_bot_set_nat_nat )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ord_less_eq_o @ C @ ( F @ I4 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ I3 ) )
= C )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1014_SUP__eq__iff,axiom,
! [I3: set_nat,C: set_nat_nat,F: nat > set_nat_nat] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ord_le9059583361652607317at_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ I3 ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I3 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1015_SUP__eq__iff,axiom,
! [I3: set_nat_nat,C: set_nat,F: ( nat > nat ) > set_nat] :
( ( I3 != bot_bot_set_nat_nat )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ I3 ) )
= C )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1016_SUP__eq__iff,axiom,
! [I3: set_nat_nat_nat,C: $o,F: ( nat > nat > nat ) > $o] :
( ( I3 != bot_bo7445843802507891576at_nat )
=> ( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ I3 )
=> ( ord_less_eq_o @ C @ ( F @ I4 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o2 @ F @ I3 ) )
= C )
= ( ! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ I3 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1017_SUP__eq__iff,axiom,
! [I3: set_nat_nat_nat2,C: $o,F: ( ( nat > nat ) > nat ) > $o] :
( ( I3 != bot_bo945813143650711160at_nat )
=> ( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ I3 )
=> ( ord_less_eq_o @ C @ ( F @ I4 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_nat_nat_o @ F @ I3 ) )
= C )
= ( ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ I3 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1018_SUP__eq__iff,axiom,
! [I3: set_nat_nat,C: set_nat_nat,F: ( nat > nat ) > set_nat_nat] :
( ( I3 != bot_bot_set_nat_nat )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ord_le9059583361652607317at_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ F @ I3 ) )
= C )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I3 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1019_SUP__eq__iff,axiom,
! [I3: set_nat_nat_nat,C: set_nat,F: ( nat > nat > nat ) > set_nat] :
( ( I3 != bot_bo7445843802507891576at_nat )
=> ( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ F @ I3 ) )
= C )
= ( ! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ I3 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1020_SUP__eq__iff,axiom,
! [I3: set_nat_nat_nat2,C: set_nat,F: ( ( nat > nat ) > nat ) > set_nat] :
( ( I3 != bot_bo945813143650711160at_nat )
=> ( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ F @ I3 ) )
= C )
= ( ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ I3 )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1021_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_1022_SUP__constant,axiom,
! [C: $o,A2: set_nat] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y2: nat] : C
@ A2 ) )
= ( ( ( A2 = bot_bot_set_nat )
=> bot_bot_o )
& ( ( A2 != bot_bot_set_nat )
=> C ) ) ) ).
% SUP_constant
thf(fact_1023_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_1024_SUP__empty,axiom,
! [F: nat > $o] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_1025_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_1026_SUP__UNION,axiom,
! [F: nat > $o,G: nat > set_nat,A2: set_nat] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y2: nat] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1027__092_060open_062card_A_I_092_060chi_062_A_096_AS_A_096_Ax_A_096_A_123_O_Ok_125_J_A_060_Acard_A_IS_A_096_Ax_A_096_A_123_O_Ok_125_J_092_060close_062,axiom,
ord_less_nat @ ( finite_card_nat @ ( image_nat_nat_nat @ chi @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) ) ) @ ( finite_card_nat_nat @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) ) ).
% \<open>card (\<chi> ` S ` x ` {..k}) < card (S ` x ` {..k})\<close>
thf(fact_1028_card_Oempty,axiom,
( ( finite_card_nat_nat @ bot_bot_set_nat_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_1029_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_1030_lhj__def,axiom,
( hales_lhj
= ( ^ [R2: nat,T3: 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 @ T3 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ K3 @ N6 @ T3 @ R2 @ Chi2 ) ) ) ) ) ) ).
% lhj_def
thf(fact_1031_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1032_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1033__092_060open_062card_A_Ix_A_096_A_123_O_Ok_125_J_A_061_Acard_A_123_O_Ok_125_092_060close_062,axiom,
( ( finite_card_nat_nat @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) )
= ( finite_card_nat @ ( set_ord_atMost_nat @ k ) ) ) ).
