TPTP Problem File: SLH0095^1.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_01904_088314__6121518_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1461 ( 640 unt; 185 typ; 0 def)
% Number of atoms : 3320 (1080 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 10434 ( 312 ~; 41 |; 184 &;8572 @)
% ( 0 <=>;1325 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 1852 (1852 >; 0 *; 0 +; 0 <<)
% Number of symbols : 179 ( 176 usr; 25 con; 0-6 aty)
% Number of variables : 3437 ( 281 ^;3079 !; 77 ?;3437 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:47:12.481
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_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_na7233567106578532785at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_na6626867396258451522at_nat: $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_nat: $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_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__Nat__Onat,type,
nat: $tType ).
% Explicit typings (176)
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_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > 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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
finite1794908990118856198at_nat: set_nat_nat_nat2 > nat ).
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_Fun_Obij__betw_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bij_be1059735840858801910at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat2 > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bij_betw_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bij_betw_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_nat > $o ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
fun_up831482295316861124at_nat: ( nat > set_nat_nat ) > nat > set_nat_nat > nat > set_nat_nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
fun_upd_nat_set_nat: ( nat > set_nat ) > nat > set_nat > nat > set_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
piE_na5223350113562215832at_nat: set_nat_nat > ( ( nat > nat ) > set_nat_nat_nat_nat ) > set_na7233567106578532785at_nat ).
thf(sy_c_FuncSet_OPiE_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
piE_na7569501297962130601at_nat: set_nat_nat > ( ( nat > nat ) > set_nat_nat_nat2 ) > set_na6626867396258451522at_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_nat: set_nat_nat > ( ( nat > nat ) > set_nat_nat ) > set_nat_nat_nat_nat ).
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_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_Orestrict_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
restri1704181820465610764at_nat: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_nat_nat > ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
restri6011711336257459485at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat > ( nat > nat ) > ( nat > nat ) > 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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
restri901343962050523125at_nat: ( nat > set_nat_nat ) > set_nat > nat > set_nat_nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
restrict_nat_set_nat: ( nat > set_nat ) > set_nat > nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
minus_9165053394918225162at_nat: set_na7233567106578532785at_nat > set_na7233567106578532785at_nat > set_na7233567106578532785at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
minus_5225787954611647771at_nat: set_na6626867396258451522at_nat > set_na6626867396258451522at_nat > set_na6626867396258451522at_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_nat > set_nat_nat_nat_nat > set_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_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_HOL_Oundefined_001t__Nat__Onat,type,
undefined_nat: nat ).
thf(sy_c_Hales__Jewett_Oclasses,type,
hales_classes: nat > nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ocube,type,
hales_cube: nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ohj,type,
hales_hj: nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__line,type,
hales_is_line: ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ois__subspace,type,
hales_is_subspace: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Ojoin_001t__Nat__Onat,type,
hales_join_nat: ( nat > nat ) > ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Hales__Jewett_Olayered__subspace_001t__Nat__Onat,type,
hales_4261547300027266985ce_nat: ( ( nat > nat ) > nat > nat ) > nat > nat > nat > nat > ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Hales__Jewett_Olhj,type,
hales_lhj: nat > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Oset__incr,type,
hales_set_incr: nat > set_nat > set_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
inf_in6008378084349164867at_nat: set_na7233567106578532785at_nat > set_na7233567106578532785at_nat > set_na7233567106578532785at_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
inf_in6213014276851238612at_nat: set_na6626867396258451522at_nat > set_na6626867396258451522at_nat > set_na6626867396258451522at_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inf_in2949407623404935909at_nat: set_nat_nat_nat_nat > set_nat_nat_nat_nat > set_nat_nat_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
inf_in7997761893158376566at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > set_nat_nat_nat2 ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inf_in5274420515160781174at_nat: set_nat_nat_nat > set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inf_inf_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
sup_su8594648213498475741at_nat: set_na7233567106578532785at_nat > set_na7233567106578532785at_nat > set_na7233567106578532785at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
sup_su481250237928500590at_nat: set_na6626867396258451522at_nat > set_na6626867396258451522at_nat > set_na6626867396258451522at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
sup_su3836648520750444671at_nat: set_nat_nat_nat_nat > set_nat_nat_nat_nat > set_nat_nat_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
sup_su6057362541959223568at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > set_nat_nat_nat2 ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
sup_su3334021163961628176at_nat: set_nat_nat_nat > set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_sup_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bot_nat_nat_o: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
bot_bo2676777031303994949at_nat: set_na7233567106578532785at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
bot_bo2074992577060541142at_nat: set_na6626867396258451522at_nat ).
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_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
bot_bo945813143650711160at_nat: set_nat_nat_nat2 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo7445843802507891576at_nat: set_nat_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_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_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le338063099783794255at_nat: ( ( nat > nat ) > ( nat > 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_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le7877100967975825120at_nat: ( ( nat > 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_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_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_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
ord_le7586516898478368261at_nat: set_na7233567106578532785at_nat > set_na7233567106578532785at_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
ord_le2785809691299232406at_nat: set_na6626867396258451522at_nat > set_na6626867396258451522at_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_nat > set_nat_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_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_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
ord_le8099187209609443857at_nat: set_na7233567106578532785at_nat > set_na7233567106578532785at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
ord_le973658574027395234at_nat: set_na6626867396258451522at_nat > set_na6626867396258451522at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le5260717879541182899at_nat: set_nat_nat_nat_nat > set_nat_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le5934964663421696068at_nat: set_nat_nat_nat2 > set_nat_nat_nat2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3211623285424100676at_nat: set_nat_nat_nat > set_nat_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
collec6535634078845029456at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o ) > set_na7233567106578532785at_nat ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
collec2410089373097230945at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ) > set_na6626867396258451522at_nat ).
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_nat ).
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_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_001t__Nat__Onat,type,
image_8194121248528334964at_nat: ( ( ( nat > nat ) > nat > nat ) > nat ) > set_nat_nat_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7482708007212282706at_nat: ( ( ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat ) > set_nat_nat_nat2 > set_nat_nat_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_8393830757900314979at_nat: ( ( ( nat > nat ) > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat2 > set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_279826485474788963at_nat: ( ( ( nat > nat ) > nat ) > nat > nat > nat ) > set_nat_nat_nat2 > set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1262493855416953332at_nat: ( ( ( nat > nat ) > nat ) > nat > nat ) > set_nat_nat_nat2 > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_7809927846809980933at_nat: ( ( ( nat > nat ) > nat ) > nat ) > set_nat_nat_nat2 > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_5348097324405788754at_nat: ( ( nat > nat > nat ) > ( nat > nat ) > nat > nat ) > set_nat_nat_nat > set_nat_nat_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_786723269765334627at_nat: ( ( nat > nat > nat ) > ( nat > nat ) > nat ) > set_nat_nat_nat > set_nat_nat_nat2 ).
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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_1896091034194584419at_nat: ( ( nat > nat > nat ) > nat > nat > nat ) > set_nat_nat_nat > set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1545173636400105204at_nat: ( ( nat > nat > nat ) > nat > nat ) > set_nat_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_913610194320715013at_nat: ( ( nat > nat > nat ) > nat ) > set_nat_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_3205354838064109189at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert_nat_nat: ( nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_or2490836252891414040at_nat: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_na7233567106578532785at_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_or6177432841829679145at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_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_nat ).
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_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_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8881365325514865170at_nat: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_na7233567106578532785at_nat > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member4402528950554000163at_nat: ( ( nat > nat ) > ( nat > nat ) > nat ) > set_na6626867396258451522at_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_nat > $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_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_001t__Nat__Onat,type,
member_nat: nat > set_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_BL____,type,
bl: nat > set_nat ).
thf(sy_v_BS____,type,
bs: nat > set_nat ).
thf(sy_v_BT____,type,
bt: nat > set_nat ).
thf(sy_v_Bstat____,type,
bstat: set_nat ).
thf(sy_v_Bvar____,type,
bvar: nat > set_nat ).
thf(sy_v_L____,type,
l: ( nat > nat ) > nat > nat ).
thf(sy_v_L__line____,type,
l_line: nat > nat > nat ).
thf(sy_v_M_H____,type,
m: nat ).
thf(sy_v_S____,type,
s: ( nat > nat ) > nat > nat ).
thf(sy_v_T_H____,type,
t: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_v_T____,type,
t2: ( nat > nat ) > nat > nat ).
thf(sy_v_T__class____,type,
t_class: nat > set_nat_nat ).
thf(sy_v_Tset____,type,
tset: set_nat_nat ).
thf(sy_v__092_060chi_062L____,type,
chi_L: ( nat > nat ) > ( nat > nat ) > nat ).
thf(sy_v__092_060chi_062L__s____,type,
chi_L_s: ( nat > nat ) > nat ).
thf(sy_v__092_060chi_062S____,type,
chi_S: ( nat > nat ) > nat ).
thf(sy_v__092_060chi_062____,type,
chi: ( nat > nat ) > nat ).
thf(sy_v__092_060phi_062____,type,
phi: ( ( nat > nat ) > nat ) > nat ).
thf(sy_v_d____,type,
d: nat ).
thf(sy_v_fL____,type,
fL: nat > nat ).
thf(sy_v_fS____,type,
fS: nat > nat ).
thf(sy_v_fT____,type,
fT: nat > nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_m____,type,
m2: nat ).
thf(sy_v_n_H____,type,
n: nat ).
thf(sy_v_n____,type,
n2: nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_s____,type,
s2: nat ).
thf(sy_v_t,type,
t3: nat ).
% Relevant facts (1269)
thf(fact_0__092_060open_062layered__subspace_AT_A_Ik_A_L_A1_J_A_In_A_L_Am_J_At_Ar_A_092_060chi_062_092_060close_062,axiom,
hales_4261547300027266985ce_nat @ t2 @ ( plus_plus_nat @ k @ one_one_nat ) @ ( plus_plus_nat @ n2 @ m2 ) @ t3 @ r @ chi ).
% \<open>layered_subspace T (k + 1) (n + m) t r \<chi>\<close>
thf(fact_1_assms_I2_J,axiom,
ord_less_eq_nat @ one_one_nat @ k ).
% assms(2)
thf(fact_2__092_060open_062n_A_L_Am_A_061_AM_H_092_060close_062,axiom,
( ( plus_plus_nat @ n2 @ m2 )
= m ) ).
% \<open>n + m = M'\<close>
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062S_O_Alayered__subspace_AS_Ak_Am_At_Ar_A_092_060chi_062S_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [S: ( nat > nat ) > nat > nat] :
~ ( hales_4261547300027266985ce_nat @ S @ k @ m2 @ t3 @ r @ chi_S ) ).
% \<open>\<And>thesis. (\<And>S. layered_subspace S k m t r \<chi>S \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_S__prop,axiom,
hales_4261547300027266985ce_nat @ s @ k @ m2 @ t3 @ r @ chi_S ).
% S_prop
thf(fact_5_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_6_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_7_n__def,axiom,
( n2
= ( plus_plus_nat @ n @ d ) ) ).
% n_def
thf(fact_8_subspace__T,axiom,
hales_is_subspace @ t2 @ ( plus_plus_nat @ k @ one_one_nat ) @ ( plus_plus_nat @ n2 @ m2 ) @ ( plus_plus_nat @ t3 @ one_one_nat ) ).
% subspace_T
thf(fact_9_assms_I4_J,axiom,
! [K: nat,R: nat] :
( ( ord_less_eq_nat @ K @ k )
=> ( hales_lhj @ R @ t3 @ K ) ) ).
% assms(4)
thf(fact_10_s__def,axiom,
( s2
= ( power_power_nat @ r @ ( power_power_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) @ m2 ) ) ) ).
% s_def
thf(fact_11_M_H__prop,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ n @ m2 ) @ m ).
% M'_prop
thf(fact_12__092_060open_062_092_060And_062i_O_Ai_A_092_060le_062_Ak_A_L_A1_A_092_060Longrightarrow_062_A_092_060exists_062c_060r_O_A_092_060forall_062x_092_060in_062classes_A_Ik_A_L_A1_J_At_Ai_O_A_092_060chi_062_A_IT_Ax_J_A_061_Ac_092_060close_062,axiom,
! [I: nat] :
( ( ord_less_eq_nat @ I @ ( plus_plus_nat @ k @ one_one_nat ) )
=> ? [C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_classes @ ( plus_plus_nat @ k @ one_one_nat ) @ t3 @ I ) )
=> ( ( chi @ ( t2 @ X ) )
= C2 ) ) ) ) ).
% \<open>\<And>i. i \<le> k + 1 \<Longrightarrow> \<exists>c<r. \<forall>x\<in>classes (k + 1) t i. \<chi> (T x) = c\<close>
thf(fact_13__092_060open_062n_H_A_092_060le_062_An_092_060close_062,axiom,
ord_less_eq_nat @ n @ n2 ).
% \<open>n' \<le> n\<close>
thf(fact_14_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ t3 ).
% assms(1)
thf(fact_15_assms_I5_J,axiom,
ord_less_nat @ zero_zero_nat @ r ).
% assms(5)
thf(fact_16_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_17_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_18_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_19_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_20_d__def,axiom,
( d
= ( minus_minus_nat @ m @ ( plus_plus_nat @ n @ m2 ) ) ) ).
% d_def
thf(fact_21_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_22_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_23_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_24_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_25_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_26_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_27_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_28_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_29_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_30_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_31_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_32_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_33_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_34_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_35_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_36_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_37_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_38_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_39_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_40_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_41_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_42_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_43_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_44_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_45_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_46_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_47_mem__Collect__eq,axiom,
! [A: ( nat > nat ) > ( nat > nat ) > nat,P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ( member4402528950554000163at_nat @ A @ ( collec2410089373097230945at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A: ( nat > nat ) > ( nat > nat ) > nat > nat,P: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o] :
( ( member8881365325514865170at_nat @ A @ ( collec6535634078845029456at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_51_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_52_Collect__mem__eq,axiom,
! [A3: set_nat_nat_nat] :
( ( collect_nat_nat_nat2
@ ^ [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A3: set_nat_nat_nat2] :
( ( collect_nat_nat_nat
@ ^ [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A3: set_nat_nat_nat_nat] :
( ( collec3567154360959927026at_nat
@ ^ [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A3: set_na6626867396258451522at_nat] :
( ( collec2410089373097230945at_nat
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A3: set_na7233567106578532785at_nat] :
( ( collec6535634078845029456at_nat
@ ^ [X2: ( nat > nat ) > ( nat > nat ) > nat > nat] : ( member8881365325514865170at_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A3: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A3: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_60_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_61_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X3: set_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_62_Collect__cong,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat_nat @ P )
= ( collect_nat_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_63_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_64_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_65_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_66_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_67_group__cancel_Oadd2,axiom,
! [B3: nat,K2: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_68_group__cancel_Oadd1,axiom,
! [A3: nat,K2: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_69_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_70_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_71_one__reorient,axiom,
! [X4: nat] :
( ( one_one_nat = X4 )
= ( X4 = one_one_nat ) ) ).
% one_reorient
thf(fact_72__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062L_O_Alayered__subspace_AL_A1_An_At_As_A_092_060chi_062L__s_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [L2: ( nat > nat ) > nat > nat] :
~ ( hales_4261547300027266985ce_nat @ L2 @ one_one_nat @ n2 @ t3 @ s2 @ chi_L_s ) ).
% \<open>\<And>thesis. (\<And>L. layered_subspace L 1 n t s \<chi>L_s \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_73_power__increasing__iff,axiom,
! [B: nat,X4: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X4 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_74_power__strict__increasing__iff,axiom,
! [B: nat,X4: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X4 @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_75__092_060open_062_092_060And_062y_Ax_Ai_O_A_092_060lbrakk_062i_A_092_060le_062_Ak_059_Ax_A_092_060in_062_AT_A_096_Aclasses_A_Ik_A_L_A1_J_At_Ai_059_Ay_A_092_060in_062_AT_A_096_Aclasses_A_Ik_A_L_A1_J_At_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_A_092_060chi_062_Ax_A_061_A_092_060chi_062_Ay_A_092_060and_062_A_092_060chi_062_Ax_A_060_Ar_092_060close_062,axiom,
! [I: nat,X4: nat > nat,Y: nat > nat] :
( ( ord_less_eq_nat @ I @ k )
=> ( ( member_nat_nat @ X4 @ ( image_3205354838064109189at_nat @ t2 @ ( hales_classes @ ( plus_plus_nat @ k @ one_one_nat ) @ t3 @ I ) ) )
=> ( ( member_nat_nat @ Y @ ( image_3205354838064109189at_nat @ t2 @ ( hales_classes @ ( plus_plus_nat @ k @ one_one_nat ) @ t3 @ I ) ) )
=> ( ( ( chi @ X4 )
= ( chi @ Y ) )
& ( ord_less_nat @ ( chi @ X4 ) @ r ) ) ) ) ) ).
% \<open>\<And>y x i. \<lbrakk>i \<le> k; x \<in> T ` classes (k + 1) t i; y \<in> T ` classes (k + 1) t i\<rbrakk> \<Longrightarrow> \<chi> x = \<chi> y \<and> \<chi> x < r\<close>
thf(fact_76__092_060open_062_092_060And_062i_O_Ai_A_092_060le_062_Ak_A_L_A1_A_092_060Longrightarrow_062_A_092_060exists_062c_060r_O_A_092_060forall_062x_092_060in_062T_A_096_Aclasses_A_Ik_A_L_A1_J_At_Ai_O_A_092_060chi_062_Ax_A_061_Ac_092_060close_062,axiom,
! [I: nat] :
( ( ord_less_eq_nat @ I @ ( plus_plus_nat @ k @ one_one_nat ) )
=> ? [C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_3205354838064109189at_nat @ t2 @ ( hales_classes @ ( plus_plus_nat @ k @ one_one_nat ) @ t3 @ I ) ) )
=> ( ( chi @ X )
= C2 ) ) ) ) ).
% \<open>\<And>i. i \<le> k + 1 \<Longrightarrow> \<exists>c<r. \<forall>x\<in>T ` classes (k + 1) t i. \<chi> x = c\<close>
thf(fact_77__092_060open_062_092_060And_062i_O_Ai_A_092_060le_062_Ak_A_092_060Longrightarrow_062_A_092_060exists_062c_060r_O_A_092_060forall_062x_092_060in_062T_A_096_Aclasses_A_Ik_A_L_A1_J_At_Ai_O_A_092_060chi_062_Ax_A_061_Ac_092_060close_062,axiom,
! [I: nat] :
( ( ord_less_eq_nat @ I @ k )
=> ? [C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_3205354838064109189at_nat @ t2 @ ( hales_classes @ ( plus_plus_nat @ k @ one_one_nat ) @ t3 @ I ) ) )
=> ( ( chi @ X )
= C2 ) ) ) ) ).
% \<open>\<And>i. i \<le> k \<Longrightarrow> \<exists>c<r. \<forall>x\<in>T ` classes (k + 1) t i. \<chi> x = c\<close>
thf(fact_78_L__prop,axiom,
hales_4261547300027266985ce_nat @ l @ one_one_nat @ n2 @ t3 @ s2 @ chi_L_s ).
% L_prop
thf(fact_79__092_060open_062_092_060exists_062c_060r_O_A_092_060forall_062x_092_060in_062T_A_096_Aclasses_A_Ik_A_L_A1_J_At_A_Ik_A_L_A1_J_O_A_092_060chi_062_Ax_A_061_Ac_092_060close_062,axiom,
? [C2: nat] :
( ( ord_less_nat @ C2 @ r )
& ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_3205354838064109189at_nat @ t2 @ ( hales_classes @ ( plus_plus_nat @ k @ one_one_nat ) @ t3 @ ( plus_plus_nat @ k @ one_one_nat ) ) ) )
=> ( ( chi @ X )
= C2 ) ) ) ).
% \<open>\<exists>c<r. \<forall>x\<in>T ` classes (k + 1) t (k + 1). \<chi> x = c\<close>
thf(fact_80_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_81_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_82_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_83_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_84__092_060open_0620_A_060_As_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ s2 ).
% \<open>0 < s\<close>
thf(fact_85_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_86_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_87_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_88_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_89_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_90_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_91_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_92_add__eq__0__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_93_zero__eq__add__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X4 @ Y ) )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_94_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_95_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_96_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_97_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_98_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_99_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_100_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_101_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_102_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_103_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_104_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_105_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_106_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_107_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_108_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_109_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_110_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_111_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_112_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_113_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_114_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_115_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_116_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_117_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_118_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_119_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_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_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_122_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_123_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_124_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_125_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_126_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_127_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_128_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_129_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_130_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_131_nat__zero__less__power__iff,axiom,
! [X4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X4 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X4 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_132_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_133_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_134_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_135_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_136_classprop,axiom,
! [J: nat] :
( ( ord_less_eq_nat @ J @ k )
=> ( ( t_class @ J )
= ( image_3205354838064109189at_nat @ t2 @ ( hales_classes @ ( plus_plus_nat @ k @ one_one_nat ) @ t3 @ J ) ) ) ) ).