% \<open>card (x ` {..k}) = card {..k}\<close>
thf(fact_1034_B,axiom,
( ( finite_card_nat_nat @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) )
= ( plus_plus_nat @ k @ one_one_nat ) ) ).
% B
thf(fact_1035__092_060open_062card_A_Ix_A_096_A_123_O_Ok_125_J_A_061_Acard_A_IS_A_096_Ax_A_096_A_123_O_Ok_125_J_092_060close_062,axiom,
( ( finite_card_nat_nat @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) )
= ( finite_card_nat_nat @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) ) ) ).
% \<open>card (x ` {..k}) = card (S ` x ` {..k})\<close>
thf(fact_1036_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1037_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1038_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1039_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_1040_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_1041_all__subset__image,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat_nat @ F @ A2 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ A2 )
=> ( P @ ( image_nat_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1042_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_1043_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_1044_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1045_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_1046_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_1047_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1048_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1049_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1050_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1051_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_1052_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1053_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1054_hj__imp__lhj__step,axiom,
! [T: nat,K: nat,R: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ( ! [R3: nat,K4: nat] :
( ( ord_less_eq_nat @ K4 @ K )
=> ( hales_lhj @ R3 @ T @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ R )
=> ( hales_lhj @ R @ T @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ).
% hj_imp_lhj_step
thf(fact_1055_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1056_is__subspace__def,axiom,
( hales_is_subspace
= ( ^ [S5: ( nat > nat ) > nat > nat,K3: nat,N3: nat,T3: 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 ) ) )
& ? [F5: nat > nat] :
( ( member_nat_nat @ F5
@ ( piE_nat_nat @ ( B6 @ K3 )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ T3 ) ) )
& ( member952132173341509300at_nat @ S5
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ K3 @ T3 )
@ ^ [I: nat > nat] : ( hales_cube @ N3 @ T3 ) ) )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ K3 @ T3 ) )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B6 @ K3 ) )
=> ( ( S5 @ X2 @ Y2 )
= ( F5 @ Y2 ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ K3 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B6 @ J3 ) )
=> ( ( S5 @ X2 @ Y2 )
= ( X2 @ J3 ) ) ) ) ) ) ) ) ) ) ).
% is_subspace_def
thf(fact_1057__092_060open_062_092_060not_062_Ainj__on_A_092_060chi_062_A_IS_A_096_Ax_A_096_A_123_O_Ok_125_J_092_060close_062,axiom,
~ ( inj_on_nat_nat_nat @ chi @ ( image_3205354838064109189at_nat @ s @ ( image_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ) ) ) ).
% \<open>\<not> inj_on \<chi> (S ` x ` {..k})\<close>
thf(fact_1058_layered__subspace__def,axiom,
( hales_4783935871306402712at_nat
= ( ^ [S5: ( nat > nat ) > nat > nat,K3: nat,N3: nat,T3: nat,R2: nat > nat,Chi2: ( nat > nat ) > nat > nat] :
( ( hales_is_subspace @ S5 @ K3 @ N3 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_atMost_nat @ K3 ) )
=> ? [C3: nat > nat] :
( ( ord_less_nat_nat @ C3 @ R2 )
& ! [Y2: nat > nat] :
( ( member_nat_nat @ Y2 @ ( hales_classes @ K3 @ T3 @ X2 ) )
=> ( ( Chi2 @ ( S5 @ Y2 ) )
= C3 ) ) ) )
& ( member952132173341509300at_nat @ Chi2
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ N3 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_or1140352010380016476at_nat @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_1059_layered__subspace__def,axiom,
( hales_4261547300027266985ce_nat
= ( ^ [S5: ( nat > nat ) > nat > nat,K3: nat,N3: nat,T3: nat,R2: nat,Chi2: ( nat > nat ) > nat] :
( ( hales_is_subspace @ S5 @ K3 @ N3 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_atMost_nat @ K3 ) )
=> ? [C3: nat] :
( ( ord_less_nat @ C3 @ R2 )
& ! [Y2: nat > nat] :
( ( member_nat_nat @ Y2 @ ( hales_classes @ K3 @ T3 @ X2 ) )
=> ( ( Chi2 @ ( S5 @ Y2 ) )
= C3 ) ) ) )
& ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N3 @ ( plus_plus_nat @ T3 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) ) ) ) ) ).