% classprop
thf(fact_137_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_138_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_139_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_140_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_141_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_142_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_143_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_144_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_145_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_146_Nat_Odiff__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_147_diff__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_148_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_149_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_150_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_151_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_152_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_153_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_154_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_155_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_156_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_157_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_158_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_159_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_160_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_161_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_162_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_163_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_164_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_165_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_166_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_167_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_168_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_169_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_170_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_171_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_172_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_173_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_174_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_175_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_176_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_177_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_178_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_179_zero__le,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_le
thf(fact_180_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_181_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_182_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_183_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_184_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_185_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_186_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_187_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_188_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_189_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_190_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_191_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_192_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_193_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_194_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_195_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_196_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_197_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_198_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_199_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_200_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K3 )
=> ~ ( P @ I2 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_201_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_202_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_214_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_215_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_216_add__nonpos__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_217_add__nonneg__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_218_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_219_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_220_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_221_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_222_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_223_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_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_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_226_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_227_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_228_dim0__subspace__ex,axiom,
! [T: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ? [S: ( nat > nat ) > nat > nat] : ( hales_is_subspace @ S @ zero_zero_nat @ N @ T ) ) ).
% dim0_subspace_ex
thf(fact_229_power__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_230_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_231_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_232_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_233_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_234_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_235_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_236_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_237_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_238_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_239_linorder__neqE__nat,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
=> ( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neqE_nat
thf(fact_240_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_241_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_242_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_243_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_244_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_245_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_246_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_247_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_248_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_249_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_250_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_251_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_252_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_253_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_254_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_255_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_256_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_257_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_258_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_259_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_260_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_261_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_262_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_263_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_264_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
| ( M3 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_265_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_266_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
& ( M3 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_267_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_268_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_269_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_270_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_271_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_272_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_273_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_274_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_275_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_276_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_277_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
? [K4: nat] :
( N4
= ( plus_plus_nat @ M3 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_278_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_279_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: 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 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_280_power__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N3 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_increasing
thf(fact_281_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( ( ord_less_nat @ N @ N3 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% power_strict_increasing
thf(fact_282_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_283_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_284_image__add__0,axiom,
! [S3: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S3 )
= S3 ) ).
% image_add_0
thf(fact_285_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_286_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_287_hj__imp__lhj__base,axiom,
! [T: nat,R: nat] :
( ( ord_less_nat @ zero_zero_nat @ T )
=> ( ! [R2: nat] : ( hales_hj @ R2 @ T )
=> ( hales_lhj @ R @ T @ one_one_nat ) ) ) ).
% hj_imp_lhj_base
thf(fact_288_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_289_image__eqI,axiom,
! [B: nat,F: nat > nat,X4: nat,A3: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A3 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_290_image__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X4: nat,A3: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A3 )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_291_image__eqI,axiom,
! [B: nat > nat,F: nat > nat > nat,X4: nat,A3: set_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A3 )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_292_image__eqI,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,X4: nat > nat,A3: set_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat @ X4 @ A3 )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_293_image__eqI,axiom,
! [B: nat > nat > nat,F: ( nat > nat > nat ) > nat > nat > nat,X4: nat > nat > nat,A3: set_nat_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat_nat2 @ X4 @ A3 )
=> ( member_nat_nat_nat2 @ B @ ( image_1896091034194584419at_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_294_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: ( nat > nat > nat ) > ( nat > nat ) > nat,X4: nat > nat > nat,A3: set_nat_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat_nat2 @ X4 @ A3 )
=> ( member_nat_nat_nat @ B @ ( image_786723269765334627at_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_295_image__eqI,axiom,
! [B: nat > nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat > nat,X4: ( nat > nat ) > nat,A3: set_nat_nat_nat2] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat_nat @ X4 @ A3 )
=> ( member_nat_nat_nat2 @ B @ ( image_279826485474788963at_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_296_image__eqI,axiom,
! [B: ( nat > nat ) > nat,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat,X4: ( nat > nat ) > nat,A3: set_nat_nat_nat2] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat_nat @ X4 @ A3 )
=> ( member_nat_nat_nat @ B @ ( image_8393830757900314979at_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_297_image__eqI,axiom,
! [B: ( nat > nat ) > nat > nat,F: ( nat > nat > nat ) > ( nat > nat ) > nat > nat,X4: nat > nat > nat,A3: set_nat_nat_nat] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat_nat2 @ X4 @ A3 )
=> ( member952132173341509300at_nat @ B @ ( image_5348097324405788754at_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_298_image__eqI,axiom,
! [B: ( nat > nat ) > nat > nat,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat,X4: ( nat > nat ) > nat,A3: set_nat_nat_nat2] :
( ( B
= ( F @ X4 ) )
=> ( ( member_nat_nat_nat @ X4 @ A3 )
=> ( member952132173341509300at_nat @ B @ ( image_7482708007212282706at_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_299_image__diff__subset,axiom,
! [F: nat > set_nat,A3: set_nat,B3: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A3 ) @ ( image_nat_set_nat @ F @ B3 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_300_image__diff__subset,axiom,
! [F: ( nat > nat ) > nat > nat,A3: set_nat_nat,B3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_3205354838064109189at_nat @ F @ A3 ) @ ( image_3205354838064109189at_nat @ F @ B3 ) ) @ ( image_3205354838064109189at_nat @ F @ ( minus_8121590178497047118at_nat @ A3 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_301_image__diff__subset,axiom,
! [F: nat > nat > nat,A3: set_nat,B3: set_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_nat_nat_nat2 @ F @ A3 ) @ ( image_nat_nat_nat2 @ F @ B3 ) ) @ ( image_nat_nat_nat2 @ F @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_302_image__diff__subset,axiom,
! [F: nat > nat,A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B3 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_303_imageI,axiom,
! [X4: nat,A3: set_nat,F: nat > nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ ( F @ X4 ) @ ( image_nat_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_304_imageI,axiom,
! [X4: nat,A3: set_nat,F: nat > set_nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_set_nat @ ( F @ X4 ) @ ( image_nat_set_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_305_imageI,axiom,
! [X4: nat,A3: set_nat,F: nat > nat > nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat_nat @ ( F @ X4 ) @ ( image_nat_nat_nat2 @ F @ A3 ) ) ) ).
% imageI
thf(fact_306_imageI,axiom,
! [X4: nat > nat,A3: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X4 @ A3 )
=> ( member_nat_nat @ ( F @ X4 ) @ ( image_3205354838064109189at_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_307_imageI,axiom,
! [X4: nat > nat > nat,A3: set_nat_nat_nat,F: ( nat > nat > nat ) > nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A3 )
=> ( member_nat_nat_nat2 @ ( F @ X4 ) @ ( image_1896091034194584419at_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_308_imageI,axiom,
! [X4: nat > nat > nat,A3: set_nat_nat_nat,F: ( nat > nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X4 @ A3 )
=> ( member_nat_nat_nat @ ( F @ X4 ) @ ( image_786723269765334627at_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_309_imageI,axiom,
! [X4: ( nat > nat ) > nat,A3: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat > nat > nat] :
( ( member_nat_nat_nat @ X4 @ A3 )
=> ( member_nat_nat_nat2 @ ( F @ X4 ) @ ( image_279826485474788963at_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_310_imageI,axiom,
! [X4: ( nat > nat ) > nat,A3: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A3 )
=> ( member_nat_nat_nat @ ( F @ X4 ) @ ( image_8393830757900314979at_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_311_imageI,axiom,
! [X4: nat > nat > nat,A3: set_nat_nat_nat,F: ( nat > nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A3 )
=> ( member952132173341509300at_nat @ ( F @ X4 ) @ ( image_5348097324405788754at_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_312_imageI,axiom,
! [X4: ( nat > nat ) > nat,A3: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat] :
( ( member_nat_nat_nat @ X4 @ A3 )
=> ( member952132173341509300at_nat @ ( F @ X4 ) @ ( image_7482708007212282706at_nat @ F @ A3 ) ) ) ).
% imageI
thf(fact_313_image__iff,axiom,
! [Z: nat > nat,F: ( nat > nat ) > nat > nat,A3: set_nat_nat] :
( ( member_nat_nat @ Z @ ( image_3205354838064109189at_nat @ F @ A3 ) )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_314_image__iff,axiom,
! [Z: set_nat,F: nat > set_nat,A3: set_nat] :
( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A3 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_315_image__iff,axiom,
! [Z: nat,F: nat > nat,A3: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A3 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_316_image__iff,axiom,
! [Z: nat > nat,F: nat > nat > nat,A3: set_nat] :
( ( member_nat_nat @ Z @ ( image_nat_nat_nat2 @ F @ A3 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_317_bex__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A3: set_nat_nat,P: ( nat > nat ) > $o] :
( ? [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_3205354838064109189at_nat @ F @ A3 ) )
& ( P @ X ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_318_bex__imageD,axiom,
! [F: nat > set_nat,A3: set_nat,P: set_nat > $o] :
( ? [X: set_nat] :
( ( member_set_nat @ X @ ( image_nat_set_nat @ F @ A3 ) )
& ( P @ X ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_319_bex__imageD,axiom,
! [F: nat > nat,A3: set_nat,P: nat > $o] :
( ? [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A3 ) )
& ( P @ X ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_320_bex__imageD,axiom,
! [F: nat > nat > nat,A3: set_nat,P: ( nat > nat ) > $o] :
( ? [X: nat > nat] :
( ( member_nat_nat @ X @ ( image_nat_nat_nat2 @ F @ A3 ) )
& ( P @ X ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_321_image__cong,axiom,
! [M5: set_nat_nat,N3: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ( M5 = N3 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_3205354838064109189at_nat @ F @ M5 )
= ( image_3205354838064109189at_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_322_image__cong,axiom,
! [M5: set_nat,N3: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M5 = N3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_set_nat @ F @ M5 )
= ( image_nat_set_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_323_image__cong,axiom,
! [M5: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
( ( M5 = N3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M5 )
= ( image_nat_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_324_image__cong,axiom,
! [M5: set_nat,N3: set_nat,F: nat > nat > nat,G: nat > nat > nat] :
( ( M5 = N3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat_nat2 @ F @ M5 )
= ( image_nat_nat_nat2 @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_325_ball__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A3: set_nat_nat,P: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_3205354838064109189at_nat @ F @ A3 ) )
=> ( P @ X3 ) )
=> ! [X: nat > nat] :
( ( member_nat_nat @ X @ A3 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_326_ball__imageD,axiom,
! [F: nat > set_nat,A3: set_nat,P: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A3 ) )
=> ( P @ X3 ) )
=> ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_327_ball__imageD,axiom,
! [F: nat > nat,A3: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A3 ) )
=> ( P @ X3 ) )
=> ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_328_ball__imageD,axiom,
! [F: nat > nat > nat,A3: set_nat,P: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( image_nat_nat_nat2 @ F @ A3 ) )
=> ( P @ X3 ) )
=> ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_329_rev__image__eqI,axiom,
! [X4: nat,A3: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X4 @ A3 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_330_rev__image__eqI,axiom,
! [X4: nat,A3: set_nat,B: set_nat,F: nat > set_nat] :
( ( member_nat @ X4 @ A3 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_331_rev__image__eqI,axiom,
! [X4: nat,A3: set_nat,B: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X4 @ A3 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_332_rev__image__eqI,axiom,
! [X4: nat > nat,A3: set_nat_nat,B: nat > nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X4 @ A3 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_333_rev__image__eqI,axiom,
! [X4: nat > nat > nat,A3: set_nat_nat_nat,B: nat > nat > nat,F: ( nat > nat > nat ) > nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A3 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat_nat2 @ B @ ( image_1896091034194584419at_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_334_rev__image__eqI,axiom,
! [X4: nat > nat > nat,A3: set_nat_nat_nat,B: ( nat > nat ) > nat,F: ( nat > nat > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X4 @ A3 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat_nat @ B @ ( image_786723269765334627at_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_335_rev__image__eqI,axiom,
! [X4: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B: nat > nat > nat,F: ( ( nat > nat ) > nat ) > nat > nat > nat] :
( ( member_nat_nat_nat @ X4 @ A3 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat_nat2 @ B @ ( image_279826485474788963at_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_336_rev__image__eqI,axiom,
! [X4: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B: ( nat > nat ) > nat,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A3 )
=> ( ( B
= ( F @ X4 ) )
=> ( member_nat_nat_nat @ B @ ( image_8393830757900314979at_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_337_rev__image__eqI,axiom,
! [X4: nat > nat > nat,A3: set_nat_nat_nat,B: ( nat > nat ) > nat > nat,F: ( nat > nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A3 )
=> ( ( B
= ( F @ X4 ) )
=> ( member952132173341509300at_nat @ B @ ( image_5348097324405788754at_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_338_rev__image__eqI,axiom,
! [X4: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B: ( nat > nat ) > nat > nat,F: ( ( nat > nat ) > nat ) > ( nat > nat ) > nat > nat] :
( ( member_nat_nat_nat @ X4 @ A3 )
=> ( ( B
= ( F @ X4 ) )
=> ( member952132173341509300at_nat @ B @ ( image_7482708007212282706at_nat @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_339_subset__image__iff,axiom,
! [B3: set_set_nat,F: nat > set_nat,A3: set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A3 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A3 )
& ( B3
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_340_subset__image__iff,axiom,
! [B3: set_nat_nat,F: ( nat > nat ) > nat > nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_3205354838064109189at_nat @ F @ A3 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A3 )
& ( B3
= ( image_3205354838064109189at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_341_subset__image__iff,axiom,
! [B3: set_nat_nat,F: nat > nat > nat,A3: set_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_nat_nat_nat2 @ F @ A3 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A3 )
& ( B3
= ( image_nat_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_342_subset__image__iff,axiom,
! [B3: set_nat,F: ( nat > nat ) > nat,A3: set_nat_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat_nat @ F @ A3 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A3 )
& ( B3
= ( image_nat_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_343_subset__image__iff,axiom,
! [B3: set_nat,F: nat > nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A3 )
& ( B3
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_344_image__subset__iff,axiom,
! [F: nat > set_nat,A3: set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A3 ) @ B3 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_set_nat @ ( F @ X2 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_345_image__subset__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A3 ) @ B3 )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
=> ( member_nat_nat @ ( F @ X2 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_346_image__subset__iff,axiom,
! [F: nat > nat > nat,A3: set_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A3 ) @ B3 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat_nat @ ( F @ X2 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_347_image__subset__iff,axiom,
! [F: nat > nat,A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B3 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ ( F @ X2 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_348_subset__imageE,axiom,
! [B3: set_set_nat,F: nat > set_nat,A3: set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A3 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A3 )
=> ( B3
!= ( image_nat_set_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_349_subset__imageE,axiom,
! [B3: set_nat_nat,F: ( nat > nat ) > nat > nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_3205354838064109189at_nat @ F @ A3 ) )
=> ~ ! [C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C4 @ A3 )
=> ( B3
!= ( image_3205354838064109189at_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_350_subset__imageE,axiom,
! [B3: set_nat_nat,F: nat > nat > nat,A3: set_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ ( image_nat_nat_nat2 @ F @ A3 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A3 )
=> ( B3
!= ( image_nat_nat_nat2 @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_351_subset__imageE,axiom,
! [B3: set_nat,F: ( nat > nat ) > nat,A3: set_nat_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat_nat @ F @ A3 ) )
=> ~ ! [C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C4 @ A3 )
=> ( B3
!= ( image_nat_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_352_subset__imageE,axiom,
! [B3: set_nat,F: nat > nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A3 )
=> ( B3
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_353_image__subsetI,axiom,
! [A3: set_nat,F: nat > nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_354_image__subsetI,axiom,
! [A3: set_nat,F: nat > set_nat,B3: set_set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_set_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_355_image__subsetI,axiom,
! [A3: set_nat,F: nat > nat > nat,B3: set_nat_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_356_image__subsetI,axiom,
! [A3: set_nat_nat,F: ( nat > nat ) > nat > nat,B3: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
=> ( member_nat_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_357_image__subsetI,axiom,
! [A3: set_nat_nat_nat,F: ( nat > nat > nat ) > nat,B3: set_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A3 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_913610194320715013at_nat @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_358_image__subsetI,axiom,
! [A3: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,B3: set_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A3 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_359_image__subsetI,axiom,
! [A3: set_nat_nat_nat,F: ( nat > nat > nat ) > nat > nat,B3: set_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A3 )
=> ( member_nat_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_1545173636400105204at_nat @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_360_image__subsetI,axiom,
! [A3: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat > nat,B3: set_nat_nat] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A3 )
=> ( member_nat_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_1262493855416953332at_nat @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_361_image__subsetI,axiom,
! [A3: set_nat_nat_nat_nat,F: ( ( nat > nat ) > nat > nat ) > nat,B3: set_nat] :
( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A3 )
=> ( member_nat @ ( F @ X3 ) @ B3 ) )
=> ( ord_less_eq_set_nat @ ( image_8194121248528334964at_nat @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_362_image__subsetI,axiom,
! [A3: set_nat_nat_nat,F: ( nat > nat > nat ) > nat > nat > nat,B3: set_nat_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A3 )
=> ( member_nat_nat_nat2 @ ( F @ X3 ) @ B3 ) )
=> ( ord_le3211623285424100676at_nat @ ( image_1896091034194584419at_nat @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_363_image__mono,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A3 ) @ ( image_3205354838064109189at_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_364_image__mono,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,F: ( nat > nat ) > nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A3 ) @ ( image_nat_nat_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_365_image__mono,axiom,
! [A3: set_nat,B3: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A3 ) @ ( image_nat_set_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_366_image__mono,axiom,
! [A3: set_nat,B3: set_nat,F: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A3 ) @ ( image_nat_nat_nat2 @ F @ B3 ) ) ) ).
% image_mono
thf(fact_367_image__mono,axiom,
! [A3: set_nat,B3: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B3 ) ) ) ).
% image_mono
thf(fact_368_bounded__Max__nat,axiom,
! [P: nat > $o,X4: nat,M5: nat] :
( ( P @ X4 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_369_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_370_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_371_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_372_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_373_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_374_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_375_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_376_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_377_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_378_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_379_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_380_add__le__imp__le__diff,axiom,
! [I: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_381_add__le__add__imp__diff__le,axiom,
! [I: nat,K2: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_382_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_383_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_384_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_385_im__T__eq__Tset,axiom,
( ( image_3205354838064109189at_nat @ t2 @ ( hales_cube @ ( plus_plus_nat @ k @ one_one_nat ) @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
= tset ) ).
% im_T_eq_Tset
thf(fact_386_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_387_order__refl,axiom,
! [X4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_388_order__refl,axiom,
! [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_389_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_390_dual__order_Orefl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_391_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_392__092_060open_062Tset_A_092_060subseteq_062_Acube_A_In_A_L_Am_J_A_It_A_L_A1_J_092_060close_062,axiom,
ord_le9059583361652607317at_nat @ tset @ ( hales_cube @ ( plus_plus_nat @ n2 @ m2 ) @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ).
% \<open>Tset \<subseteq> cube (n + m) (t + 1)\<close>
thf(fact_393_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K3 @ I2 )
=> ( P @ I2 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_394_subsetI,axiom,
! [A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A3 )
=> ( member_nat_nat_nat2 @ X3 @ B3 ) )
=> ( ord_le3211623285424100676at_nat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_395_subsetI,axiom,
! [A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A3 )
=> ( member_nat_nat_nat @ X3 @ B3 ) )
=> ( ord_le5934964663421696068at_nat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_396_subsetI,axiom,
! [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A3 )
=> ( member952132173341509300at_nat @ X3 @ B3 ) )
=> ( ord_le5260717879541182899at_nat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_397_subsetI,axiom,
! [A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A3 )
=> ( member4402528950554000163at_nat @ X3 @ B3 ) )
=> ( ord_le973658574027395234at_nat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_398_subsetI,axiom,
! [A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ! [X3: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X3 @ A3 )
=> ( member8881365325514865170at_nat @ X3 @ B3 ) )
=> ( ord_le8099187209609443857at_nat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_399_subsetI,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
=> ( member_nat_nat @ X3 @ B3 ) )
=> ( ord_le9059583361652607317at_nat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_400_subsetI,axiom,
! [A3: set_nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B3 ) )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_401_psubsetI,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_set_nat_nat @ A3 @ B3 ) ) ) ).