% layered_subspace_def
thf(fact_1060_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_1061_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_1062_dual__order_Orefl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_1063_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_1064_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_1065_order__refl,axiom,
! [X: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X @ X ) ).
% order_refl
thf(fact_1066_inj__on__restrict__eq,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( inj_on_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ A2 ) @ A2 )
= ( inj_on_nat_nat_nat @ F @ A2 ) ) ).
% inj_on_restrict_eq
thf(fact_1067_inj__on__restrict__eq,axiom,
! [F: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ ( restrict_nat_nat @ F @ A2 ) @ A2 )
= ( inj_on_nat_nat @ F @ A2 ) ) ).
% inj_on_restrict_eq
thf(fact_1068_inj__on__restrict__eq,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( inj_on_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ A2 ) @ A2 )
= ( inj_on_nat_nat_nat2 @ F @ A2 ) ) ).
% inj_on_restrict_eq
thf(fact_1069_inj__on__restrict__eq,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ ( restri4446420529079022766at_nat @ F @ A2 ) @ A2 )
= ( inj_on2461717442902640625at_nat @ F @ A2 ) ) ).
% inj_on_restrict_eq
thf(fact_1070_inj__on__image,axiom,
! [F: ( nat > nat ) > nat,A2: set_set_nat_nat] :
( ( inj_on_nat_nat_nat @ F @ ( comple5448282615319421384at_nat @ A2 ) )
=> ( inj_on1908564424730267886et_nat @ ( image_nat_nat_nat @ F ) @ A2 ) ) ).
% inj_on_image
thf(fact_1071_inj__on__image,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ ( comple5448282615319421384at_nat @ A2 ) )
=> ( inj_on290067182627368541at_nat @ ( image_3205354838064109189at_nat @ F ) @ A2 ) ) ).
% inj_on_image
thf(fact_1072_inj__on__image,axiom,
! [F: nat > set_nat,A2: set_set_nat] :
( ( inj_on_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( inj_on2776966659131765557et_nat @ ( image_nat_set_nat @ F ) @ A2 ) ) ).
% inj_on_image
thf(fact_1073_inj__on__image,axiom,
! [F: nat > nat > nat,A2: set_set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( inj_on2031908246727262830at_nat @ ( image_nat_nat_nat2 @ F ) @ A2 ) ) ).
% inj_on_image
thf(fact_1074_inj__on__image,axiom,
! [F: nat > nat,A2: set_set_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ A2 ) ) ).
% inj_on_image
thf(fact_1075_card__image,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( inj_on_nat_set_nat @ F @ A2 )
=> ( ( finite_card_set_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ).
% card_image
thf(fact_1076_card__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ).
% card_image
thf(fact_1077_card__image,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( inj_on_nat_nat_nat @ F @ A2 )
=> ( ( finite_card_nat @ ( image_nat_nat_nat @ F @ A2 ) )
= ( finite_card_nat_nat @ A2 ) ) ) ).
% card_image
thf(fact_1078_card__image,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( inj_on_nat_nat_nat2 @ F @ A2 )
=> ( ( finite_card_nat_nat @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( finite_card_nat @ A2 ) ) ) ).
% card_image
thf(fact_1079_card__image,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ A2 )
=> ( ( finite_card_nat_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) )
= ( finite_card_nat_nat @ A2 ) ) ) ).