% psubsetI
thf(fact_402_psubsetI,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_set_nat @ A3 @ B3 ) ) ) ).
% psubsetI
thf(fact_403_subset__antisym,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_404_subset__antisym,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_405_Diff__iff,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A3 @ B3 ) )
= ( ( member_nat_nat_nat2 @ C @ A3 )
& ~ ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_406_Diff__iff,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A3 @ B3 ) )
= ( ( member_nat_nat_nat @ C @ A3 )
& ~ ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_407_Diff__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A3 @ B3 ) )
= ( ( member952132173341509300at_nat @ C @ A3 )
& ~ ( member952132173341509300at_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_408_Diff__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A3 @ B3 ) )
= ( ( member4402528950554000163at_nat @ C @ A3 )
& ~ ( member4402528950554000163at_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_409_Diff__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( minus_9165053394918225162at_nat @ A3 @ B3 ) )
= ( ( member8881365325514865170at_nat @ C @ A3 )
& ~ ( member8881365325514865170at_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_410_DiffI,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ A3 )
=> ( ~ ( member_nat_nat_nat2 @ C @ B3 )
=> ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_411_DiffI,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ A3 )
=> ( ~ ( member_nat_nat_nat @ C @ B3 )
=> ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_412_DiffI,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ A3 )
=> ( ~ ( member952132173341509300at_nat @ C @ B3 )
=> ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_413_DiffI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ A3 )
=> ( ~ ( member4402528950554000163at_nat @ C @ B3 )
=> ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_414_DiffI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ A3 )
=> ( ~ ( member8881365325514865170at_nat @ C @ B3 )
=> ( member8881365325514865170at_nat @ C @ ( minus_9165053394918225162at_nat @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_415_in__mono,axiom,
! [A3: set_nat_nat_nat,B3: set_nat_nat_nat,X4: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A3 @ B3 )
=> ( ( member_nat_nat_nat2 @ X4 @ A3 )
=> ( member_nat_nat_nat2 @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_416_in__mono,axiom,
! [A3: set_nat_nat_nat2,B3: set_nat_nat_nat2,X4: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A3 @ B3 )
=> ( ( member_nat_nat_nat @ X4 @ A3 )
=> ( member_nat_nat_nat @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_417_in__mono,axiom,
! [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat,X4: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A3 @ B3 )
=> ( ( member952132173341509300at_nat @ X4 @ A3 )
=> ( member952132173341509300at_nat @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_418_in__mono,axiom,
! [A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat,X4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le973658574027395234at_nat @ A3 @ B3 )
=> ( ( member4402528950554000163at_nat @ X4 @ A3 )
=> ( member4402528950554000163at_nat @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_419_in__mono,axiom,
! [A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat,X4: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( ord_le8099187209609443857at_nat @ A3 @ B3 )
=> ( ( member8881365325514865170at_nat @ X4 @ A3 )
=> ( member8881365325514865170at_nat @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_420_in__mono,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,X4: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ( ( member_nat_nat @ X4 @ A3 )
=> ( member_nat_nat @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_421_in__mono,axiom,
! [A3: set_nat,B3: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( member_nat @ X4 @ A3 )
=> ( member_nat @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_422_subsetD,axiom,
! [A3: set_nat_nat_nat,B3: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A3 @ B3 )
=> ( ( member_nat_nat_nat2 @ C @ A3 )
=> ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% subsetD
thf(fact_423_subsetD,axiom,
! [A3: set_nat_nat_nat2,B3: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A3 @ B3 )
=> ( ( member_nat_nat_nat @ C @ A3 )
=> ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_424_subsetD,axiom,
! [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat,C: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A3 @ B3 )
=> ( ( member952132173341509300at_nat @ C @ A3 )
=> ( member952132173341509300at_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_425_subsetD,axiom,
! [A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat,C: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le973658574027395234at_nat @ A3 @ B3 )
=> ( ( member4402528950554000163at_nat @ C @ A3 )
=> ( member4402528950554000163at_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_426_subsetD,axiom,
! [A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat,C: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( ord_le8099187209609443857at_nat @ A3 @ B3 )
=> ( ( member8881365325514865170at_nat @ C @ A3 )
=> ( member8881365325514865170at_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_427_subsetD,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ( ( member_nat_nat @ C @ A3 )
=> ( member_nat_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_428_subsetD,axiom,
! [A3: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_429_psubsetE,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B3 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ).
% psubsetE
thf(fact_430_psubsetE,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ~ ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ).
% psubsetE
thf(fact_431_Diff__mono,axiom,
! [A3: set_nat_nat,C5: set_nat_nat,D3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ C5 )
=> ( ( ord_le9059583361652607317at_nat @ D3 @ B3 )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A3 @ B3 ) @ ( minus_8121590178497047118at_nat @ C5 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_432_Diff__mono,axiom,
! [A3: set_nat,C5: set_nat,D3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C5 )
=> ( ( ord_less_eq_set_nat @ D3 @ B3 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ C5 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_433_equalityE,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( A3 = B3 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ~ ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_434_equalityE,axiom,
! [A3: set_nat,B3: set_nat] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_435_subset__eq,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A4: set_nat_nat_nat,B4: set_nat_nat_nat] :
! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A4 )
=> ( member_nat_nat_nat2 @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_436_subset__eq,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A4: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A4 )
=> ( member_nat_nat_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_437_subset__eq,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A4: set_nat_nat_nat_nat,B4: set_nat_nat_nat_nat] :
! [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A4 )
=> ( member952132173341509300at_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_438_subset__eq,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A4: set_na6626867396258451522at_nat,B4: set_na6626867396258451522at_nat] :
! [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ A4 )
=> ( member4402528950554000163at_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_439_subset__eq,axiom,
( ord_le8099187209609443857at_nat
= ( ^ [A4: set_na7233567106578532785at_nat,B4: set_na7233567106578532785at_nat] :
! [X2: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X2 @ A4 )
=> ( member8881365325514865170at_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_440_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A4 )
=> ( member_nat_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_441_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ( member_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_442_equalityD1,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( A3 = B3 )
=> ( ord_le9059583361652607317at_nat @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_443_equalityD1,axiom,
! [A3: set_nat,B3: set_nat] :
( ( A3 = B3 )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_444_equalityD2,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( A3 = B3 )
=> ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_445_equalityD2,axiom,
! [A3: set_nat,B3: set_nat] :
( ( A3 = B3 )
=> ( ord_less_eq_set_nat @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_446_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_447_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_448_subset__iff,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A4: set_nat_nat_nat,B4: set_nat_nat_nat] :
! [T2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ T2 @ A4 )
=> ( member_nat_nat_nat2 @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_449_subset__iff,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A4: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
! [T2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ T2 @ A4 )
=> ( member_nat_nat_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_450_subset__iff,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A4: set_nat_nat_nat_nat,B4: set_nat_nat_nat_nat] :
! [T2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ T2 @ A4 )
=> ( member952132173341509300at_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_451_subset__iff,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A4: set_na6626867396258451522at_nat,B4: set_na6626867396258451522at_nat] :
! [T2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ T2 @ A4 )
=> ( member4402528950554000163at_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_452_subset__iff,axiom,
( ord_le8099187209609443857at_nat
= ( ^ [A4: set_na7233567106578532785at_nat,B4: set_na7233567106578532785at_nat] :
! [T2: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ T2 @ A4 )
=> ( member8881365325514865170at_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_453_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [T2: nat > nat] :
( ( member_nat_nat @ T2 @ A4 )
=> ( member_nat_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_454_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A4 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_455_Diff__subset,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A3 @ B3 ) @ A3 ) ).
% Diff_subset
thf(fact_456_Diff__subset,axiom,
! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ A3 ) ).
% Diff_subset
thf(fact_457_double__diff,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ C5 )
=> ( ( minus_8121590178497047118at_nat @ B3 @ ( minus_8121590178497047118at_nat @ C5 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_458_double__diff,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C5 )
=> ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C5 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_459_subset__refl,axiom,
! [A3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A3 @ A3 ) ).
% subset_refl
thf(fact_460_subset__refl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% subset_refl
thf(fact_461_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_462_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_463_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_464_subset__trans,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ C5 )
=> ( ord_le9059583361652607317at_nat @ A3 @ C5 ) ) ) ).
% subset_trans
thf(fact_465_subset__trans,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C5 )
=> ( ord_less_eq_set_nat @ A3 @ C5 ) ) ) ).
% subset_trans
thf(fact_466_set__eq__subset,axiom,
( ( ^ [Y4: set_nat_nat,Z2: set_nat_nat] : ( Y4 = Z2 ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_467_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_468_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X2: set_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_469_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_470_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_471_psubset__imp__ex__mem,axiom,
! [A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( ord_le6871433888996735800at_nat @ A3 @ B3 )
=> ? [B5: nat > nat > nat] : ( member_nat_nat_nat2 @ B5 @ ( minus_7721066311745899709at_nat @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_472_psubset__imp__ex__mem,axiom,
! [A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( ord_le371403230139555384at_nat @ A3 @ B3 )
=> ? [B5: ( nat > nat ) > nat] : ( member_nat_nat_nat @ B5 @ ( minus_1221035652888719293at_nat @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_473_psubset__imp__ex__mem,axiom,
! [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( ord_le6177938698872215975at_nat @ A3 @ B3 )
=> ? [B5: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ B5 @ ( minus_4646100876039749548at_nat @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_474_psubset__imp__ex__mem,axiom,
! [A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( ord_le2785809691299232406at_nat @ A3 @ B3 )
=> ? [B5: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ B5 @ ( minus_5225787954611647771at_nat @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_475_psubset__imp__ex__mem,axiom,
! [A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( ord_le7586516898478368261at_nat @ A3 @ B3 )
=> ? [B5: ( nat > nat ) > ( nat > nat ) > nat > nat] : ( member8881365325514865170at_nat @ B5 @ ( minus_9165053394918225162at_nat @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_476_psubset__imp__subset,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B3 )
=> ( ord_le9059583361652607317at_nat @ A3 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_477_psubset__imp__subset,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_478_psubset__subset__trans,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,C5: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ C5 )
=> ( ord_less_set_nat_nat @ A3 @ C5 ) ) ) ).
% psubset_subset_trans
thf(fact_479_psubset__subset__trans,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C5 )
=> ( ord_less_set_nat @ A3 @ C5 ) ) ) ).
% psubset_subset_trans
thf(fact_480_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ~ ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_481_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_482_subset__psubset__trans,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ( ( ord_less_set_nat_nat @ B3 @ C5 )
=> ( ord_less_set_nat_nat @ A3 @ C5 ) ) ) ).
% subset_psubset_trans
thf(fact_483_subset__psubset__trans,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C5 )
=> ( ord_less_set_nat @ A3 @ C5 ) ) ) ).
% subset_psubset_trans
thf(fact_484_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_485_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_486_DiffD2,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A3 @ B3 ) )
=> ~ ( member_nat_nat_nat2 @ C @ B3 ) ) ).
% DiffD2
thf(fact_487_DiffD2,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A3 @ B3 ) )
=> ~ ( member_nat_nat_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_488_DiffD2,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A3 @ B3 ) )
=> ~ ( member952132173341509300at_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_489_DiffD2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A3 @ B3 ) )
=> ~ ( member4402528950554000163at_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_490_DiffD2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( minus_9165053394918225162at_nat @ A3 @ B3 ) )
=> ~ ( member8881365325514865170at_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_491_DiffD1,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A3 @ B3 ) )
=> ( member_nat_nat_nat2 @ C @ A3 ) ) ).
% DiffD1
thf(fact_492_DiffD1,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A3 @ B3 ) )
=> ( member_nat_nat_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_493_DiffD1,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A3 @ B3 ) )
=> ( member952132173341509300at_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_494_DiffD1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A3 @ B3 ) )
=> ( member4402528950554000163at_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_495_DiffD1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( minus_9165053394918225162at_nat @ A3 @ B3 ) )
=> ( member8881365325514865170at_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_496_DiffE,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( minus_7721066311745899709at_nat @ A3 @ B3 ) )
=> ~ ( ( member_nat_nat_nat2 @ C @ A3 )
=> ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% DiffE
thf(fact_497_DiffE,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( minus_1221035652888719293at_nat @ A3 @ B3 ) )
=> ~ ( ( member_nat_nat_nat @ C @ A3 )
=> ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_498_DiffE,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A3 @ B3 ) )
=> ~ ( ( member952132173341509300at_nat @ C @ A3 )
=> ( member952132173341509300at_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_499_DiffE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( minus_5225787954611647771at_nat @ A3 @ B3 ) )
=> ~ ( ( member4402528950554000163at_nat @ C @ A3 )
=> ( member4402528950554000163at_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_500_DiffE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( minus_9165053394918225162at_nat @ A3 @ B3 ) )
=> ~ ( ( member8881365325514865170at_nat @ C @ A3 )
=> ( member8881365325514865170at_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_501_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_502_subspace__elems__embed,axiom,
! [S3: ( nat > nat ) > nat > nat,K2: nat,N: nat,T: nat] :
( ( hales_is_subspace @ S3 @ K2 @ N @ T )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ S3 @ ( hales_cube @ K2 @ T ) ) @ ( hales_cube @ N @ T ) ) ) ).
% subspace_elems_embed
thf(fact_503_cube__props_I1_J,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
& ( ( X3 @ zero_zero_nat )
= S2 ) ) ) ).
% cube_props(1)
thf(fact_504_classes__subset__cube,axiom,
! [N: nat,T: nat,I: nat] : ( ord_le9059583361652607317at_nat @ ( hales_classes @ N @ T @ I ) @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) ) ) ).
% classes_subset_cube
thf(fact_505_order__antisym__conv,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_506_order__antisym__conv,axiom,
! [Y: set_nat_nat,X4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X4 )
=> ( ( ord_le9059583361652607317at_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_507_order__antisym__conv,axiom,
! [Y: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X4 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_508_linorder__le__cases,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_le_cases
thf(fact_509_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,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_510_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,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_511_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,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_512_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,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_513_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,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_514_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,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_515_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,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_516_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,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_517_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,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_518_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,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_519_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,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_520_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,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_521_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,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_522_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,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_523_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,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_524_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,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_525_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,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_526_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,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_527_linorder__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
| ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_linear
thf(fact_528_order__eq__refl,axiom,
! [X4: nat,Y: nat] :
( ( X4 = Y )
=> ( ord_less_eq_nat @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_529_order__eq__refl,axiom,
! [X4: set_nat_nat,Y: set_nat_nat] :
( ( X4 = Y )
=> ( ord_le9059583361652607317at_nat @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_530_order__eq__refl,axiom,
! [X4: set_nat,Y: set_nat] :
( ( X4 = Y )
=> ( ord_less_eq_set_nat @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_531_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,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_532_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,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_533_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,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_534_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,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_535_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,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_536_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,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_537_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,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_538_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,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_539_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,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_540_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,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_541_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,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_542_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,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_543_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,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_544_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,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_545_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,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le9059583361652607317at_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_546_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,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_547_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,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_548_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,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_549_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_550_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat_nat,Z2: set_nat_nat] : ( Y4 = Z2 ) )
= ( ^ [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
& ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_551_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_552_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_553_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_554_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_555_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_556_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_557_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_558_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_559_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_560_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_561_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_562_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat_nat,Z2: set_nat_nat] : ( Y4 = Z2 ) )
= ( ^ [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
& ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_563_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_564_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_565_order__trans,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_566_order__trans,axiom,
! [X4: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ Z )
=> ( ord_le9059583361652607317at_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_567_order__trans,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_568_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_569_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_570_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_571_order__antisym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_572_order__antisym,axiom,
! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_573_order__antisym,axiom,
! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_574_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_575_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_576_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_577_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_578_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_579_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_580_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_581_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat_nat,Z2: set_nat_nat] : ( Y4 = Z2 ) )
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y5 )
& ( ord_le9059583361652607317at_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_582_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_583_le__cases3,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_584_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_585_order__less__imp__not__less,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_586_order__less__imp__not__eq2,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( Y != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_587_order__less__imp__not__eq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( X4 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_588_linorder__less__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
| ( X4 = Y )
| ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_less_linear
thf(fact_589_order__less__imp__triv,axiom,
! [X4: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_590_order__less__not__sym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_not_sym
thf(fact_591_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,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_592_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,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_593_order__less__irrefl,axiom,
! [X4: nat] :
~ ( ord_less_nat @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_594_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_595_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_596_order__less__trans,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_less_trans
thf(fact_597_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_598_linorder__neq__iff,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
= ( ( ord_less_nat @ X4 @ Y )
| ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_599_order__less__asym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_asym
thf(fact_600_linorder__neqE,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
=> ( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neqE
thf(fact_601_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_602_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_603_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_604_not__less__iff__gr__or__eq,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( ( ord_less_nat @ Y @ X4 )
| ( X4 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_605_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_606_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_607_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_608_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_609_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_610_linorder__cases,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ( X4 != Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_cases
thf(fact_611_antisym__conv3,axiom,
! [Y: nat,X4: nat] :
( ~ ( ord_less_nat @ Y @ X4 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv3
thf(fact_612_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X3 )
=> ( P @ Y3 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_613_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_614_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_615_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_616_less__imp__neq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( X4 != Y ) ) ).
% less_imp_neq
thf(fact_617_gt__ex,axiom,
! [X4: nat] :
? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% gt_ex
thf(fact_618_order__le__imp__less__or__eq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_nat @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_619_order__le__imp__less__or__eq,axiom,
! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ( ord_less_set_nat_nat @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_620_order__le__imp__less__or__eq,axiom,
! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ( ord_less_set_nat @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_621_linorder__le__less__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
| ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_622_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_623_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_624_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_625_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_626_order__less__le__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_less_set_nat_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_627_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_628_order__less__le__subst1,axiom,
! [A: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_629_order__less__le__subst1,axiom,
! [A: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_630_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat_nat > set_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ! [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_631_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_632_order__less__le__subst1,axiom,
! [A: set_nat_nat,F: set_nat > set_nat_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_633_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_634_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_635_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_636_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_637_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_638_order__le__less__subst2,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_639_order__le__less__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_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_640_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_641_order__le__less__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_less_set_nat_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_642_order__le__less__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_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_643_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_644_order__le__less__subst1,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_645_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_646_order__less__le__trans,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_647_order__less__le__trans,axiom,
! [X4: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_less_set_nat_nat @ X4 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ Z )
=> ( ord_less_set_nat_nat @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_648_order__less__le__trans,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X4 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X4 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_649_order__le__less__trans,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_650_order__le__less__trans,axiom,
! [X4: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ( ord_less_set_nat_nat @ Y @ Z )
=> ( ord_less_set_nat_nat @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_651_order__le__less__trans,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X4 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_652_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_653_order__neq__le__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A != B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_654_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_655_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_656_order__le__neq__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_657_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_658_order__less__imp__le,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_659_order__less__imp__le,axiom,
! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_less_set_nat_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_660_order__less__imp__le,axiom,
! [X4: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_661_linorder__not__less,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_not_less
thf(fact_662_linorder__not__le,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X4 @ Y ) )
= ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_not_le
thf(fact_663_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_664_order__less__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_665_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_666_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_667_order__le__less,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_668_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_669_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_670_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_671_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_672_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_673_order_Ostrict__implies__order,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_674_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_675_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_676_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
& ~ ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_677_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ~ ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_678_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_679_dual__order_Ostrict__trans2,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_less_set_nat_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_680_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_681_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_682_dual__order_Ostrict__trans1,axiom,
! [B: set_nat_nat,A: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_less_set_nat_nat @ C @ B )
=> ( ord_less_set_nat_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_683_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_684_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_685_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_686_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_687_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_688_dual__order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_689_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_690_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_691_order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
& ~ ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_692_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_693_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_694_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_695_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_696_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_697_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_698_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_699_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_700_order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_701_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_702_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_703_order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_704_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_705_not__le__imp__less,axiom,
! [Y: nat,X4: nat] :
( ~ ( ord_less_eq_nat @ Y @ X4 )
=> ( ord_less_nat @ X4 @ Y ) ) ).
% not_le_imp_less
thf(fact_706_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_707_less__le__not__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y5 )
& ~ ( ord_le9059583361652607317at_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_708_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
& ~ ( ord_less_eq_set_nat @ Y5 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_709_antisym__conv2,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_710_antisym__conv2,axiom,
! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ( ~ ( ord_less_set_nat_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_711_antisym__conv2,axiom,
! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ( ~ ( ord_less_set_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_712_antisym__conv1,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_713_antisym__conv1,axiom,
! [X4: set_nat_nat,Y: set_nat_nat] :
( ~ ( ord_less_set_nat_nat @ X4 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_714_antisym__conv1,axiom,
! [X4: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X4 @ Y )
=> ( ( ord_less_eq_set_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_715_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_716_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_717_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_718_leI,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) ) ).