% card_image
thf(fact_1080_inj__on__restrict__iff,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inj_on_nat_nat @ ( restrict_nat_nat @ F @ B2 ) @ A2 )
= ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% inj_on_restrict_iff
thf(fact_1081_inj__on__restrict__iff,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inj_on_nat_nat_nat2 @ ( restrict_nat_nat_nat2 @ F @ B2 ) @ A2 )
= ( inj_on_nat_nat_nat2 @ F @ A2 ) ) ) ).
% inj_on_restrict_iff
thf(fact_1082_inj__on__restrict__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( inj_on_nat_nat_nat @ ( restrict_nat_nat_nat @ F @ B2 ) @ A2 )
= ( inj_on_nat_nat_nat @ F @ A2 ) ) ) ).
% inj_on_restrict_iff
thf(fact_1083_inj__on__restrict__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( inj_on2461717442902640625at_nat @ ( restri4446420529079022766at_nat @ F @ B2 ) @ A2 )
= ( inj_on2461717442902640625at_nat @ F @ A2 ) ) ) ).
% inj_on_restrict_iff
thf(fact_1084_pigeonhole,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( ord_less_nat @ ( finite_card_set_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) )
=> ~ ( inj_on_nat_set_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_1085_pigeonhole,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) )
=> ~ ( inj_on_nat_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_1086_pigeonhole,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_nat_nat @ F @ A2 ) ) @ ( finite_card_nat_nat @ A2 ) )
=> ~ ( inj_on_nat_nat_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_1087_pigeonhole,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( ord_less_nat @ ( finite_card_nat_nat @ ( image_nat_nat_nat2 @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) )
=> ~ ( inj_on_nat_nat_nat2 @ F @ A2 ) ) ).
% pigeonhole
thf(fact_1088_pigeonhole,axiom,
! [F: ( nat > nat ) > nat > nat,A2: set_nat_nat] :
( ( ord_less_nat @ ( finite_card_nat_nat @ ( image_3205354838064109189at_nat @ F @ A2 ) ) @ ( finite_card_nat_nat @ A2 ) )
=> ~ ( inj_on2461717442902640625at_nat @ F @ A2 ) ) ).
% pigeonhole
thf(fact_1089_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_1090_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_1091_order__antisym__conv,axiom,
! [Y: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ( ( ord_le9059583361652607317at_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_1092_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_1093_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1094_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1095_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1096_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1097_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1098_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1099_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 )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1100_ord__le__eq__subst,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat,C: set_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1101_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 )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1102_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1103_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1104_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1105_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1106_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1107_ord__eq__le__subst,axiom,
! [A: set_nat_nat,F: set_nat > set_nat_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1108_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 )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1109_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat_nat > set_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1110_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 )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1111_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_1112_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1113_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1114_order__eq__refl,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( X = Y )
=> ( ord_le9059583361652607317at_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1115_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1116_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1117_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1118_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1119_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1120_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1121_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 )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1122_order__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat,C: set_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1123_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 )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1124_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1125_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1126_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 )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1127_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1128_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1129_order__subst1,axiom,
! [A: set_nat,F: set_nat_nat > set_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1130_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1131_order__subst1,axiom,
! [A: set_nat_nat,F: set_nat > set_nat_nat,B: set_nat,C: set_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1132_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 )
=> ( ! [X3: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1133_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1134_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z2: set_nat] : ( Y6 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1135_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat_nat,Z2: set_nat_nat] : ( Y6 = Z2 ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1136_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_1137_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_1138_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_1139_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_1140_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_1141_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_1142_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_1143_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_1144_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_1145_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1146_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_nat,Z2: set_nat] : ( Y6 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1147_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_nat_nat,Z2: set_nat_nat] : ( Y6 = Z2 ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ A3 )
& ( ord_le9059583361652607317at_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1148_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_1149_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_1150_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_1151_order__trans,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ Z )
=> ( ord_le9059583361652607317at_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_1152_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_1153_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_1154_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_1155_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1156_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1157_order__antisym,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1158_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_1159_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1160_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_1161_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_1162_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1163_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_1164_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1165_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z2: set_nat] : ( Y6 = Z2 ) )
= ( ^ [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
& ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1166_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat_nat,Z2: set_nat_nat] : ( Y6 = Z2 ) )
= ( ^ [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
& ( ord_le9059583361652607317at_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1167_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1168_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_1169_inj__on__UNION__chain,axiom,
! [I3: set_nat,A2: nat > set_nat,F: nat > nat] :
( ! [I4: nat,J4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( member_nat @ J4 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A2 @ I4 ) @ ( A2 @ J4 ) )
| ( ord_less_eq_set_nat @ ( A2 @ J4 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( inj_on_nat_nat @ F @ ( A2 @ I4 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1170_inj__on__UNION__chain,axiom,
! [I3: set_nat,A2: nat > set_nat_nat,F: ( nat > nat ) > nat] :
( ! [I4: nat,J4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( member_nat @ J4 @ I3 )
=> ( ( ord_le9059583361652607317at_nat @ ( A2 @ I4 ) @ ( A2 @ J4 ) )
| ( ord_le9059583361652607317at_nat @ ( A2 @ J4 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( inj_on_nat_nat_nat @ F @ ( A2 @ I4 ) ) )
=> ( inj_on_nat_nat_nat @ F @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1171_inj__on__UNION__chain,axiom,
! [I3: set_nat,A2: nat > set_nat,F: nat > nat > nat] :
( ! [I4: nat,J4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( member_nat @ J4 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A2 @ I4 ) @ ( A2 @ J4 ) )
| ( ord_less_eq_set_nat @ ( A2 @ J4 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( inj_on_nat_nat_nat2 @ F @ ( A2 @ I4 ) ) )
=> ( inj_on_nat_nat_nat2 @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1172_inj__on__UNION__chain,axiom,
! [I3: set_nat_nat,A2: ( nat > nat ) > set_nat,F: nat > nat] :
( ! [I4: nat > nat,J4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( member_nat_nat @ J4 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A2 @ I4 ) @ ( A2 @ J4 ) )
| ( ord_less_eq_set_nat @ ( A2 @ J4 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( inj_on_nat_nat @ F @ ( A2 @ I4 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1173_inj__on__UNION__chain,axiom,
! [I3: set_nat,A2: nat > set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ! [I4: nat,J4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( member_nat @ J4 @ I3 )
=> ( ( ord_le9059583361652607317at_nat @ ( A2 @ I4 ) @ ( A2 @ J4 ) )
| ( ord_le9059583361652607317at_nat @ ( A2 @ J4 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( inj_on2461717442902640625at_nat @ F @ ( A2 @ I4 ) ) )
=> ( inj_on2461717442902640625at_nat @ F @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1174_inj__on__UNION__chain,axiom,
! [I3: set_nat_nat,A2: ( nat > nat ) > set_nat_nat,F: ( nat > nat ) > nat] :
( ! [I4: nat > nat,J4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( member_nat_nat @ J4 @ I3 )
=> ( ( ord_le9059583361652607317at_nat @ ( A2 @ I4 ) @ ( A2 @ J4 ) )
| ( ord_le9059583361652607317at_nat @ ( A2 @ J4 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( inj_on_nat_nat_nat @ F @ ( A2 @ I4 ) ) )
=> ( inj_on_nat_nat_nat @ F @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1175_inj__on__UNION__chain,axiom,
! [I3: set_nat_nat,A2: ( nat > nat ) > set_nat,F: nat > nat > nat] :
( ! [I4: nat > nat,J4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( member_nat_nat @ J4 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A2 @ I4 ) @ ( A2 @ J4 ) )
| ( ord_less_eq_set_nat @ ( A2 @ J4 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( inj_on_nat_nat_nat2 @ F @ ( A2 @ I4 ) ) )
=> ( inj_on_nat_nat_nat2 @ F @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1176_inj__on__UNION__chain,axiom,
! [I3: set_nat_nat_nat,A2: ( nat > nat > nat ) > set_nat,F: nat > nat] :
( ! [I4: nat > nat > nat,J4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ I3 )
=> ( ( member_nat_nat_nat2 @ J4 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A2 @ I4 ) @ ( A2 @ J4 ) )
| ( ord_less_eq_set_nat @ ( A2 @ J4 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I4 @ I3 )