% leI
thf(fact_719_leD,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ~ ( ord_less_nat @ X4 @ Y ) ) ).
% leD
thf(fact_720_leD,axiom,
! [Y: set_nat_nat,X4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X4 )
=> ~ ( ord_less_set_nat_nat @ X4 @ Y ) ) ).
% leD
thf(fact_721_leD,axiom,
! [Y: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X4 )
=> ~ ( ord_less_set_nat @ X4 @ Y ) ) ).
% leD
thf(fact_722_BfS__props_I6_J,axiom,
! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( bs @ k ) )
=> ( ( s @ X @ Xa )
= ( fS @ Xa ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ k )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( bs @ J3 ) )
=> ( ( s @ X @ Xa )
= ( X @ J3 ) ) ) ) ) ) ).
% BfS_props(6)
thf(fact_723_BfL__props_I6_J,axiom,
! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( bl @ one_one_nat ) )
=> ( ( l @ X @ Xa )
= ( fL @ Xa ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ one_one_nat )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( bl @ J3 ) )
=> ( ( l @ X @ Xa )
= ( X @ J3 ) ) ) ) ) ) ).
% BfL_props(6)
thf(fact_724__092_060open_062r_A_094_Acard_A_Icube_Am_A_It_A_L_A1_J_J_A_061_Ar_A_094_A_It_A_L_A1_J_A_094_Am_092_060close_062,axiom,
( ( power_power_nat @ r @ ( finite_card_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) )
= ( power_power_nat @ r @ ( power_power_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) @ m2 ) ) ) ).
% \<open>r ^ card (cube m (t + 1)) = r ^ (t + 1) ^ m\<close>
thf(fact_725_F5,axiom,
! [Y: nat > nat] :
( ( member_nat_nat @ Y @ ( hales_cube @ ( plus_plus_nat @ k @ one_one_nat ) @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( bt @ ( plus_plus_nat @ k @ one_one_nat ) ) )
=> ( ( t2 @ Y @ X )
= ( fT @ X ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( plus_plus_nat @ k @ one_one_nat ) )
=> ! [X: nat] :
( ( member_nat @ X @ ( bt @ J3 ) )
=> ( ( t2 @ Y @ X )
= ( Y @ J3 ) ) ) ) ) ) ).
% F5
thf(fact_726_A,axiom,
! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( member_nat @ ( chi @ ( hales_join_nat @ X @ Xa @ n2 @ m2 ) ) @ ( set_ord_lessThan_nat @ r ) ) ) ) ).
% A
thf(fact_727_lessThan__eq__iff,axiom,
! [X4: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X4 )
= ( set_ord_lessThan_nat @ Y ) )
= ( X4 = Y ) ) ).
% lessThan_eq_iff
thf(fact_728_lessThan__iff,axiom,
! [I: nat > nat > nat,K2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ ( set_or3808701207811398603at_nat @ K2 ) )
= ( ord_less_nat_nat_nat2 @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_729_lessThan__iff,axiom,
! [I: ( nat > nat ) > nat,K2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ ( set_or2699333443382148811at_nat @ K2 ) )
= ( ord_less_nat_nat_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_730_lessThan__iff,axiom,
! [I: ( nat > nat ) > nat > nat,K2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I @ ( set_or7562748684798938298at_nat @ K2 ) )
= ( ord_le4629963735342356977at_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_731_lessThan__iff,axiom,
! [I: ( nat > nat ) > ( nat > nat ) > nat,K2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I @ ( set_or6177432841829679145at_nat @ K2 ) )
= ( ord_le7877100967975825120at_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_732_lessThan__iff,axiom,
! [I: ( nat > nat ) > ( nat > nat ) > nat > nat,K2: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ I @ ( set_or2490836252891414040at_nat @ K2 ) )
= ( ord_le338063099783794255at_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_733_lessThan__iff,axiom,
! [I: nat,K2: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K2 ) )
= ( ord_less_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_734_BfL__props_I3_J,axiom,
~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ bl @ ( set_ord_lessThan_nat @ one_one_nat ) ) ) ).
% BfL_props(3)
thf(fact_735_lessThan__subset__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X4 ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X4 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_736_BfS__props_I3_J,axiom,
~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ bs @ ( set_ord_lessThan_nat @ k ) ) ) ).
% BfS_props(3)
thf(fact_737_psubsetD,axiom,
! [A3: set_nat_nat_nat,B3: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le6871433888996735800at_nat @ A3 @ B3 )
=> ( ( member_nat_nat_nat2 @ C @ A3 )
=> ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_738_psubsetD,axiom,
! [A3: set_nat_nat_nat2,B3: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le371403230139555384at_nat @ A3 @ B3 )
=> ( ( member_nat_nat_nat @ C @ A3 )
=> ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_739_psubsetD,axiom,
! [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat,C: ( nat > nat ) > nat > nat] :
( ( ord_le6177938698872215975at_nat @ A3 @ B3 )
=> ( ( member952132173341509300at_nat @ C @ A3 )
=> ( member952132173341509300at_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_740_psubsetD,axiom,
! [A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat,C: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le2785809691299232406at_nat @ A3 @ B3 )
=> ( ( member4402528950554000163at_nat @ C @ A3 )
=> ( member4402528950554000163at_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_741_psubsetD,axiom,
! [A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat,C: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( ord_le7586516898478368261at_nat @ A3 @ B3 )
=> ( ( member8881365325514865170at_nat @ C @ A3 )
=> ( member8881365325514865170at_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_742_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_743_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_744_L__line__base__prop,axiom,
! [X: nat] :
( ( member_nat @ X @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( member_nat_nat @ ( l_line @ X ) @ ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% L_line_base_prop
thf(fact_745_fT__def,axiom,
( fT
= ( ^ [X2: nat] : ( if_nat @ ( member_nat @ X2 @ ( bl @ one_one_nat ) ) @ ( fL @ X2 ) @ ( if_nat @ ( member_nat @ X2 @ ( hales_set_incr @ n2 @ ( bs @ k ) ) ) @ ( fS @ ( minus_minus_nat @ X2 @ n2 ) ) @ undefined_nat ) ) ) ) ).
% fT_def
thf(fact_746_Bvar__def,axiom,
( bvar
= ( ^ [I4: nat] : ( if_set_nat @ ( I4 = zero_zero_nat ) @ ( bl @ zero_zero_nat ) @ ( hales_set_incr @ n2 @ ( bs @ ( minus_minus_nat @ I4 @ one_one_nat ) ) ) ) ) ) ).
% Bvar_def
thf(fact_747__092_060open_062card_A_123_O_O_060r_125_A_094_Acard_A_Icube_Am_A_It_A_L_A1_J_J_A_061_Ar_A_094_Acard_A_Icube_Am_A_It_A_L_A1_J_J_092_060close_062,axiom,
( ( power_power_nat @ ( finite_card_nat @ ( set_ord_lessThan_nat @ r ) ) @ ( finite_card_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) )
= ( power_power_nat @ r @ ( finite_card_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ) ).
% \<open>card {..<r} ^ card (cube m (t + 1)) = r ^ card (cube m (t + 1))\<close>
thf(fact_748_empty__iff,axiom,
! [C: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ C @ bot_bo7445843802507891576at_nat ) ).
% empty_iff
thf(fact_749_empty__iff,axiom,
! [C: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ C @ bot_bo945813143650711160at_nat ) ).
% empty_iff
thf(fact_750_empty__iff,axiom,
! [C: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ C @ bot_bo3919185967433191911at_nat ) ).
% empty_iff
thf(fact_751_empty__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat] :
~ ( member4402528950554000163at_nat @ C @ bot_bo2074992577060541142at_nat ) ).
% empty_iff
thf(fact_752_empty__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( member8881365325514865170at_nat @ C @ bot_bo2676777031303994949at_nat ) ).
% empty_iff
thf(fact_753_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_754_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_755_all__not__in__conv,axiom,
! [A3: set_nat_nat_nat] :
( ( ! [X2: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ X2 @ A3 ) )
= ( A3 = bot_bo7445843802507891576at_nat ) ) ).
% all_not_in_conv
thf(fact_756_all__not__in__conv,axiom,
! [A3: set_nat_nat_nat2] :
( ( ! [X2: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ X2 @ A3 ) )
= ( A3 = bot_bo945813143650711160at_nat ) ) ).
% all_not_in_conv
thf(fact_757_all__not__in__conv,axiom,
! [A3: set_nat_nat_nat_nat] :
( ( ! [X2: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ X2 @ A3 ) )
= ( A3 = bot_bo3919185967433191911at_nat ) ) ).
% all_not_in_conv
thf(fact_758_all__not__in__conv,axiom,
! [A3: set_na6626867396258451522at_nat] :
( ( ! [X2: ( nat > nat ) > ( nat > nat ) > nat] :
~ ( member4402528950554000163at_nat @ X2 @ A3 ) )
= ( A3 = bot_bo2074992577060541142at_nat ) ) ).
% all_not_in_conv
thf(fact_759_all__not__in__conv,axiom,
! [A3: set_na7233567106578532785at_nat] :
( ( ! [X2: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( member8881365325514865170at_nat @ X2 @ A3 ) )
= ( A3 = bot_bo2676777031303994949at_nat ) ) ).
% all_not_in_conv
thf(fact_760_all__not__in__conv,axiom,
! [A3: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_761_all__not__in__conv,axiom,
! [A3: set_nat_nat] :
( ( ! [X2: nat > nat] :
~ ( member_nat_nat @ X2 @ A3 ) )
= ( A3 = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_762_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X2: set_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_763_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_764_Collect__empty__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( ! [X2: nat > nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_765_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X2: set_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_766_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_767_empty__Collect__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P ) )
= ( ! [X2: nat > nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_768_image__is__empty,axiom,
! [F: nat > set_nat,A3: set_nat] :
( ( ( image_nat_set_nat @ F @ A3 )
= bot_bot_set_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_769_image__is__empty,axiom,
! [F: nat > nat,A3: set_nat] :
( ( ( image_nat_nat @ F @ A3 )
= bot_bot_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_770_image__is__empty,axiom,
! [F: ( nat > nat ) > nat,A3: set_nat_nat] :
( ( ( image_nat_nat_nat @ F @ A3 )
= bot_bot_set_nat )
= ( A3 = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_771_image__is__empty,axiom,
! [F: nat > nat > nat,A3: set_nat] :
( ( ( image_nat_nat_nat2 @ F @ A3 )
= bot_bot_set_nat_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_772_image__is__empty,axiom,
! [F: ( nat > nat ) > nat > nat,A3: set_nat_nat] :
( ( ( image_3205354838064109189at_nat @ F @ A3 )
= bot_bot_set_nat_nat )
= ( A3 = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_773_empty__is__image,axiom,
! [F: nat > set_nat,A3: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_774_empty__is__image,axiom,
! [F: nat > nat,A3: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_775_empty__is__image,axiom,
! [F: ( nat > nat ) > nat,A3: set_nat_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat_nat @ F @ A3 ) )
= ( A3 = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_776_empty__is__image,axiom,
! [F: nat > nat > nat,A3: set_nat] :
( ( bot_bot_set_nat_nat
= ( image_nat_nat_nat2 @ F @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_777_empty__is__image,axiom,
! [F: ( nat > nat ) > nat > nat,A3: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( image_3205354838064109189at_nat @ F @ A3 ) )
= ( A3 = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_778_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_779_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_780_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_781_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_782_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_783_subset__empty,axiom,
! [A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ bot_bot_set_nat_nat )
= ( A3 = bot_bot_set_nat_nat ) ) ).
% subset_empty
thf(fact_784_subset__empty,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_785_empty__subsetI,axiom,
! [A3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A3 ) ).
% empty_subsetI
thf(fact_786_empty__subsetI,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).
% empty_subsetI
thf(fact_787_Diff__cancel,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ A3 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_788_Diff__cancel,axiom,
! [A3: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A3 @ A3 )
= bot_bot_set_nat_nat ) ).
% Diff_cancel
thf(fact_789_empty__Diff,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_790_empty__Diff,axiom,
! [A3: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ bot_bot_set_nat_nat @ A3 )
= bot_bot_set_nat_nat ) ).
% empty_Diff
thf(fact_791_Diff__empty,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
= A3 ) ).
% Diff_empty
thf(fact_792_Diff__empty,axiom,
! [A3: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A3 @ bot_bot_set_nat_nat )
= A3 ) ).
% Diff_empty
thf(fact_793__092_060open_062_123_125_A_092_060notin_062_ABvar_A_096_A_123_O_O_060k_A_L_A1_125_092_060close_062,axiom,
~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ bvar @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ) ) ).
% \<open>{} \<notin> Bvar ` {..<k + 1}\<close>
thf(fact_794_F1,axiom,
~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ bt @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ) ) ).
% F1
thf(fact_795_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_796_Diff__eq__empty__iff,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( ( minus_8121590178497047118at_nat @ A3 @ B3 )
= bot_bot_set_nat_nat )
= ( ord_le9059583361652607317at_nat @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_797_Diff__eq__empty__iff,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( minus_minus_set_nat @ A3 @ B3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_798_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_799_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_800_bot_Oextremum,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% bot.extremum
thf(fact_801_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_802_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_803_bot_Oextremum__unique,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% bot.extremum_unique
thf(fact_804_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_805_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_806_bot_Oextremum__uniqueI,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
=> ( A = bot_bot_set_nat_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_807_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_808_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_809_bot_Oextremum__strict,axiom,
! [A: set_nat_nat] :
~ ( ord_less_set_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% bot.extremum_strict
thf(fact_810_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_811_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_812_bot_Onot__eq__extremum,axiom,
! [A: set_nat_nat] :
( ( A != bot_bot_set_nat_nat )
= ( ord_less_set_nat_nat @ bot_bot_set_nat_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_813_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_814_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_815_not__psubset__empty,axiom,
! [A3: set_nat] :
~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_816_not__psubset__empty,axiom,
! [A3: set_nat_nat] :
~ ( ord_less_set_nat_nat @ A3 @ bot_bot_set_nat_nat ) ).
% not_psubset_empty
thf(fact_817_emptyE,axiom,
! [A: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ A @ bot_bo7445843802507891576at_nat ) ).
% emptyE
thf(fact_818_emptyE,axiom,
! [A: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ A @ bot_bo945813143650711160at_nat ) ).
% emptyE
thf(fact_819_emptyE,axiom,
! [A: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ A @ bot_bo3919185967433191911at_nat ) ).
% emptyE
thf(fact_820_emptyE,axiom,
! [A: ( nat > nat ) > ( nat > nat ) > nat] :
~ ( member4402528950554000163at_nat @ A @ bot_bo2074992577060541142at_nat ) ).
% emptyE
thf(fact_821_emptyE,axiom,
! [A: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( member8881365325514865170at_nat @ A @ bot_bo2676777031303994949at_nat ) ).
% emptyE
thf(fact_822_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_823_emptyE,axiom,
! [A: nat > nat] :
~ ( member_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_824_equals0D,axiom,
! [A3: set_nat_nat_nat,A: nat > nat > nat] :
( ( A3 = bot_bo7445843802507891576at_nat )
=> ~ ( member_nat_nat_nat2 @ A @ A3 ) ) ).
% equals0D
thf(fact_825_equals0D,axiom,
! [A3: set_nat_nat_nat2,A: ( nat > nat ) > nat] :
( ( A3 = bot_bo945813143650711160at_nat )
=> ~ ( member_nat_nat_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_826_equals0D,axiom,
! [A3: set_nat_nat_nat_nat,A: ( nat > nat ) > nat > nat] :
( ( A3 = bot_bo3919185967433191911at_nat )
=> ~ ( member952132173341509300at_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_827_equals0D,axiom,
! [A3: set_na6626867396258451522at_nat,A: ( nat > nat ) > ( nat > nat ) > nat] :
( ( A3 = bot_bo2074992577060541142at_nat )
=> ~ ( member4402528950554000163at_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_828_equals0D,axiom,
! [A3: set_na7233567106578532785at_nat,A: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( A3 = bot_bo2676777031303994949at_nat )
=> ~ ( member8881365325514865170at_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_829_equals0D,axiom,
! [A3: set_nat,A: nat] :
( ( A3 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_830_equals0D,axiom,
! [A3: set_nat_nat,A: nat > nat] :
( ( A3 = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_831_equals0I,axiom,
! [A3: set_nat_nat_nat] :
( ! [Y2: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ Y2 @ A3 )
=> ( A3 = bot_bo7445843802507891576at_nat ) ) ).
% equals0I
thf(fact_832_equals0I,axiom,
! [A3: set_nat_nat_nat2] :
( ! [Y2: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ Y2 @ A3 )
=> ( A3 = bot_bo945813143650711160at_nat ) ) ).
% equals0I
thf(fact_833_equals0I,axiom,
! [A3: set_nat_nat_nat_nat] :
( ! [Y2: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ Y2 @ A3 )
=> ( A3 = bot_bo3919185967433191911at_nat ) ) ).
% equals0I
thf(fact_834_equals0I,axiom,
! [A3: set_na6626867396258451522at_nat] :
( ! [Y2: ( nat > nat ) > ( nat > nat ) > nat] :
~ ( member4402528950554000163at_nat @ Y2 @ A3 )
=> ( A3 = bot_bo2074992577060541142at_nat ) ) ).
% equals0I
thf(fact_835_equals0I,axiom,
! [A3: set_na7233567106578532785at_nat] :
( ! [Y2: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( member8881365325514865170at_nat @ Y2 @ A3 )
=> ( A3 = bot_bo2676777031303994949at_nat ) ) ).
% equals0I
thf(fact_836_equals0I,axiom,
! [A3: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A3 )
=> ( A3 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_837_equals0I,axiom,
! [A3: set_nat_nat] :
( ! [Y2: nat > nat] :
~ ( member_nat_nat @ Y2 @ A3 )
=> ( A3 = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_838_ex__in__conv,axiom,
! [A3: set_nat_nat_nat] :
( ( ? [X2: nat > nat > nat] : ( member_nat_nat_nat2 @ X2 @ A3 ) )
= ( A3 != bot_bo7445843802507891576at_nat ) ) ).
% ex_in_conv
thf(fact_839_ex__in__conv,axiom,
! [A3: set_nat_nat_nat2] :
( ( ? [X2: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X2 @ A3 ) )
= ( A3 != bot_bo945813143650711160at_nat ) ) ).
% ex_in_conv
thf(fact_840_ex__in__conv,axiom,
! [A3: set_nat_nat_nat_nat] :
( ( ? [X2: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X2 @ A3 ) )
= ( A3 != bot_bo3919185967433191911at_nat ) ) ).
% ex_in_conv
thf(fact_841_ex__in__conv,axiom,
! [A3: set_na6626867396258451522at_nat] :
( ( ? [X2: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X2 @ A3 ) )
= ( A3 != bot_bo2074992577060541142at_nat ) ) ).
% ex_in_conv
thf(fact_842_ex__in__conv,axiom,
! [A3: set_na7233567106578532785at_nat] :
( ( ? [X2: ( nat > nat ) > ( nat > nat ) > nat > nat] : ( member8881365325514865170at_nat @ X2 @ A3 ) )
= ( A3 != bot_bo2676777031303994949at_nat ) ) ).
% ex_in_conv
thf(fact_843_ex__in__conv,axiom,
! [A3: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A3 ) )
= ( A3 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_844_ex__in__conv,axiom,
! [A3: set_nat_nat] :
( ( ? [X2: nat > nat] : ( member_nat_nat @ X2 @ A3 ) )
= ( A3 != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_845_set__incr__altdef,axiom,
( hales_set_incr
= ( ^ [N4: nat] : ( image_nat_nat @ ( plus_plus_nat @ N4 ) ) ) ) ).
% set_incr_altdef
thf(fact_846_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_847_fact3,axiom,
! [X: nat] :
( ( member_nat @ X @ ( set_ord_lessThan_nat @ k ) )
=> ( ( inf_inf_set_nat @ ( bl @ zero_zero_nat ) @ ( hales_set_incr @ n2 @ ( bs @ X ) ) )
= bot_bot_set_nat ) ) ).
% fact3
thf(fact_848_Bstat__def,axiom,
( bstat
= ( sup_sup_set_nat @ ( hales_set_incr @ n2 @ ( bs @ k ) ) @ ( bl @ one_one_nat ) ) ) ).
% Bstat_def
thf(fact_849_fact4,axiom,
! [X: nat] :
( ( member_nat @ X @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) )
=> ( ( X != Xa )
=> ( ( inf_inf_set_nat @ ( hales_set_incr @ n2 @ ( bs @ X ) ) @ ( hales_set_incr @ n2 @ ( bs @ Xa ) ) )
= bot_bot_set_nat ) ) ) ) ).