=> ( inj_on_nat_nat @ F @ ( A2 @ I4 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_6782468043973903547et_nat @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1177_inj__on__UNION__chain,axiom,
! [I3: set_nat_nat_nat2,A2: ( ( nat > nat ) > nat ) > set_nat,F: nat > nat] :
( ! [I4: ( nat > nat ) > nat,J4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ I3 )
=> ( ( member_nat_nat_nat @ J4 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A2 @ I4 ) @ ( A2 @ J4 ) )
| ( ord_less_eq_set_nat @ ( A2 @ J4 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I4 @ I3 )
=> ( inj_on_nat_nat @ F @ ( A2 @ I4 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_7565631143590340539et_nat @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1178_inj__on__UNION__chain,axiom,
! [I3: set_nat_nat,A2: ( nat > nat ) > set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ! [I4: nat > nat,J4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( ( member_nat_nat @ J4 @ I3 )
=> ( ( ord_le9059583361652607317at_nat @ ( A2 @ I4 ) @ ( A2 @ J4 ) )
| ( ord_le9059583361652607317at_nat @ ( A2 @ J4 ) @ ( A2 @ I4 ) ) ) ) )
=> ( ! [I4: nat > nat] :
( ( member_nat_nat @ I4 @ I3 )
=> ( inj_on2461717442902640625at_nat @ F @ ( A2 @ I4 ) ) )
=> ( inj_on2461717442902640625at_nat @ F @ ( comple5448282615319421384at_nat @ ( image_6905811865970898491at_nat @ A2 @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_1179_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_1180_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1181_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_1182_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_1183_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_1184_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_1185_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_1186_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_1187_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_1188_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_1189_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1190_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_1191_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_1192_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1193_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_1194_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_1195_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_1196_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_1197_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_1198_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_1199_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_1200_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_1201_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1202_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1203_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_1204_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_1205_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_1206_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_1207_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_1208_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_1209_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1210_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_1211_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_1212_dim0__subspace__ex,axiom,
! [T: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ? [S4: ( nat > nat ) > nat > nat] : ( hales_is_subspace @ S4 @ zero_zero_nat @ N @ T ) ) ).
% dim0_subspace_ex
thf(fact_1213_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1214_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_1215_leD,axiom,
! [Y: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ~ ( ord_less_set_nat_nat @ X @ Y ) ) ).
% leD
thf(fact_1216_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1217_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1218_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1219_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_1220_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1221_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1222_antisym__conv1,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ~ ( ord_less_set_nat_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1223_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1224_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1225_antisym__conv2,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1226_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1227_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1228_less__le__not__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
& ~ ( ord_le9059583361652607317at_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1229_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1230_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1231_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1232_order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1233_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1234_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1235_order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1236_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_1237_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1238_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_1239_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_1240_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1241_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_1242_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1243_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1244_order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ~ ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1245_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1246_dual__order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B3: set_nat_nat,A3: set_nat_nat] :
( ( ord_less_set_nat_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1247_subspace__elems__embed,axiom,
! [S3: ( nat > nat ) > nat > nat,K: nat,N: nat,T: nat] :
( ( hales_is_subspace @ S3 @ K @ N @ T )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ S3 @ ( hales_cube @ K @ T ) ) @ ( hales_cube @ N @ T ) ) ) ).