% fact4
thf(fact_850_fact1,axiom,
( ( inf_inf_set_nat @ ( hales_set_incr @ n2 @ ( bs @ k ) ) @ ( bl @ one_one_nat ) )
= bot_bot_set_nat ) ).
% fact1
thf(fact_851_inf__right__idem,axiom,
! [X4: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X4 @ Y ) @ Y )
= ( inf_inf_set_nat @ X4 @ Y ) ) ).
% inf_right_idem
thf(fact_852_inf_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_853_inf__left__idem,axiom,
! [X4: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( inf_inf_set_nat @ X4 @ Y ) )
= ( inf_inf_set_nat @ X4 @ Y ) ) ).
% inf_left_idem
thf(fact_854_inf_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_855_inf__idem,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_856_inf_Oidem,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_857_sup_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ B )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_858_sup_Oright__idem,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ B )
= ( sup_sup_set_set_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_859_sup__left__idem,axiom,
! [X4: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ X4 @ Y ) )
= ( sup_sup_set_nat @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_860_sup__left__idem,axiom,
! [X4: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ X4 @ Y ) )
= ( sup_sup_set_set_nat @ X4 @ Y ) ) ).
% sup_left_idem
thf(fact_861_sup_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_862_sup_Oleft__idem,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_set_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_863_sup__idem,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_864_sup__idem,axiom,
! [X4: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_865_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_866_sup_Oidem,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_867_Int__iff,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( inf_in5274420515160781174at_nat @ A3 @ B3 ) )
= ( ( member_nat_nat_nat2 @ C @ A3 )
& ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_868_Int__iff,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( inf_in7997761893158376566at_nat @ A3 @ B3 ) )
= ( ( member_nat_nat_nat @ C @ A3 )
& ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_869_Int__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A3 @ B3 ) )
= ( ( member952132173341509300at_nat @ C @ A3 )
& ( member952132173341509300at_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_870_Int__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( inf_in6213014276851238612at_nat @ A3 @ B3 ) )
= ( ( member4402528950554000163at_nat @ C @ A3 )
& ( member4402528950554000163at_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_871_Int__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A3 @ B3 ) )
= ( ( member8881365325514865170at_nat @ C @ A3 )
& ( member8881365325514865170at_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_872_Int__iff,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
= ( ( member_nat @ C @ A3 )
& ( member_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_873_IntI,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ A3 )
=> ( ( member_nat_nat_nat2 @ C @ B3 )
=> ( member_nat_nat_nat2 @ C @ ( inf_in5274420515160781174at_nat @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_874_IntI,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ A3 )
=> ( ( member_nat_nat_nat @ C @ B3 )
=> ( member_nat_nat_nat @ C @ ( inf_in7997761893158376566at_nat @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_875_IntI,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ A3 )
=> ( ( member952132173341509300at_nat @ C @ B3 )
=> ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_876_IntI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ A3 )
=> ( ( member4402528950554000163at_nat @ C @ B3 )
=> ( member4402528950554000163at_nat @ C @ ( inf_in6213014276851238612at_nat @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_877_IntI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ A3 )
=> ( ( member8881365325514865170at_nat @ C @ B3 )
=> ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_878_IntI,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_879_Un__iff,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( sup_su3334021163961628176at_nat @ A3 @ B3 ) )
= ( ( member_nat_nat_nat2 @ C @ A3 )
| ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_880_Un__iff,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( sup_su6057362541959223568at_nat @ A3 @ B3 ) )
= ( ( member_nat_nat_nat @ C @ A3 )
| ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_881_Un__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A3 @ B3 ) )
= ( ( member952132173341509300at_nat @ C @ A3 )
| ( member952132173341509300at_nat @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_882_Un__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( sup_su481250237928500590at_nat @ A3 @ B3 ) )
= ( ( member4402528950554000163at_nat @ C @ A3 )
| ( member4402528950554000163at_nat @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_883_Un__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A3 @ B3 ) )
= ( ( member8881365325514865170at_nat @ C @ A3 )
| ( member8881365325514865170at_nat @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_884_Un__iff,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) )
= ( ( member_nat @ C @ A3 )
| ( member_nat @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_885_Un__iff,axiom,
! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B3 ) )
= ( ( member_set_nat @ C @ A3 )
| ( member_set_nat @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_886_UnCI,axiom,
! [C: nat > nat > nat,B3: set_nat_nat_nat,A3: set_nat_nat_nat] :
( ( ~ ( member_nat_nat_nat2 @ C @ B3 )
=> ( member_nat_nat_nat2 @ C @ A3 ) )
=> ( member_nat_nat_nat2 @ C @ ( sup_su3334021163961628176at_nat @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_887_UnCI,axiom,
! [C: ( nat > nat ) > nat,B3: set_nat_nat_nat2,A3: set_nat_nat_nat2] :
( ( ~ ( member_nat_nat_nat @ C @ B3 )
=> ( member_nat_nat_nat @ C @ A3 ) )
=> ( member_nat_nat_nat @ C @ ( sup_su6057362541959223568at_nat @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_888_UnCI,axiom,
! [C: ( nat > nat ) > nat > nat,B3: set_nat_nat_nat_nat,A3: set_nat_nat_nat_nat] :
( ( ~ ( member952132173341509300at_nat @ C @ B3 )
=> ( member952132173341509300at_nat @ C @ A3 ) )
=> ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_889_UnCI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,B3: set_na6626867396258451522at_nat,A3: set_na6626867396258451522at_nat] :
( ( ~ ( member4402528950554000163at_nat @ C @ B3 )
=> ( member4402528950554000163at_nat @ C @ A3 ) )
=> ( member4402528950554000163at_nat @ C @ ( sup_su481250237928500590at_nat @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_890_UnCI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,B3: set_na7233567106578532785at_nat,A3: set_na7233567106578532785at_nat] :
( ( ~ ( member8881365325514865170at_nat @ C @ B3 )
=> ( member8881365325514865170at_nat @ C @ A3 ) )
=> ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_891_UnCI,axiom,
! [C: nat,B3: set_nat,A3: set_nat] :
( ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ A3 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_892_UnCI,axiom,
! [C: set_nat,B3: set_set_nat,A3: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B3 )
=> ( member_set_nat @ C @ A3 ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_893_le__inf__iff,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X4 @ Y )
& ( ord_less_eq_nat @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_894_le__inf__iff,axiom,
! [X4: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ ( inf_inf_set_nat_nat @ Y @ Z ) )
= ( ( ord_le9059583361652607317at_nat @ X4 @ Y )
& ( ord_le9059583361652607317at_nat @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_895_le__inf__iff,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ ( inf_inf_set_nat @ Y @ Z ) )
= ( ( ord_less_eq_set_nat @ X4 @ Y )
& ( ord_less_eq_set_nat @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_896_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_897_inf_Obounded__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( inf_inf_set_nat_nat @ B @ C ) )
= ( ( ord_le9059583361652607317at_nat @ A @ B )
& ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_898_inf_Obounded__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A @ B )
& ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_899_le__sup__iff,axiom,
! [X4: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X4 @ Y ) @ Z )
= ( ( ord_le6893508408891458716et_nat @ X4 @ Z )
& ( ord_le6893508408891458716et_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_900_le__sup__iff,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X4 @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_901_le__sup__iff,axiom,
! [X4: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X4 @ Y ) @ Z )
= ( ( ord_le9059583361652607317at_nat @ X4 @ Z )
& ( ord_le9059583361652607317at_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_902_le__sup__iff,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X4 @ Y ) @ Z )
= ( ( ord_less_eq_set_nat @ X4 @ Z )
& ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_903_sup_Obounded__iff,axiom,
! [B: set_set_nat,C: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B @ C ) @ A )
= ( ( ord_le6893508408891458716et_nat @ B @ A )
& ( ord_le6893508408891458716et_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_904_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_905_sup_Obounded__iff,axiom,
! [B: set_nat_nat,C: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B @ C ) @ A )
= ( ( ord_le9059583361652607317at_nat @ B @ A )
& ( ord_le9059583361652607317at_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_906_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_907_inf__bot__left,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X4 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_908_inf__bot__left,axiom,
! [X4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X4 )
= bot_bot_set_nat_nat ) ).
% inf_bot_left
thf(fact_909_inf__bot__right,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ X4 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_910_inf__bot__right,axiom,
! [X4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X4 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% inf_bot_right
thf(fact_911_sup__bot__left,axiom,
! [X4: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_912_sup__bot__left,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_913_sup__bot__left,axiom,
! [X4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_914_sup__bot__right,axiom,
! [X4: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ bot_bot_set_set_nat )
= X4 ) ).
% sup_bot_right
thf(fact_915_sup__bot__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
= X4 ) ).
% sup_bot_right
thf(fact_916_sup__bot__right,axiom,
! [X4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X4 @ bot_bot_set_nat_nat )
= X4 ) ).
% sup_bot_right
thf(fact_917_bot__eq__sup__iff,axiom,
! [X4: set_set_nat,Y: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ X4 @ Y ) )
= ( ( X4 = bot_bot_set_set_nat )
& ( Y = bot_bot_set_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_918_bot__eq__sup__iff,axiom,
! [X4: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X4 @ Y ) )
= ( ( X4 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_919_bot__eq__sup__iff,axiom,
! [X4: set_nat_nat,Y: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ X4 @ Y ) )
= ( ( X4 = bot_bot_set_nat_nat )
& ( Y = bot_bot_set_nat_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_920_sup__eq__bot__iff,axiom,
! [X4: set_set_nat,Y: set_set_nat] :
( ( ( sup_sup_set_set_nat @ X4 @ Y )
= bot_bot_set_set_nat )
= ( ( X4 = bot_bot_set_set_nat )
& ( Y = bot_bot_set_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_921_sup__eq__bot__iff,axiom,
! [X4: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X4 @ Y )
= bot_bot_set_nat )
= ( ( X4 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_922_sup__eq__bot__iff,axiom,
! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ X4 @ Y )
= bot_bot_set_nat_nat )
= ( ( X4 = bot_bot_set_nat_nat )
& ( Y = bot_bot_set_nat_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_923_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ( A = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_924_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_925_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ( A = bot_bot_set_nat_nat )
& ( B = bot_bot_set_nat_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_926_sup__bot_Oleft__neutral,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_927_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_928_sup__bot_Oleft__neutral,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_929_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_930_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_931_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat_nat )
& ( B = bot_bot_set_nat_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_932_sup__bot_Oright__neutral,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ bot_bot_set_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_933_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_934_sup__bot_Oright__neutral,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ bot_bot_set_nat_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_935_sup__inf__absorb,axiom,
! [X4: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( inf_inf_set_nat @ X4 @ Y ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_936_sup__inf__absorb,axiom,
! [X4: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ ( inf_inf_set_set_nat @ X4 @ Y ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_937_inf__sup__absorb,axiom,
! [X4: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( sup_sup_set_nat @ X4 @ Y ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_938_inf__sup__absorb,axiom,
! [X4: set_set_nat,Y: set_set_nat] :
( ( inf_inf_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ X4 @ Y ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_939_Un__empty,axiom,
! [A3: set_set_nat,B3: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A3 @ B3 )
= bot_bot_set_set_nat )
= ( ( A3 = bot_bot_set_set_nat )
& ( B3 = bot_bot_set_set_nat ) ) ) ).
% Un_empty
thf(fact_940_Un__empty,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( sup_sup_set_nat @ A3 @ B3 )
= bot_bot_set_nat )
= ( ( A3 = bot_bot_set_nat )
& ( B3 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_941_Un__empty,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A3 @ B3 )
= bot_bot_set_nat_nat )
= ( ( A3 = bot_bot_set_nat_nat )
& ( B3 = bot_bot_set_nat_nat ) ) ) ).
% Un_empty
thf(fact_942_Int__subset__iff,axiom,
! [C5: set_nat_nat,A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ ( inf_inf_set_nat_nat @ A3 @ B3 ) )
= ( ( ord_le9059583361652607317at_nat @ C5 @ A3 )
& ( ord_le9059583361652607317at_nat @ C5 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_943_Int__subset__iff,axiom,
! [C5: set_nat,A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ ( inf_inf_set_nat @ A3 @ B3 ) )
= ( ( ord_less_eq_set_nat @ C5 @ A3 )
& ( ord_less_eq_set_nat @ C5 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_944_Un__subset__iff,axiom,
! [A3: set_set_nat,B3: set_set_nat,C5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A3 @ B3 ) @ C5 )
= ( ( ord_le6893508408891458716et_nat @ A3 @ C5 )
& ( ord_le6893508408891458716et_nat @ B3 @ C5 ) ) ) ).
% Un_subset_iff
thf(fact_945_Un__subset__iff,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A3 @ B3 ) @ C5 )
= ( ( ord_le9059583361652607317at_nat @ A3 @ C5 )
& ( ord_le9059583361652607317at_nat @ B3 @ C5 ) ) ) ).
% Un_subset_iff
thf(fact_946_Un__subset__iff,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C5 )
= ( ( ord_less_eq_set_nat @ A3 @ C5 )
& ( ord_less_eq_set_nat @ B3 @ C5 ) ) ) ).
% Un_subset_iff
thf(fact_947_Un__Int__eq_I1_J,axiom,
! [S3: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S3 @ T3 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_948_Un__Int__eq_I1_J,axiom,
! [S3: set_set_nat,T3: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S3 @ T3 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_949_Un__Int__eq_I2_J,axiom,
! [S3: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S3 @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_950_Un__Int__eq_I2_J,axiom,
! [S3: set_set_nat,T3: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S3 @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_951_Un__Int__eq_I3_J,axiom,
! [S3: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ S3 @ ( sup_sup_set_nat @ S3 @ T3 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_952_Un__Int__eq_I3_J,axiom,
! [S3: set_set_nat,T3: set_set_nat] :
( ( inf_inf_set_set_nat @ S3 @ ( sup_sup_set_set_nat @ S3 @ T3 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_953_Un__Int__eq_I4_J,axiom,
! [T3: set_nat,S3: set_nat] :
( ( inf_inf_set_nat @ T3 @ ( sup_sup_set_nat @ S3 @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_954_Un__Int__eq_I4_J,axiom,
! [T3: set_set_nat,S3: set_set_nat] :
( ( inf_inf_set_set_nat @ T3 @ ( sup_sup_set_set_nat @ S3 @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_955_Int__Un__eq_I1_J,axiom,
! [S3: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S3 @ T3 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_956_Int__Un__eq_I1_J,axiom,
! [S3: set_set_nat,T3: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S3 @ T3 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_957_Int__Un__eq_I2_J,axiom,
! [S3: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S3 @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_958_Int__Un__eq_I2_J,axiom,
! [S3: set_set_nat,T3: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S3 @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_959_Int__Un__eq_I3_J,axiom,
! [S3: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ S3 @ ( inf_inf_set_nat @ S3 @ T3 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_960_Int__Un__eq_I3_J,axiom,
! [S3: set_set_nat,T3: set_set_nat] :
( ( sup_sup_set_set_nat @ S3 @ ( inf_inf_set_set_nat @ S3 @ T3 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_961_Int__Un__eq_I4_J,axiom,
! [T3: set_nat,S3: set_nat] :
( ( sup_sup_set_nat @ T3 @ ( inf_inf_set_nat @ S3 @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_962_Int__Un__eq_I4_J,axiom,
! [T3: set_set_nat,S3: set_set_nat] :
( ( sup_sup_set_set_nat @ T3 @ ( inf_inf_set_set_nat @ S3 @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_963_Un__Diff__cancel2,axiom,
! [B3: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B3 @ A3 ) @ A3 )
= ( sup_sup_set_nat @ B3 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_964_Un__Diff__cancel2,axiom,
! [B3: set_set_nat,A3: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ B3 @ A3 ) @ A3 )
= ( sup_sup_set_set_nat @ B3 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_965_Un__Diff__cancel,axiom,
! [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ A3 ) )
= ( sup_sup_set_nat @ A3 @ B3 ) ) ).
% Un_Diff_cancel
thf(fact_966_Un__Diff__cancel,axiom,
! [A3: set_set_nat,B3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ ( minus_2163939370556025621et_nat @ B3 @ A3 ) )
= ( sup_sup_set_set_nat @ A3 @ B3 ) ) ).
% Un_Diff_cancel
thf(fact_967_Diff__disjoint,axiom,
! [A3: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ A3 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_968_Diff__disjoint,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A3 @ ( minus_8121590178497047118at_nat @ B3 @ A3 ) )
= bot_bot_set_nat_nat ) ).
% Diff_disjoint
thf(fact_969_fact5,axiom,
! [X: nat] :
( ( member_nat @ X @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) )
=> ( ( inf_inf_set_nat @ ( bvar @ X ) @ bstat )
= bot_bot_set_nat ) ) ).
% fact5
thf(fact_970_less__supI1,axiom,
! [X4: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ X4 @ A )
=> ( ord_less_set_nat @ X4 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_971_less__supI1,axiom,
! [X4: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ A )
=> ( ord_less_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_972_less__supI1,axiom,
! [X4: nat,A: nat,B: nat] :
( ( ord_less_nat @ X4 @ A )
=> ( ord_less_nat @ X4 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_973_less__supI2,axiom,
! [X4: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X4 @ B )
=> ( ord_less_set_nat @ X4 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_974_less__supI2,axiom,
! [X4: set_set_nat,B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ X4 @ B )
=> ( ord_less_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_975_less__supI2,axiom,
! [X4: nat,B: nat,A: nat] :
( ( ord_less_nat @ X4 @ B )
=> ( ord_less_nat @ X4 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_976_sup_Oabsorb3,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_977_sup_Oabsorb3,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ B @ A )
=> ( ( sup_sup_set_set_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_978_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_979_sup_Oabsorb4,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_980_sup_Oabsorb4,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( sup_sup_set_set_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_981_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_982_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_983_sup_Ostrict__boundedE,axiom,
! [B: set_set_nat,C: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ ( sup_sup_set_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_set_set_nat @ B @ A )
=> ~ ( ord_less_set_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_984_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_985_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_986_sup_Ostrict__order__iff,axiom,
( ord_less_set_set_nat
= ( ^ [B2: set_set_nat,A2: set_set_nat] :
( ( A2
= ( sup_sup_set_set_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_987_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_988_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_989_sup_Ostrict__coboundedI1,axiom,
! [C: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ C @ A )
=> ( ord_less_set_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_990_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_991_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_992_sup_Ostrict__coboundedI2,axiom,
! [C: set_set_nat,B: set_set_nat,A: set_set_nat] :
( ( ord_less_set_set_nat @ C @ B )
=> ( ord_less_set_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_993_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_994_less__infI1,axiom,
! [A: set_nat,X4: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ X4 )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X4 ) ) ).
% less_infI1
thf(fact_995_less__infI1,axiom,
! [A: nat,X4: nat,B: nat] :
( ( ord_less_nat @ A @ X4 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X4 ) ) ).
% less_infI1
thf(fact_996_less__infI2,axiom,
! [B: set_nat,X4: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ X4 )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X4 ) ) ).
% less_infI2
thf(fact_997_less__infI2,axiom,
! [B: nat,X4: nat,A: nat] :
( ( ord_less_nat @ B @ X4 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X4 ) ) ).
% less_infI2
thf(fact_998_inf_Oabsorb3,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_999_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_1000_inf_Oabsorb4,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_1001_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_1002_inf_Ostrict__boundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_set_nat @ A @ B )
=> ~ ( ord_less_set_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1003_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1004_inf_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1005_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1006_inf_Ostrict__coboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1007_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1008_inf_Ostrict__coboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1009_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1010_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1011_image__Un,axiom,
! [F: ( nat > nat ) > nat > nat,A3: set_nat_nat,B3: set_nat_nat] :
( ( image_3205354838064109189at_nat @ F @ ( sup_sup_set_nat_nat @ A3 @ B3 ) )
= ( sup_sup_set_nat_nat @ ( image_3205354838064109189at_nat @ F @ A3 ) @ ( image_3205354838064109189at_nat @ F @ B3 ) ) ) ).
% image_Un
thf(fact_1012_image__Un,axiom,
! [F: nat > nat > nat,A3: set_nat,B3: set_nat] :
( ( image_nat_nat_nat2 @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
= ( sup_sup_set_nat_nat @ ( image_nat_nat_nat2 @ F @ A3 ) @ ( image_nat_nat_nat2 @ F @ B3 ) ) ) ).
% image_Un
thf(fact_1013_image__Un,axiom,
! [F: nat > nat,A3: set_nat,B3: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A3 ) @ ( image_nat_nat @ F @ B3 ) ) ) ).
% image_Un
thf(fact_1014_image__Un,axiom,
! [F: nat > set_nat,A3: set_nat,B3: set_nat] :
( ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ A3 ) @ ( image_nat_set_nat @ F @ B3 ) ) ) ).
% image_Un
thf(fact_1015_image__Un,axiom,
! [F: set_nat > nat,A3: set_set_nat,B3: set_set_nat] :
( ( image_set_nat_nat @ F @ ( sup_sup_set_set_nat @ A3 @ B3 ) )
= ( sup_sup_set_nat @ ( image_set_nat_nat @ F @ A3 ) @ ( image_set_nat_nat @ F @ B3 ) ) ) ).
% image_Un
thf(fact_1016_image__Un,axiom,
! [F: set_nat > set_nat,A3: set_set_nat,B3: set_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( sup_sup_set_set_nat @ A3 @ B3 ) )
= ( sup_sup_set_set_nat @ ( image_7916887816326733075et_nat @ F @ A3 ) @ ( image_7916887816326733075et_nat @ F @ B3 ) ) ) ).
% image_Un
thf(fact_1017_Int__emptyI,axiom,
! [A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A3 )
=> ~ ( member_nat_nat_nat2 @ X3 @ B3 ) )
=> ( ( inf_in5274420515160781174at_nat @ A3 @ B3 )
= bot_bo7445843802507891576at_nat ) ) ).
% Int_emptyI
thf(fact_1018_Int__emptyI,axiom,
! [A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A3 )
=> ~ ( member_nat_nat_nat @ X3 @ B3 ) )
=> ( ( inf_in7997761893158376566at_nat @ A3 @ B3 )
= bot_bo945813143650711160at_nat ) ) ).
% Int_emptyI
thf(fact_1019_Int__emptyI,axiom,
! [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A3 )
=> ~ ( member952132173341509300at_nat @ X3 @ B3 ) )
=> ( ( inf_in2949407623404935909at_nat @ A3 @ B3 )
= bot_bo3919185967433191911at_nat ) ) ).
% Int_emptyI
thf(fact_1020_Int__emptyI,axiom,
! [A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A3 )
=> ~ ( member4402528950554000163at_nat @ X3 @ B3 ) )
=> ( ( inf_in6213014276851238612at_nat @ A3 @ B3 )
= bot_bo2074992577060541142at_nat ) ) ).
% Int_emptyI
thf(fact_1021_Int__emptyI,axiom,
! [A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ! [X3: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X3 @ A3 )
=> ~ ( member8881365325514865170at_nat @ X3 @ B3 ) )
=> ( ( inf_in6008378084349164867at_nat @ A3 @ B3 )
= bot_bo2676777031303994949at_nat ) ) ).
% Int_emptyI
thf(fact_1022_Int__emptyI,axiom,
! [A3: set_nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ~ ( member_nat @ X3 @ B3 ) )
=> ( ( inf_inf_set_nat @ A3 @ B3 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1023_Int__emptyI,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
=> ~ ( member_nat_nat @ X3 @ B3 ) )
=> ( ( inf_inf_set_nat_nat @ A3 @ B3 )
= bot_bot_set_nat_nat ) ) ).
% Int_emptyI
thf(fact_1024_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_1025_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1026_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_1027_disjoint__iff,axiom,
! [A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( ( inf_in5274420515160781174at_nat @ A3 @ B3 )
= bot_bo7445843802507891576at_nat )
= ( ! [X2: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X2 @ A3 )
=> ~ ( member_nat_nat_nat2 @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_1028_disjoint__iff,axiom,
! [A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( ( inf_in7997761893158376566at_nat @ A3 @ B3 )
= bot_bo945813143650711160at_nat )
= ( ! [X2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X2 @ A3 )
=> ~ ( member_nat_nat_nat @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_1029_disjoint__iff,axiom,
! [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( ( inf_in2949407623404935909at_nat @ A3 @ B3 )
= bot_bo3919185967433191911at_nat )
= ( ! [X2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X2 @ A3 )
=> ~ ( member952132173341509300at_nat @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_1030_disjoint__iff,axiom,
! [A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( ( inf_in6213014276851238612at_nat @ A3 @ B3 )
= bot_bo2074992577060541142at_nat )
= ( ! [X2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X2 @ A3 )
=> ~ ( member4402528950554000163at_nat @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_1031_disjoint__iff,axiom,
! [A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( ( inf_in6008378084349164867at_nat @ A3 @ B3 )
= bot_bo2676777031303994949at_nat )
= ( ! [X2: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X2 @ A3 )
=> ~ ( member8881365325514865170at_nat @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_1032_disjoint__iff,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B3 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ~ ( member_nat @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_1033_disjoint__iff,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A3 @ B3 )
= bot_bot_set_nat_nat )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
=> ~ ( member_nat_nat @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_1034_Un__empty__left,axiom,
! [B3: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_1035_Un__empty__left,axiom,
! [B3: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_1036_Un__empty__left,axiom,
! [B3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_1037_Int__empty__left,axiom,
! [B3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B3 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_1038_Int__empty__left,axiom,
! [B3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ B3 )
= bot_bot_set_nat_nat ) ).
% Int_empty_left
thf(fact_1039_Un__empty__right,axiom,
! [A3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ bot_bot_set_set_nat )
= A3 ) ).
% Un_empty_right
thf(fact_1040_Un__empty__right,axiom,
! [A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ bot_bot_set_nat )
= A3 ) ).
% Un_empty_right
thf(fact_1041_Un__empty__right,axiom,
! [A3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A3 @ bot_bot_set_nat_nat )
= A3 ) ).
% Un_empty_right
thf(fact_1042_Int__empty__right,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_1043_Int__empty__right,axiom,
! [A3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A3 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% Int_empty_right
thf(fact_1044_disjoint__iff__not__equal,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B3 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ B3 )
=> ( X2 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1045_disjoint__iff__not__equal,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A3 @ B3 )
= bot_bot_set_nat_nat )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
=> ! [Y5: nat > nat] :
( ( member_nat_nat @ Y5 @ B3 )
=> ( X2 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1046_Un__Diff__Int,axiom,
! [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ A3 @ B3 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1047_Un__Diff__Int,axiom,
! [A3: set_set_nat,B3: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B3 ) @ ( inf_inf_set_set_nat @ A3 @ B3 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1048_Int__Diff__Un,axiom,
! [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ B3 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1049_Int__Diff__Un,axiom,
! [A3: set_set_nat,B3: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A3 @ B3 ) @ ( minus_2163939370556025621et_nat @ A3 @ B3 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1050_Diff__Int,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C5 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ C5 ) ) ) ).
% Diff_Int
thf(fact_1051_Diff__Int,axiom,
! [A3: set_set_nat,B3: set_set_nat,C5: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A3 @ ( inf_inf_set_set_nat @ B3 @ C5 ) )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B3 ) @ ( minus_2163939370556025621et_nat @ A3 @ C5 ) ) ) ).
% Diff_Int
thf(fact_1052_Diff__Un,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C5 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ C5 ) ) ) ).
% Diff_Un
thf(fact_1053_Diff__Un,axiom,
! [A3: set_set_nat,B3: set_set_nat,C5: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A3 @ ( sup_sup_set_set_nat @ B3 @ C5 ) )
= ( inf_inf_set_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B3 ) @ ( minus_2163939370556025621et_nat @ A3 @ C5 ) ) ) ).
% Diff_Un
thf(fact_1054_sup__left__commute,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X4 @ Z ) ) ) ).
% sup_left_commute
thf(fact_1055_sup__left__commute,axiom,
! [X4: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ Y @ ( sup_sup_set_set_nat @ X4 @ Z ) ) ) ).
% sup_left_commute
thf(fact_1056_sup_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_1057_sup_Oleft__commute,axiom,
! [B: set_set_nat,A: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ B @ ( sup_sup_set_set_nat @ A @ C ) )
= ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_1058_inf__left__commute,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X4 @ Z ) ) ) ).
% inf_left_commute
thf(fact_1059_inf_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C ) )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_1060_sup__inf__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X4: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X4 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X4 ) @ ( sup_sup_set_nat @ Z @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_1061_sup__inf__distrib2,axiom,
! [Y: set_set_nat,Z: set_set_nat,X4: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y @ Z ) @ X4 )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y @ X4 ) @ ( sup_sup_set_set_nat @ Z @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_1062_sup__inf__distrib1,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X4 @ Y ) @ ( sup_sup_set_nat @ X4 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1063_sup__inf__distrib1,axiom,
! [X4: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X4 @ Y ) @ ( sup_sup_set_set_nat @ X4 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1064_inf__sup__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X4: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X4 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X4 ) @ ( inf_inf_set_nat @ Z @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_1065_inf__sup__distrib2,axiom,
! [Y: set_set_nat,Z: set_set_nat,X4: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y @ Z ) @ X4 )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y @ X4 ) @ ( inf_inf_set_set_nat @ Z @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_1066_inf__sup__distrib1,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X4 @ Y ) @ ( inf_inf_set_nat @ X4 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1067_inf__sup__distrib1,axiom,
! [X4: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X4 @ Y ) @ ( inf_inf_set_set_nat @ X4 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1068_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] : ( sup_sup_set_nat @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_1069_sup__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [X2: set_set_nat,Y5: set_set_nat] : ( sup_sup_set_set_nat @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_1070_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A2: set_nat,B2: set_nat] : ( sup_sup_set_nat @ B2 @ A2 ) ) ) ).
% sup.commute
thf(fact_1071_sup_Ocommute,axiom,
( sup_sup_set_set_nat
= ( ^ [A2: set_set_nat,B2: set_set_nat] : ( sup_sup_set_set_nat @ B2 @ A2 ) ) ) ).
% sup.commute
thf(fact_1072_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] : ( inf_inf_set_nat @ Y5 @ X2 ) ) ) ).
% inf_commute
thf(fact_1073_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A2: set_nat,B2: set_nat] : ( inf_inf_set_nat @ B2 @ A2 ) ) ) ).
% inf.commute
thf(fact_1074_sup__assoc,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X4 @ Y ) @ Z )
= ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_1075_sup__assoc,axiom,
! [X4: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X4 @ Y ) @ Z )
= ( sup_sup_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_1076_sup_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_1077_sup_Oassoc,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C )
= ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_1078_inf__assoc,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X4 @ Y ) @ Z )
= ( inf_inf_set_nat @ X4 @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_1079_inf_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_1080_distrib__imp2,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ! [X3: set_nat,Y2: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z3 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ ( sup_sup_set_nat @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_nat @ X4 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X4 @ Y ) @ ( inf_inf_set_nat @ X4 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1081_distrib__imp2,axiom,
! [X4: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ! [X3: set_set_nat,Y2: set_set_nat,Z3: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( inf_inf_set_set_nat @ Y2 @ Z3 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X3 @ Y2 ) @ ( sup_sup_set_set_nat @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X4 @ Y ) @ ( inf_inf_set_set_nat @ X4 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1082_distrib__imp1,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ! [X3: set_nat,Y2: set_nat,Z3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z3 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ ( inf_inf_set_nat @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_nat @ X4 @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X4 @ Y ) @ ( sup_sup_set_nat @ X4 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1083_distrib__imp1,axiom,
! [X4: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ! [X3: set_set_nat,Y2: set_set_nat,Z3: set_set_nat] :
( ( inf_inf_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ Y2 @ Z3 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X3 @ Y2 ) @ ( inf_inf_set_set_nat @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_set_nat @ X4 @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X4 @ Y ) @ ( sup_sup_set_set_nat @ X4 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1084_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] : ( inf_inf_set_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_1085_inf__sup__aci_I2_J,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X4 @ Y ) @ Z )
= ( inf_inf_set_nat @ X4 @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_1086_inf__sup__aci_I3_J,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_1087_inf__sup__aci_I4_J,axiom,
! [X4: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( inf_inf_set_nat @ X4 @ Y ) )
= ( inf_inf_set_nat @ X4 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_1088_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] : ( sup_sup_set_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1089_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_nat
= ( ^ [X2: set_set_nat,Y5: set_set_nat] : ( sup_sup_set_set_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1090_inf__sup__aci_I6_J,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X4 @ Y ) @ Z )
= ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_1091_inf__sup__aci_I6_J,axiom,
! [X4: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X4 @ Y ) @ Z )
= ( sup_sup_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_1092_inf__sup__aci_I7_J,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X4 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_1093_inf__sup__aci_I7_J,axiom,
! [X4: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ Y @ ( sup_sup_set_set_nat @ X4 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_1094_inf__sup__aci_I8_J,axiom,
! [X4: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ X4 @ Y ) )
= ( sup_sup_set_nat @ X4 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_1095_inf__sup__aci_I8_J,axiom,
! [X4: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ X4 @ Y ) )
= ( sup_sup_set_set_nat @ X4 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_1096_Int__left__commute,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C5 ) )
= ( inf_inf_set_nat @ B3 @ ( inf_inf_set_nat @ A3 @ C5 ) ) ) ).
% Int_left_commute
thf(fact_1097_Un__left__commute,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C5 ) )
= ( sup_sup_set_nat @ B3 @ ( sup_sup_set_nat @ A3 @ C5 ) ) ) ).
% Un_left_commute
thf(fact_1098_Un__left__commute,axiom,
! [A3: set_set_nat,B3: set_set_nat,C5: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ ( sup_sup_set_set_nat @ B3 @ C5 ) )
= ( sup_sup_set_set_nat @ B3 @ ( sup_sup_set_set_nat @ A3 @ C5 ) ) ) ).
% Un_left_commute
thf(fact_1099_Un__Int__distrib2,axiom,
! [B3: set_nat,C5: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B3 @ C5 ) @ A3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B3 @ A3 ) @ ( sup_sup_set_nat @ C5 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_1100_Un__Int__distrib2,axiom,
! [B3: set_set_nat,C5: set_set_nat,A3: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B3 @ C5 ) @ A3 )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B3 @ A3 ) @ ( sup_sup_set_set_nat @ C5 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_1101_Int__left__absorb,axiom,
! [A3: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
= ( inf_inf_set_nat @ A3 @ B3 ) ) ).
% Int_left_absorb
thf(fact_1102_Int__Un__distrib2,axiom,
! [B3: set_nat,C5: set_nat,A3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B3 @ C5 ) @ A3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B3 @ A3 ) @ ( inf_inf_set_nat @ C5 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_1103_Int__Un__distrib2,axiom,
! [B3: set_set_nat,C5: set_set_nat,A3: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B3 @ C5 ) @ A3 )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B3 @ A3 ) @ ( inf_inf_set_set_nat @ C5 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_1104_Un__left__absorb,axiom,
! [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B3 ) )
= ( sup_sup_set_nat @ A3 @ B3 ) ) ).
% Un_left_absorb
thf(fact_1105_Un__left__absorb,axiom,
! [A3: set_set_nat,B3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ ( sup_sup_set_set_nat @ A3 @ B3 ) )
= ( sup_sup_set_set_nat @ A3 @ B3 ) ) ).
% Un_left_absorb
thf(fact_1106_Un__Int__distrib,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C5 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ ( sup_sup_set_nat @ A3 @ C5 ) ) ) ).
% Un_Int_distrib
thf(fact_1107_Un__Int__distrib,axiom,
! [A3: set_set_nat,B3: set_set_nat,C5: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ ( inf_inf_set_set_nat @ B3 @ C5 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A3 @ B3 ) @ ( sup_sup_set_set_nat @ A3 @ C5 ) ) ) ).
% Un_Int_distrib
thf(fact_1108_Int__Un__distrib,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C5 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ A3 @ C5 ) ) ) ).
% Int_Un_distrib
thf(fact_1109_Int__Un__distrib,axiom,
! [A3: set_set_nat,B3: set_set_nat,C5: set_set_nat] :
( ( inf_inf_set_set_nat @ A3 @ ( sup_sup_set_set_nat @ B3 @ C5 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A3 @ B3 ) @ ( inf_inf_set_set_nat @ A3 @ C5 ) ) ) ).
% Int_Un_distrib
thf(fact_1110_Un__Int__crazy,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ B3 @ C5 ) ) @ ( inf_inf_set_nat @ C5 @ A3 ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ ( sup_sup_set_nat @ B3 @ C5 ) ) @ ( sup_sup_set_nat @ C5 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_1111_Un__Int__crazy,axiom,
! [A3: set_set_nat,B3: set_set_nat,C5: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A3 @ B3 ) @ ( inf_inf_set_set_nat @ B3 @ C5 ) ) @ ( inf_inf_set_set_nat @ C5 @ A3 ) )
= ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A3 @ B3 ) @ ( sup_sup_set_set_nat @ B3 @ C5 ) ) @ ( sup_sup_set_set_nat @ C5 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_1112_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_1113_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_1114_Un__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] : ( sup_sup_set_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_1115_Int__absorb,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_1116_Un__absorb,axiom,
! [A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_1117_Un__absorb,axiom,
! [A3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_1118_Int__assoc,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C5 )
= ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C5 ) ) ) ).
% Int_assoc
thf(fact_1119_Un__assoc,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C5 )
= ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C5 ) ) ) ).
% Un_assoc
thf(fact_1120_Un__assoc,axiom,
! [A3: set_set_nat,B3: set_set_nat,C5: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A3 @ B3 ) @ C5 )
= ( sup_sup_set_set_nat @ A3 @ ( sup_sup_set_set_nat @ B3 @ C5 ) ) ) ).
% Un_assoc
thf(fact_1121_ball__Un,axiom,
! [A3: set_nat,B3: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A3 @ B3 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_1122_ball__Un,axiom,
! [A3: set_set_nat,B3: set_set_nat,P: set_nat > $o] :
( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( sup_sup_set_set_nat @ A3 @ B3 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_1123_bex__Un,axiom,
! [A3: set_nat,B3: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A3 @ B3 ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: nat] :
( ( member_nat @ X2 @ B3 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_1124_bex__Un,axiom,
! [A3: set_set_nat,B3: set_set_nat,P: set_nat > $o] :
( ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ ( sup_sup_set_set_nat @ A3 @ B3 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: set_nat] :
( ( member_set_nat @ X2 @ B3 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_1125_IntD2,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( inf_in5274420515160781174at_nat @ A3 @ B3 ) )
=> ( member_nat_nat_nat2 @ C @ B3 ) ) ).
% IntD2
thf(fact_1126_IntD2,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( inf_in7997761893158376566at_nat @ A3 @ B3 ) )
=> ( member_nat_nat_nat @ C @ B3 ) ) ).
% IntD2
thf(fact_1127_IntD2,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A3 @ B3 ) )
=> ( member952132173341509300at_nat @ C @ B3 ) ) ).
% IntD2
thf(fact_1128_IntD2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( inf_in6213014276851238612at_nat @ A3 @ B3 ) )
=> ( member4402528950554000163at_nat @ C @ B3 ) ) ).
% IntD2
thf(fact_1129_IntD2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A3 @ B3 ) )
=> ( member8881365325514865170at_nat @ C @ B3 ) ) ).
% IntD2
thf(fact_1130_IntD2,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
=> ( member_nat @ C @ B3 ) ) ).
% IntD2
thf(fact_1131_IntD1,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( inf_in5274420515160781174at_nat @ A3 @ B3 ) )
=> ( member_nat_nat_nat2 @ C @ A3 ) ) ).
% IntD1
thf(fact_1132_IntD1,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( inf_in7997761893158376566at_nat @ A3 @ B3 ) )
=> ( member_nat_nat_nat @ C @ A3 ) ) ).
% IntD1
thf(fact_1133_IntD1,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A3 @ B3 ) )
=> ( member952132173341509300at_nat @ C @ A3 ) ) ).
% IntD1
thf(fact_1134_IntD1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( inf_in6213014276851238612at_nat @ A3 @ B3 ) )
=> ( member4402528950554000163at_nat @ C @ A3 ) ) ).
% IntD1
thf(fact_1135_IntD1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A3 @ B3 ) )
=> ( member8881365325514865170at_nat @ C @ A3 ) ) ).
% IntD1
thf(fact_1136_IntD1,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
=> ( member_nat @ C @ A3 ) ) ).