% subspace_elems_embed
thf(fact_1248_dim1__subspace__elims_I2_J,axiom,
! [B2: nat > set_nat,N: nat,F: nat > nat,T: nat,S3: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B2 @ one_one_nat )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S3
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ T )
@ ^ [I: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ one_one_nat ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J4: nat] :
( ( ord_less_nat @ J4 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ J4 ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( X3 @ J4 ) ) ) ) ) )
=> ( ( inf_inf_set_nat @ ( B2 @ zero_zero_nat ) @ ( B2 @ one_one_nat ) )
= bot_bot_set_nat ) ) ) ) ) ) ) ).
% dim1_subspace_elims(2)
thf(fact_1249_dim1__subspace__elims_I1_J,axiom,
! [B2: nat > set_nat,N: nat,F: nat > nat,T: nat,S3: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B2 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B2 @ one_one_nat )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S3
@ ( piE_nat_nat_nat_nat3 @ ( hales_cube @ one_one_nat @ T )
@ ^ [I: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ one_one_nat ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J4: nat] :
( ( ord_less_nat @ J4 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B2 @ J4 ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( X3 @ J4 ) ) ) ) ) )
=> ( ( sup_sup_set_nat @ ( B2 @ zero_zero_nat ) @ ( B2 @ one_one_nat ) )
= ( set_ord_lessThan_nat @ N ) ) ) ) ) ) ) ) ).
% dim1_subspace_elims(1)
thf(fact_1250_hj__imp__lhj__base,axiom,
! [T: nat,R: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ! [R4: nat] : ( hales_hj @ R4 @ T )
=> ( hales_lhj @ R @ T @ one_one_nat ) ) ) ).
% hj_imp_lhj_base
thf(fact_1251__092_060open_062inj__on_Ax_A_123_O_Ok_125_092_060close_062,axiom,
inj_on_nat_nat_nat2 @ x @ ( set_ord_atMost_nat @ k ) ).
% \<open>inj_on x {..k}\<close>
thf(fact_1252_subspace__inj__on__cube,axiom,
! [S3: ( nat > nat ) > nat > nat,K: nat,N: nat,T: nat] :
( ( hales_is_subspace @ S3 @ K @ N @ T )
=> ( inj_on2461717442902640625at_nat @ S3 @ ( hales_cube @ K @ T ) ) ) ).
% subspace_inj_on_cube
thf(fact_1253_inj__on__diff__nat,axiom,
! [N4: set_nat,K: nat] :
( ! [N2: nat] :
( ( member_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ K @ N2 ) )
=> ( inj_on_nat_nat
@ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
@ N4 ) ) ).
% inj_on_diff_nat
thf(fact_1254_hj__imp__lhj,axiom,
! [T: nat,R: nat,K: nat] :
( ! [R4: nat] : ( hales_hj @ R4 @ T )
=> ( hales_lhj @ R @ T @ K ) ) ).
% hj_imp_lhj
thf(fact_1255_hj__def,axiom,
( hales_hj
= ( ^ [R2: nat,T3: 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 @ T3 )
@ ^ [I: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) )
=> ? [L2: nat > nat > nat,C3: nat] :
( ( ord_less_nat @ C3 @ R2 )
& ( hales_is_line @ L2 @ N6 @ T3 )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ L2 @ ( set_ord_lessThan_nat @ T3 ) ) )
=> ( ( Chi2 @ X2 )
= C3 ) ) ) ) ) ) ) ) ).
% hj_def
thf(fact_1256_line__points__in__cube__unfolded,axiom,
! [L3: nat > nat > nat,N: nat,T: nat,S: nat,J: nat] :
( ( hales_is_line @ L3 @ N @ T )
=> ( ( ord_less_nat @ S @ T )
=> ( ( ord_less_nat @ J @ N )
=> ( member_nat @ ( L3 @ S @ J ) @ ( set_ord_lessThan_nat @ T ) ) ) ) ) ).
% line_points_in_cube_unfolded
thf(fact_1257_line__points__in__cube,axiom,
! [L3: nat > nat > nat,N: nat,T: nat,S: nat] :
( ( hales_is_line @ L3 @ N @ T )
=> ( ( ord_less_nat @ S @ T )
=> ( member_nat_nat @ ( L3 @ S ) @ ( hales_cube @ N @ T ) ) ) ) ).