% IntD1
thf(fact_1137_UnI2,axiom,
! [C: nat > nat > nat,B3: set_nat_nat_nat,A3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ B3 )
=> ( member_nat_nat_nat2 @ C @ ( sup_su3334021163961628176at_nat @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_1138_UnI2,axiom,
! [C: ( nat > nat ) > nat,B3: set_nat_nat_nat2,A3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ B3 )
=> ( member_nat_nat_nat @ C @ ( sup_su6057362541959223568at_nat @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_1139_UnI2,axiom,
! [C: ( nat > nat ) > nat > nat,B3: set_nat_nat_nat_nat,A3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ B3 )
=> ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_1140_UnI2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,B3: set_na6626867396258451522at_nat,A3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ B3 )
=> ( member4402528950554000163at_nat @ C @ ( sup_su481250237928500590at_nat @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_1141_UnI2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,B3: set_na7233567106578532785at_nat,A3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ B3 )
=> ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_1142_UnI2,axiom,
! [C: nat,B3: set_nat,A3: set_nat] :
( ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_1143_UnI2,axiom,
! [C: set_nat,B3: set_set_nat,A3: set_set_nat] :
( ( member_set_nat @ C @ B3 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_1144_UnI1,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ A3 )
=> ( member_nat_nat_nat2 @ C @ ( sup_su3334021163961628176at_nat @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_1145_UnI1,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ A3 )
=> ( member_nat_nat_nat @ C @ ( sup_su6057362541959223568at_nat @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_1146_UnI1,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ A3 )
=> ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_1147_UnI1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ A3 )
=> ( member4402528950554000163at_nat @ C @ ( sup_su481250237928500590at_nat @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_1148_UnI1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ A3 )
=> ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_1149_UnI1,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_1150_UnI1,axiom,
! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ A3 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_1151_IntE,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( inf_in5274420515160781174at_nat @ A3 @ B3 ) )
=> ~ ( ( member_nat_nat_nat2 @ C @ A3 )
=> ~ ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% IntE
thf(fact_1152_IntE,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( inf_in7997761893158376566at_nat @ A3 @ B3 ) )
=> ~ ( ( member_nat_nat_nat @ C @ A3 )
=> ~ ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% IntE
thf(fact_1153_IntE,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A3 @ B3 ) )
=> ~ ( ( member952132173341509300at_nat @ C @ A3 )
=> ~ ( member952132173341509300at_nat @ C @ B3 ) ) ) ).
% IntE
thf(fact_1154_IntE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( inf_in6213014276851238612at_nat @ A3 @ B3 ) )
=> ~ ( ( member4402528950554000163at_nat @ C @ A3 )
=> ~ ( member4402528950554000163at_nat @ C @ B3 ) ) ) ).
% IntE
thf(fact_1155_IntE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A3 @ B3 ) )
=> ~ ( ( member8881365325514865170at_nat @ C @ A3 )
=> ~ ( member8881365325514865170at_nat @ C @ B3 ) ) ) ).
% IntE
thf(fact_1156_IntE,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ~ ( member_nat @ C @ B3 ) ) ) ).
% IntE
thf(fact_1157_UnE,axiom,
! [C: nat > nat > nat,A3: set_nat_nat_nat,B3: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( sup_su3334021163961628176at_nat @ A3 @ B3 ) )
=> ( ~ ( member_nat_nat_nat2 @ C @ A3 )
=> ( member_nat_nat_nat2 @ C @ B3 ) ) ) ).
% UnE
thf(fact_1158_UnE,axiom,
! [C: ( nat > nat ) > nat,A3: set_nat_nat_nat2,B3: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( sup_su6057362541959223568at_nat @ A3 @ B3 ) )
=> ( ~ ( member_nat_nat_nat @ C @ A3 )
=> ( member_nat_nat_nat @ C @ B3 ) ) ) ).
% UnE
thf(fact_1159_UnE,axiom,
! [C: ( nat > nat ) > nat > nat,A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A3 @ B3 ) )
=> ( ~ ( member952132173341509300at_nat @ C @ A3 )
=> ( member952132173341509300at_nat @ C @ B3 ) ) ) ).
% UnE
thf(fact_1160_UnE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( sup_su481250237928500590at_nat @ A3 @ B3 ) )
=> ( ~ ( member4402528950554000163at_nat @ C @ A3 )
=> ( member4402528950554000163at_nat @ C @ B3 ) ) ) ).
% UnE
thf(fact_1161_UnE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A3 @ B3 ) )
=> ( ~ ( member8881365325514865170at_nat @ C @ A3 )
=> ( member8881365325514865170at_nat @ C @ B3 ) ) ) ).
% UnE
thf(fact_1162_UnE,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) )
=> ( ~ ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% UnE
thf(fact_1163_UnE,axiom,
! [C: set_nat,A3: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B3 ) )
=> ( ~ ( member_set_nat @ C @ A3 )
=> ( member_set_nat @ C @ B3 ) ) ) ).
% UnE
thf(fact_1164_Diff__Int__distrib2,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ C5 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C5 ) @ ( inf_inf_set_nat @ B3 @ C5 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1165_Diff__Int__distrib,axiom,
! [C5: set_nat,A3: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ C5 @ ( minus_minus_set_nat @ A3 @ B3 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C5 @ A3 ) @ ( inf_inf_set_nat @ C5 @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_1166_Diff__Diff__Int,axiom,
! [A3: set_nat,B3: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( minus_minus_set_nat @ A3 @ B3 ) )
= ( inf_inf_set_nat @ A3 @ B3 ) ) ).
% Diff_Diff_Int
thf(fact_1167_Diff__Int2,axiom,
! [A3: set_nat,C5: set_nat,B3: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C5 ) @ ( inf_inf_set_nat @ B3 @ C5 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C5 ) @ B3 ) ) ).
% Diff_Int2
thf(fact_1168_Int__Diff,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C5 )
= ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ C5 ) ) ) ).
% Int_Diff
thf(fact_1169_Un__Diff,axiom,
! [A3: set_nat,B3: set_nat,C5: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C5 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ C5 ) @ ( minus_minus_set_nat @ B3 @ C5 ) ) ) ).
% Un_Diff
thf(fact_1170_Un__Diff,axiom,
! [A3: set_set_nat,B3: set_set_nat,C5: set_set_nat] :
( ( minus_2163939370556025621et_nat @ ( sup_sup_set_set_nat @ A3 @ B3 ) @ C5 )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ C5 ) @ ( minus_2163939370556025621et_nat @ B3 @ C5 ) ) ) ).
% Un_Diff
thf(fact_1171_Int__mono,axiom,
! [A3: set_nat_nat,C5: set_nat_nat,B3: set_nat_nat,D3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ C5 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ D3 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A3 @ B3 ) @ ( inf_inf_set_nat_nat @ C5 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_1172_Int__mono,axiom,
! [A3: set_nat,C5: set_nat,B3: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C5 )
=> ( ( ord_less_eq_set_nat @ B3 @ D3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ C5 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_1173_Int__lower1,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A3 @ B3 ) @ A3 ) ).
% Int_lower1
thf(fact_1174_Int__lower1,axiom,
! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ A3 ) ).
% Int_lower1
thf(fact_1175_Int__lower2,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A3 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_1176_Int__lower2,axiom,
! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_1177_Int__absorb1,axiom,
! [B3: set_nat_nat,A3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ A3 )
=> ( ( inf_inf_set_nat_nat @ A3 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_1178_Int__absorb1,axiom,
! [B3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_1179_Int__absorb2,axiom,
! [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ( ( inf_inf_set_nat_nat @ A3 @ B3 )
= A3 ) ) ).
% Int_absorb2
thf(fact_1180_Int__absorb2,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( inf_inf_set_nat @ A3 @ B3 )
= A3 ) ) ).
% Int_absorb2
thf(fact_1181_Int__greatest,axiom,
! [C5: set_nat_nat,A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A3 )
=> ( ( ord_le9059583361652607317at_nat @ C5 @ B3 )
=> ( ord_le9059583361652607317at_nat @ C5 @ ( inf_inf_set_nat_nat @ A3 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_1182_Int__greatest,axiom,
! [C5: set_nat,A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A3 )
=> ( ( ord_less_eq_set_nat @ C5 @ B3 )
=> ( ord_less_eq_set_nat @ C5 @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_1183_Int__Collect__mono,axiom,
! [A3: set_nat_nat_nat,B3: set_nat_nat_nat,P: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ( ord_le3211623285424100676at_nat @ A3 @ B3 )
=> ( ! [X3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3211623285424100676at_nat @ ( inf_in5274420515160781174at_nat @ A3 @ ( collect_nat_nat_nat2 @ P ) ) @ ( inf_in5274420515160781174at_nat @ B3 @ ( collect_nat_nat_nat2 @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1184_Int__Collect__mono,axiom,
! [A3: set_nat_nat_nat2,B3: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ( ord_le5934964663421696068at_nat @ A3 @ B3 )
=> ( ! [X3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le5934964663421696068at_nat @ ( inf_in7997761893158376566at_nat @ A3 @ ( collect_nat_nat_nat @ P ) ) @ ( inf_in7997761893158376566at_nat @ B3 @ ( collect_nat_nat_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1185_Int__Collect__mono,axiom,
! [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat,P: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
( ( ord_le5260717879541182899at_nat @ A3 @ B3 )
=> ( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le5260717879541182899at_nat @ ( inf_in2949407623404935909at_nat @ A3 @ ( collec3567154360959927026at_nat @ P ) ) @ ( inf_in2949407623404935909at_nat @ B3 @ ( collec3567154360959927026at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1186_Int__Collect__mono,axiom,
! [A3: set_na6626867396258451522at_nat,B3: set_na6626867396258451522at_nat,P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ( ord_le973658574027395234at_nat @ A3 @ B3 )
=> ( ! [X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le973658574027395234at_nat @ ( inf_in6213014276851238612at_nat @ A3 @ ( collec2410089373097230945at_nat @ P ) ) @ ( inf_in6213014276851238612at_nat @ B3 @ ( collec2410089373097230945at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1187_Int__Collect__mono,axiom,
! [A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat,P: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o] :
( ( ord_le8099187209609443857at_nat @ A3 @ B3 )
=> ( ! [X3: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8099187209609443857at_nat @ ( inf_in6008378084349164867at_nat @ A3 @ ( collec6535634078845029456at_nat @ P ) ) @ ( inf_in6008378084349164867at_nat @ B3 @ ( collec6535634078845029456at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1188_Int__Collect__mono,axiom,
! [A3: set_set_nat,B3: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A3 @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B3 @ ( collect_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1189_Int__Collect__mono,axiom,
! [A3: set_nat_nat,B3: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A3 @ ( collect_nat_nat @ P ) ) @ ( inf_inf_set_nat_nat @ B3 @ ( collect_nat_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1190_Int__Collect__mono,axiom,
! [A3: set_nat,B3: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B3 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1191_Un__mono,axiom,
! [A3: set_set_nat,C5: set_set_nat,B3: set_set_nat,D3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ C5 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ D3 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A3 @ B3 ) @ ( sup_sup_set_set_nat @ C5 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_1192_Un__mono,axiom,
! [A3: set_nat_nat,C5: set_nat_nat,B3: set_nat_nat,D3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ C5 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ D3 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A3 @ B3 ) @ ( sup_sup_set_nat_nat @ C5 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_1193_Un__mono,axiom,
! [A3: set_nat,C5: set_nat,B3: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C5 )
=> ( ( ord_less_eq_set_nat @ B3 @ D3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ ( sup_sup_set_nat @ C5 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_1194_Un__least,axiom,
! [A3: set_nat,C5: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C5 )
=> ( ( ord_less_eq_set_nat @ B3 @ C5 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C5 ) ) ) ).
% Un_least
thf(fact_1195_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 )
& ! [X: nat] :
( ( member_nat @ X @ B_1 )
=> ! [Xa: nat] :
( ( ord_less_nat @ Xa @ T )
=> ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ T )
=> ( ( L3 @ Xa @ X )
= ( L3 @ Y3 @ X ) ) ) ) )
& ! [X: nat] :
( ( member_nat @ X @ B_0 )
=> ! [S4: nat] :
( ( ord_less_nat @ S4 @ T )
=> ( ( L3 @ S4 @ X )
= S4 ) ) ) ) ) ) ).
% is_line_elim_t_1
thf(fact_1196_line__points__in__cube__unfolded,axiom,
! [L3: nat > nat > nat,N: nat,T: nat,S2: nat,J: nat] :
( ( hales_is_line @ L3 @ N @ T )
=> ( ( ord_less_nat @ S2 @ T )
=> ( ( ord_less_nat @ J @ N )
=> ( member_nat @ ( L3 @ S2 @ J ) @ ( set_ord_lessThan_nat @ T ) ) ) ) ) ).
% line_points_in_cube_unfolded
thf(fact_1197_line__points__in__cube,axiom,
! [L3: nat > nat > nat,N: nat,T: nat,S2: nat] :
( ( hales_is_line @ L3 @ N @ T )
=> ( ( ord_less_nat @ S2 @ T )
=> ( member_nat_nat @ ( L3 @ S2 ) @ ( hales_cube @ N @ T ) ) ) ) ).
% line_points_in_cube
thf(fact_1198__092_060open_062Bvar_A_096_A_123_O_O_060k_A_L_A1_125_A_061_ABL_A_096_A_123_O_O_0601_125_A_092_060union_062_ABvar_A_096_A_1231_O_O_060k_A_L_A1_125_092_060close_062,axiom,
( ( image_nat_set_nat @ bvar @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ bl @ ( set_ord_lessThan_nat @ one_one_nat ) ) @ ( image_nat_set_nat @ bvar @ ( set_or4665077453230672383an_nat @ one_one_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ) ) ) ).
% \<open>Bvar ` {..<k + 1} = BL ` {..<1} \<union> Bvar ` {1..<k + 1}\<close>
thf(fact_1199_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_1200_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_1201_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1202_subset__card__intvl__is__intvl,axiom,
! [A3: set_nat,K2: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A3 ) ) ) )
=> ( A3
= ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_1203_subset__eq__atLeast0__lessThan__card,axiom,
! [N3: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_1204_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K2 ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K2 ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_1205_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1206_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1207_BfS__props_I2_J,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ bs @ ( set_ord_atMost_nat @ k ) ) )
= ( set_ord_lessThan_nat @ m2 ) ) ).
% BfS_props(2)
thf(fact_1208__092_060chi_062S__def,axiom,
( chi_S
= ( restrict_nat_nat_nat
@ ^ [Y5: nat > nat] : ( chi @ ( hales_join_nat @ ( l_line @ zero_zero_nat ) @ Y5 @ n2 @ m2 ) )
@ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% \<chi>S_def
thf(fact_1209_BfL__props_I2_J,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ bl @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ n2 ) ) ).
% BfL_props(2)
thf(fact_1210_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1211_BfL__props_I1_J,axiom,
disjoi6798895846410478970at_nat @ bl @ ( set_ord_atMost_nat @ one_one_nat ) ).
% BfL_props(1)
thf(fact_1212_BfS__props_I1_J,axiom,
disjoi6798895846410478970at_nat @ bs @ ( set_ord_atMost_nat @ k ) ).
% BfS_props(1)
thf(fact_1213_F3,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ bt @ ( set_ord_atMost_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ) )
= ( set_ord_lessThan_nat @ ( plus_plus_nat @ n2 @ m2 ) ) ) ).
% F3
thf(fact_1214_fact2,axiom,
( ( inf_inf_set_nat @ ( bl @ zero_zero_nat )
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I4: nat] : ( hales_set_incr @ n2 @ ( bs @ I4 ) )
@ ( set_ord_lessThan_nat @ k ) ) ) )
= bot_bot_set_nat ) ).
% fact2
thf(fact_1215__092_060chi_062L__s__def,axiom,
( chi_L_s
= ( restrict_nat_nat_nat
@ ^ [X2: nat > nat] : ( phi @ ( chi_L @ X2 ) )
@ ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% \<chi>L_s_def
thf(fact_1216_BfL__props_I4_J,axiom,
( member_nat_nat @ fL
@ ( piE_nat_nat @ ( bl @ one_one_nat )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% BfL_props(4)
thf(fact_1217_BfS__props_I4_J,axiom,
( member_nat_nat @ fS
@ ( piE_nat_nat @ ( bs @ k )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% BfS_props(4)
thf(fact_1218_set__incr__def,axiom,
( hales_set_incr
= ( ^ [N4: nat] :
( image_nat_nat
@ ^ [A2: nat] : ( plus_plus_nat @ A2 @ N4 ) ) ) ) ).
% set_incr_def
thf(fact_1219_BfS__props_I5_J,axiom,
( member952132173341509300at_nat @ s
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% BfS_props(5)
thf(fact_1220_BfL__props_I5_J,axiom,
( member952132173341509300at_nat @ l
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% BfL_props(5)
thf(fact_1221_F4,axiom,
( member_nat_nat @ fT
@ ( piE_nat_nat @ ( bt @ ( plus_plus_nat @ k @ one_one_nat ) )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% F4
thf(fact_1222_T__prop,axiom,
( member952132173341509300at_nat @ t2
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ ( plus_plus_nat @ k @ one_one_nat ) @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( hales_cube @ ( plus_plus_nat @ n2 @ m2 ) @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% T_prop
thf(fact_1223__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062BL_AfL_O_A_092_060lbrakk_062disjoint__family__on_ABL_A_123_O_O1_125_059_A_092_060Union_062_A_IBL_A_096_A_123_O_O1_125_J_A_061_A_123_O_O_060n_125_059_A_123_125_A_092_060notin_062_ABL_A_096_A_123_O_O_0601_125_059_AfL_A_092_060in_062_ABL_A1_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060t_A_L_A1_125_059_AL_A_092_060in_062_Acube_A1_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_A1_A_It_A_L_A1_J_O_A_I_092_060forall_062i_092_060in_062BL_A1_O_AL_Ay_Ai_A_061_AfL_Ai_J_A_092_060and_062_A_I_092_060forall_062j_0601_O_A_092_060forall_062i_092_060in_062BL_Aj_O_AL_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,
~ ! [BL: nat > set_nat] :
( ( disjoi6798895846410478970at_nat @ BL @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ BL @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ n2 ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ BL @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ! [FL: nat > nat] :
( ( member_nat_nat @ FL
@ ( piE_nat_nat @ ( BL @ one_one_nat )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) )
=> ( ( member952132173341509300at_nat @ l
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) )
=> ~ ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( BL @ one_one_nat ) )
=> ( ( l @ X @ Xa )
= ( FL @ Xa ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ one_one_nat )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( BL @ J3 ) )
=> ( ( l @ X @ Xa )
= ( X @ J3 ) ) ) ) ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>BL fL. \<lbrakk>disjoint_family_on BL {..1}; \<Union> (BL ` {..1}) = {..<n}; {} \<notin> BL ` {..<1}; fL \<in> BL 1 \<rightarrow>\<^sub>E {..<t + 1}; L \<in> cube 1 (t + 1) \<rightarrow>\<^sub>E cube n (t + 1); \<forall>y\<in>cube 1 (t + 1). (\<forall>i\<in>BL 1. L y i = fL i) \<and> (\<forall>j<1. \<forall>i\<in>BL j. L y i = y j)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1224__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062BS_AfS_O_A_092_060lbrakk_062disjoint__family__on_ABS_A_123_O_Ok_125_059_A_092_060Union_062_A_IBS_A_096_A_123_O_Ok_125_J_A_061_A_123_O_O_060m_125_059_A_123_125_A_092_060notin_062_ABS_A_096_A_123_O_O_060k_125_059_AfS_A_092_060in_062_ABS_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_Am_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_062BS_Ak_O_AS_Ay_Ai_A_061_AfS_Ai_J_A_092_060and_062_A_I_092_060forall_062j_060k_O_A_092_060forall_062i_092_060in_062BS_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,
~ ! [BS: nat > set_nat] :
( ( disjoi6798895846410478970at_nat @ BS @ ( set_ord_atMost_nat @ k ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ BS @ ( set_ord_atMost_nat @ k ) ) )
= ( set_ord_lessThan_nat @ m2 ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ BS @ ( set_ord_lessThan_nat @ k ) ) )
=> ! [FS: nat > nat] :
( ( member_nat_nat @ FS
@ ( piE_nat_nat @ ( BS @ k )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) )
=> ( ( member952132173341509300at_nat @ s
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) )
=> ~ ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( BS @ k ) )
=> ( ( s @ X @ Xa )
= ( FS @ Xa ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ k )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( BS @ J3 ) )
=> ( ( s @ X @ Xa )
= ( X @ J3 ) ) ) ) ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>BS fS. \<lbrakk>disjoint_family_on BS {..k}; \<Union> (BS ` {..k}) = {..<m}; {} \<notin> BS ` {..<k}; fS \<in> BS k \<rightarrow>\<^sub>E {..<t + 1}; S \<in> cube k (t + 1) \<rightarrow>\<^sub>E cube m (t + 1); \<forall>y\<in>cube k (t + 1). (\<forall>i\<in>BS k. S y i = fS i) \<and> (\<forall>j<k. \<forall>i\<in>BS j. S y i = y j)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1225__092_060chi_062L__def,axiom,
( chi_L
= ( restri6011711336257459485at_nat
@ ^ [X2: nat > nat] :
( restrict_nat_nat_nat
@ ^ [Y5: nat > nat] : ( chi @ ( hales_join_nat @ X2 @ Y5 @ n2 @ m2 ) )
@ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
@ ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% \<chi>L_def
thf(fact_1226_cube__def,axiom,
( hales_cube
= ( ^ [N4: nat,T2: nat] :
( piE_nat_nat @ ( set_ord_lessThan_nat @ N4 )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ T2 ) ) ) ) ).