% line_points_in_cube
thf(fact_1258_is__line__def,axiom,
( hales_is_line
= ( ^ [L2: nat > nat > nat,N3: nat,T3: nat] :
( ( member_nat_nat_nat2 @ L2
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ T3 )
@ ^ [I: nat] : ( hales_cube @ N3 @ T3 ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ N3 )
=> ( ! [X2: nat] :
( ( ord_less_nat @ X2 @ T3 )
=> ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ T3 )
=> ( ( L2 @ X2 @ J3 )
= ( L2 @ Y2 @ J3 ) ) ) )
| ! [S2: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( ( L2 @ S2 @ J3 )
= S2 ) ) ) )
& ? [J3: nat] :
( ( ord_less_nat @ J3 @ N3 )
& ! [S2: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( ( L2 @ S2 @ J3 )
= S2 ) ) ) ) ) ) ).
% is_line_def
thf(fact_1259_line__is__dim1__subspace,axiom,
! [N: nat,T: nat,L3: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_is_line @ L3 @ N @ T )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y2: nat > nat] : ( L3 @ ( Y2 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace
thf(fact_1260_line__is__dim1__subspace__t__1,axiom,
! [N: nat,L3: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( hales_is_line @ L3 @ N @ one_one_nat )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y2: nat > nat] : ( L3 @ ( 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_1261_line__is__dim1__subspace__t__ge__1,axiom,
! [N: nat,T: nat,L3: nat > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ one_one_nat @ T )
=> ( ( hales_is_line @ L3 @ N @ T )
=> ( hales_is_subspace
@ ( restri4446420529079022766at_nat
@ ^ [Y2: nat > nat] : ( L3 @ ( Y2 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace_t_ge_1
thf(fact_1262_is__line__elim__t__1,axiom,
! [L3: nat > nat > nat,N: nat,T: nat] :
( ( hales_is_line @ L3 @ 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 )
=> ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ T )
=> ( ( L3 @ Xa @ X4 )
= ( L3 @ Y5 @ X4 ) ) ) ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ B_0 )
=> ! [S6: nat] :
( ( ord_less_nat @ S6 @ T )
=> ( ( L3 @ S6 @ X4 )
= S6 ) ) ) ) ) ) ).
% is_line_elim_t_1
thf(fact_1263_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_1264_dim1__subspace__is__line,axiom,
! [T: nat,S3: ( nat > nat ) > nat > nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ( hales_is_subspace @ S3 @ one_one_nat @ N @ T )
=> ( hales_is_line
@ ( restrict_nat_nat_nat2
@ ^ [S2: nat] :
( S3
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S2 ) ) ) )
@ ( set_ord_lessThan_nat @ T ) )
@ N
@ T ) ) ) ).
% dim1_subspace_is_line
thf(fact_1265_cube__props_I2_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S ) )
@ zero_zero_nat )
= S ) ) ).
% cube_props(2)
thf(fact_1266_cube__props_I4_J,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( member_nat_nat
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( P4 @ zero_zero_nat )
= S ) ) )
@ ( hales_cube @ one_one_nat @ T ) ) ) ).
% cube_props(4)
thf(fact_1267_some__inv__into__2,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ T @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= S ) ) )
= ( the_in5300466440149791684at_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [F5: nat > nat] : ( F5 @ zero_zero_nat )
@ S ) ) ) ).
% some_inv_into_2
% Helper facts (4)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
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 ) )
= ( ? [X8: nat > nat] : ( P @ X8 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( s
@ ( restrict_nat_nat
@ ^ [I: nat] : ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ v ) ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ I @ ( minus_minus_nat @ k @ u ) ) @ a @ t ) )
@ ( set_ord_lessThan_nat @ k ) )
@ j )
= ( f @ j ) ) ).
%------------------------------------------------------------------------------