% cube_def
thf(fact_1227_is__subspace__def,axiom,
( hales_is_subspace
= ( ^ [S5: ( nat > nat ) > nat > nat,K4: nat,N4: nat,T2: nat] :
? [B4: nat > set_nat] :
( ( disjoi6798895846410478970at_nat @ B4 @ ( set_ord_atMost_nat @ K4 ) )
& ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ ( set_ord_atMost_nat @ K4 ) ) )
= ( set_ord_lessThan_nat @ N4 ) )
& ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B4 @ ( set_ord_lessThan_nat @ K4 ) ) )
& ? [F2: nat > nat] :
( ( member_nat_nat @ F2
@ ( piE_nat_nat @ ( B4 @ K4 )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ T2 ) ) )
& ( member952132173341509300at_nat @ S5
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ K4 @ T2 )
@ ^ [I4: nat > nat] : ( hales_cube @ N4 @ T2 ) ) )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ K4 @ T2 ) )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ ( B4 @ K4 ) )
=> ( ( S5 @ X2 @ Y5 )
= ( F2 @ Y5 ) ) )
& ! [J4: nat] :
( ( ord_less_nat @ J4 @ K4 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ ( B4 @ J4 ) )
=> ( ( S5 @ X2 @ Y5 )
= ( X2 @ J4 ) ) ) ) ) ) ) ) ) ) ).
% is_subspace_def
thf(fact_1228_dim1__subspace__elims_I3_J,axiom,
! [B3: nat > set_nat,N: nat,F: nat > nat,T: nat,S3: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B3 @ one_one_nat )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S3
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [I4: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ one_one_nat ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ J2 ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( X3 @ J2 ) ) ) ) ) )
=> ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( B3 @ one_one_nat ) )
=> ( ( S3 @ X @ Xa )
= ( F @ Xa ) ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( B3 @ zero_zero_nat ) )
=> ( ( S3 @ X @ Xa )
= ( X @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ).
% dim1_subspace_elims(3)
thf(fact_1229_dim1__subspace__elims_I4_J,axiom,
! [B3: nat > set_nat,N: nat,F: nat > nat,T: nat,S3: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B3 @ one_one_nat )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S3
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [I4: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ one_one_nat ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ J2 ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( X3 @ J2 ) ) ) ) ) )
=> ( ( B3 @ zero_zero_nat )
!= bot_bot_set_nat ) ) ) ) ) ) ) ).
% dim1_subspace_elims(4)
thf(fact_1230_dim1__subspace__elims_I2_J,axiom,
! [B3: nat > set_nat,N: nat,F: nat > nat,T: nat,S3: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B3 @ one_one_nat )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S3
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [I4: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ one_one_nat ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ J2 ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( X3 @ J2 ) ) ) ) ) )
=> ( ( inf_inf_set_nat @ ( B3 @ zero_zero_nat ) @ ( B3 @ one_one_nat ) )
= bot_bot_set_nat ) ) ) ) ) ) ) ).
% dim1_subspace_elims(2)
thf(fact_1231_dim1__subspace__elims_I1_J,axiom,
! [B3: nat > set_nat,N: nat,F: nat > nat,T: nat,S3: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B3 @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B3 @ one_one_nat )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ T ) ) )
=> ( ( member952132173341509300at_nat @ S3
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ one_one_nat @ T )
@ ^ [I4: nat > nat] : ( hales_cube @ N @ T ) ) )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ one_one_nat @ T ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ one_one_nat ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B3 @ J2 ) )
=> ( ( S3 @ X3 @ Xa2 )
= ( X3 @ J2 ) ) ) ) ) )
=> ( ( sup_sup_set_nat @ ( B3 @ zero_zero_nat ) @ ( B3 @ one_one_nat ) )
= ( set_ord_lessThan_nat @ N ) ) ) ) ) ) ) ) ).
% dim1_subspace_elims(1)
thf(fact_1232_cube__card,axiom,
! [N: nat,T: nat] :
( ( finite_card_nat_nat
@ ( piE_nat_nat @ ( set_ord_lessThan_nat @ N )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ T ) ) )
= ( power_power_nat @ T @ N ) ) ).
% cube_card
thf(fact_1233_m__props,axiom,
( ( ord_less_nat @ zero_zero_nat @ m2 )
& ! [M6: nat] :
( ( ord_less_eq_nat @ m2 @ M6 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ M6 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [S: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S @ k @ M6 @ t3 @ r @ Chi ) ) ) ) ).
% m_props
thf(fact_1234_n_H__props,axiom,
( ( ord_less_nat @ zero_zero_nat @ n )
& ! [N5: nat] :
( ( ord_less_eq_nat @ n @ N5 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N5 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ s2 ) ) )
=> ? [S: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S @ one_one_nat @ N5 @ t3 @ s2 @ Chi ) ) ) ) ).
% n'_props
thf(fact_1235__092_060chi_062__prop,axiom,
( member_nat_nat_nat @ chi
@ ( piE_nat_nat_nat @ ( hales_cube @ m @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% \<chi>_prop
thf(fact_1236__092_060open_062card_A_Icube_Am_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_J_A_061_Ar_A_094_A_It_A_L_A1_J_A_094_Am_092_060close_062,axiom,
( ( finite1794908990118856198at_nat
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
= ( power_power_nat @ r @ ( power_power_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) @ m2 ) ) ) ).
% \<open>card (cube m (t + 1) \<rightarrow>\<^sub>E {..<r}) = r ^ (t + 1) ^ m\<close>
thf(fact_1237_s__coloured,axiom,
( ( finite1794908990118856198at_nat
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
= s2 ) ).
% s_coloured
thf(fact_1238__092_060open_062_092_060chi_062S_A_092_060in_062_Acube_Am_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_092_060close_062,axiom,
( member_nat_nat_nat @ chi_S
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% \<open>\<chi>S \<in> cube m (t + 1) \<rightarrow>\<^sub>E {..<r}\<close>
thf(fact_1239__092_060open_062_092_060chi_062L__s_A_092_060in_062_Acube_An_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060s_125_092_060close_062,axiom,
( member_nat_nat_nat @ chi_L_s
@ ( piE_nat_nat_nat @ ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ s2 ) ) ) ).
% \<open>\<chi>L_s \<in> cube n (t + 1) \<rightarrow>\<^sub>E {..<s}\<close>
thf(fact_1240__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_H_O_A0_A_060_An_H_A_092_060and_062_A_I_092_060forall_062N_092_060ge_062n_H_O_A_092_060forall_062_092_060chi_062_O_A_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_060s_125_A_092_060longrightarrow_062_A_I_092_060exists_062S_O_Alayered__subspace_AS_A1_AN_At_As_A_092_060chi_062_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N6: nat] :
~ ( ( ord_less_nat @ zero_zero_nat @ N6 )
& ! [N5: nat] :
( ( ord_less_eq_nat @ N6 @ N5 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ N5 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ s2 ) ) )
=> ? [S: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S @ one_one_nat @ N5 @ t3 @ s2 @ Chi ) ) ) ) ).
% \<open>\<And>thesis. (\<And>n'. 0 < n' \<and> (\<forall>N\<ge>n'. \<forall>\<chi>. \<chi> \<in> cube N (t + 1) \<rightarrow>\<^sub>E {..<s} \<longrightarrow> (\<exists>S. layered_subspace S 1 N t s \<chi>)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1241__092_060open_062card_A_Icube_Am_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_J_A_061_Acard_A_123_O_O_060r_125_A_094_Acard_A_Icube_Am_A_It_A_L_A1_J_J_092_060close_062,axiom,
( ( finite1794908990118856198at_nat
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
= ( power_power_nat @ ( finite_card_nat @ ( set_ord_lessThan_nat @ r ) ) @ ( finite_card_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ) ).
% \<open>card (cube m (t + 1) \<rightarrow>\<^sub>E {..<r}) = card {..<r} ^ card (cube m (t + 1))\<close>
thf(fact_1242__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_A0_A_060_Am_A_092_060and_062_A_I_092_060forall_062M_H_092_060ge_062m_O_A_092_060forall_062_092_060chi_062_O_A_092_060chi_062_A_092_060in_062_Acube_AM_H_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_A_092_060longrightarrow_062_A_I_092_060exists_062S_O_Alayered__subspace_AS_Ak_AM_H_At_Ar_A_092_060chi_062_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [M4: nat] :
~ ( ( ord_less_nat @ zero_zero_nat @ M4 )
& ! [M6: nat] :
( ( ord_less_eq_nat @ M4 @ M6 )
=> ! [Chi: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi
@ ( piE_nat_nat_nat @ ( hales_cube @ M6 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [S: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S @ k @ M6 @ t3 @ r @ Chi ) ) ) ) ).
% \<open>\<And>thesis. (\<And>m. 0 < m \<and> (\<forall>M'\<ge>m. \<forall>\<chi>. \<chi> \<in> cube M' (t + 1) \<rightarrow>\<^sub>E {..<r} \<longrightarrow> (\<exists>S. layered_subspace S k M' t r \<chi>)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1243__092_060chi_062L__prop,axiom,
( member4402528950554000163at_nat @ chi_L
@ ( piE_na7569501297962130601at_nat @ ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] :
( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [J4: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ) ).
% \<chi>L_prop
thf(fact_1244_T_H__prop,axiom,
( member8881365325514865170at_nat @ t
@ ( piE_na5223350113562215832at_nat @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] :
( piE_nat_nat_nat_nat @ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [J4: nat > nat] : ( hales_cube @ ( plus_plus_nat @ n2 @ m2 ) @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ) ).
% T'_prop
thf(fact_1245_is__line__def,axiom,
( hales_is_line
= ( ^ [L4: nat > nat > nat,N4: nat,T2: nat] :
( ( member_nat_nat_nat2 @ L4
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ T2 )
@ ^ [I4: nat] : ( hales_cube @ N4 @ T2 ) ) )
& ! [J4: nat] :
( ( ord_less_nat @ J4 @ N4 )
=> ( ! [X2: nat] :
( ( ord_less_nat @ X2 @ T2 )
=> ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ T2 )
=> ( ( L4 @ X2 @ J4 )
= ( L4 @ Y5 @ J4 ) ) ) )
| ! [S6: nat] :
( ( ord_less_nat @ S6 @ T2 )
=> ( ( L4 @ S6 @ J4 )
= S6 ) ) ) )
& ? [J4: nat] :
( ( ord_less_nat @ J4 @ N4 )
& ! [S6: nat] :
( ( ord_less_nat @ S6 @ T2 )
=> ( ( L4 @ S6 @ J4 )
= S6 ) ) ) ) ) ) ).
% is_line_def
thf(fact_1246_dim0__layered__subspace__ex,axiom,
! [Chi2: ( nat > nat ) > nat,N: nat,T: nat,R: nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N @ ( plus_plus_nat @ T @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ R ) ) )
=> ? [S: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S @ zero_zero_nat @ N @ T @ R @ Chi2 ) ) ).
% dim0_layered_subspace_ex
thf(fact_1247_hj__def,axiom,
( hales_hj
= ( ^ [R3: nat,T2: nat] :
? [N7: nat] :
( ( ord_less_nat @ zero_zero_nat @ N7 )
& ! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ! [Chi3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N8 @ T2 )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ R3 ) ) )
=> ? [L4: nat > nat > nat,C3: nat] :
( ( ord_less_nat @ C3 @ R3 )
& ( hales_is_line @ L4 @ N8 @ T2 )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ L4 @ ( set_ord_lessThan_nat @ T2 ) ) )
=> ( ( Chi3 @ X2 )
= C3 ) ) ) ) ) ) ) ) ).
% hj_def
thf(fact_1248_lhj__def,axiom,
( hales_lhj
= ( ^ [R3: nat,T2: nat,K4: nat] :
? [N7: nat] :
( ( ord_less_nat @ zero_zero_nat @ N7 )
& ! [N8: nat] :
( ( ord_less_eq_nat @ N7 @ N8 )
=> ! [Chi3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi3
@ ( piE_nat_nat_nat @ ( hales_cube @ N8 @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ R3 ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ K4 @ N8 @ T2 @ R3 @ Chi3 ) ) ) ) ) ) ).
% lhj_def
thf(fact_1249__092_060phi_062__prop,axiom,
( bij_be1059735840858801910at_nat @ phi
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) )
@ ( set_ord_lessThan_nat @ s2 ) ) ).
% \<phi>_prop
thf(fact_1250__092_060open_062BL_A_096_A_123_O_O_0601_125_A_092_060union_062_ABvar_A_096_A_1231_O_O_060k_A_L_A1_125_A_061_ABL_A_096_A_123_O_O_0601_125_A_092_060union_062_A_123set__incr_An_A_IBS_Ai_J_A_124i_O_Ai_A_092_060in_062_A_123_O_O_060k_125_125_092_060close_062,axiom,
( ( sup_sup_set_set_nat @ ( image_nat_set_nat @ bl @ ( set_ord_lessThan_nat @ one_one_nat ) ) @ ( image_nat_set_nat @ bvar @ ( set_or4665077453230672383an_nat @ one_one_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ) )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ bl @ ( set_ord_lessThan_nat @ one_one_nat ) )
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [I4: nat] :
( ( Uu
= ( hales_set_incr @ n2 @ ( bs @ I4 ) ) )
& ( member_nat @ I4 @ ( set_ord_lessThan_nat @ k ) ) ) ) ) ) ).
% \<open>BL ` {..<1} \<union> Bvar ` {1..<k + 1} = BL ` {..<1} \<union> {set_incr n (BS i) |i. i \<in> {..<k}}\<close>
thf(fact_1251__092_060open_062_123_125_A_092_060notin_062_A_123set__incr_An_A_IBS_Ai_J_A_124i_O_Ai_A_092_060in_062_A_123_O_O_060k_125_125_092_060close_062,axiom,
~ ( member_set_nat @ bot_bot_set_nat
@ ( collect_set_nat
@ ^ [Uu: set_nat] :
? [I4: nat] :
( ( Uu
= ( hales_set_incr @ n2 @ ( bs @ I4 ) ) )
& ( member_nat @ I4 @ ( set_ord_lessThan_nat @ k ) ) ) ) ) ).
% \<open>{} \<notin> {set_incr n (BS i) |i. i \<in> {..<k}}\<close>
thf(fact_1252__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060phi_062_O_Abij__betw_A_092_060phi_062_A_Icube_Am_A_It_A_L_A1_J_A_092_060rightarrow_062_092_060_094sub_062E_A_123_O_O_060r_125_J_A_123_O_O_060s_125_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Phi: ( ( nat > nat ) > nat ) > nat] :
~ ( bij_be1059735840858801910at_nat @ Phi
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ r ) )
@ ( set_ord_lessThan_nat @ s2 ) ) ).
% \<open>\<And>thesis. (\<And>\<phi>. bij_betw \<phi> (cube m (t + 1) \<rightarrow>\<^sub>E {..<r}) {..<s} \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1253_Tset__def,axiom,
( tset
= ( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [I4: nat,S6: nat > nat] :
( ( Uu
= ( hales_join_nat @ ( l_line @ I4 ) @ S6 @ n2 @ m2 ) )
& ( member_nat @ I4 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
& ( member_nat_nat @ S6 @ ( image_3205354838064109189at_nat @ s @ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ) ) ) ).
% Tset_def
thf(fact_1254_one__dim__cube__eq__nat__set,axiom,
! [K2: nat] :
( bij_betw_nat_nat_nat
@ ^ [F2: nat > nat] : ( F2 @ zero_zero_nat )
@ ( hales_cube @ one_one_nat @ K2 )
@ ( set_ord_lessThan_nat @ K2 ) ) ).
% one_dim_cube_eq_nat_set
thf(fact_1255_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1256_T_H__def,axiom,
( t
= ( restri1704181820465610764at_nat
@ ^ [X2: nat > nat] :
( restri4446420529079022766at_nat
@ ^ [Y5: nat > nat] : ( hales_join_nat @ ( l_line @ ( X2 @ zero_zero_nat ) ) @ ( s @ Y5 ) @ n2 @ m2 )
@ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
@ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% T'_def
thf(fact_1257_T__def,axiom,
( t2
= ( restri4446420529079022766at_nat
@ ^ [X2: nat > nat] :
( t @ ( restrict_nat_nat @ X2 @ ( set_ord_lessThan_nat @ one_one_nat ) )
@ ( restrict_nat_nat
@ ^ [Y5: nat] : ( X2 @ ( plus_plus_nat @ Y5 @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ k ) ) )
@ ( hales_cube @ ( plus_plus_nat @ k @ one_one_nat ) @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% T_def
thf(fact_1258_BT__def,axiom,
( bt
= ( fun_upd_nat_set_nat @ ( restrict_nat_set_nat @ bvar @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ) @ ( plus_plus_nat @ k @ one_one_nat ) @ bstat ) ) ).
% BT_def
thf(fact_1259_L__line__def,axiom,
( l_line
= ( restrict_nat_nat_nat2
@ ^ [S6: nat] :
( l
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= S6 ) ) ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% L_line_def
thf(fact_1260_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_1261_nat__set__eq__one__dim__cube,axiom,
! [K2: nat] :
( bij_betw_nat_nat_nat2
@ ^ [X2: nat] :
( restrict_nat_nat
@ ^ [Y5: nat] : X2
@ ( set_ord_lessThan_nat @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K2 )
@ ( hales_cube @ one_one_nat @ K2 ) ) ).
% nat_set_eq_one_dim_cube
thf(fact_1262_split__cube_I2_J,axiom,
! [X4: nat > nat,K2: nat,T: nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ ( plus_plus_nat @ K2 @ one_one_nat ) @ T ) )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [Y5: nat] : ( X4 @ ( plus_plus_nat @ Y5 @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K2 ) )
@ ( hales_cube @ K2 @ T ) ) ) ).
% split_cube(2)
thf(fact_1263_split__cube_I1_J,axiom,
! [X4: nat > nat,K2: nat,T: nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ ( plus_plus_nat @ K2 @ one_one_nat ) @ T ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ X4 @ ( set_ord_lessThan_nat @ one_one_nat ) ) @ ( hales_cube @ one_one_nat @ T ) ) ) ).
% split_cube(1)
thf(fact_1264_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
@ ^ [Y5: nat > nat] : ( L3 @ ( Y5 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace
thf(fact_1265_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
@ ^ [Y5: nat > nat] : ( L3 @ ( Y5 @ 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_1266_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
@ ^ [Y5: nat > nat] : ( L3 @ ( Y5 @ zero_zero_nat ) )
@ ( hales_cube @ one_one_nat @ T ) )
@ one_one_nat
@ N
@ T ) ) ) ) ).
% line_is_dim1_subspace_t_ge_1
thf(fact_1267_line__subspace__s,axiom,
! [Chi2: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I4: nat > nat] : ( set_ord_lessThan_nat @ s2 ) ) )
=> ? [S: ( nat > nat ) > nat > nat] :
( ( hales_4261547300027266985ce_nat @ S @ one_one_nat @ n2 @ t3 @ s2 @ Chi2 )
& ( hales_is_line
@ ( restrict_nat_nat_nat2
@ ^ [S6: nat] :
( S
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
& ( ( P4 @ zero_zero_nat )
= S6 ) ) ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
@ n2
@ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% line_subspace_s
thf(fact_1268_T__class__def,axiom,
( t_class
= ( fun_up831482295316861124at_nat
@ ( restri901343962050523125at_nat
@ ^ [J4: nat] :
( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [I4: nat,S6: nat > nat] :
( ( Uu
= ( hales_join_nat @ ( l_line @ I4 ) @ S6 @ n2 @ m2 ) )
& ( member_nat @ I4 @ ( set_ord_lessThan_nat @ t3 ) )
& ( member_nat_nat @ S6 @ ( image_3205354838064109189at_nat @ s @ ( hales_classes @ k @ t3 @ J4 ) ) ) ) )
@ ( set_ord_atMost_nat @ k ) )
@ ( plus_plus_nat @ k @ one_one_nat )
@ ( insert_nat_nat
@ ( hales_join_nat @ ( l_line @ t3 )
@ ( fChoice_nat_nat
@ ^ [S6: nat > nat] : ( member_nat_nat @ S6 @ ( image_3205354838064109189at_nat @ s @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) )
@ n2
@ m2 )
@ bot_bot_set_nat_nat ) ) ) ).
% T_class_def
% Helper facts (6)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $true @ X4 @ Y )
= X4 ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X4: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X4: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X4 @ Y )
= X4 ) ).
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 ) )
= ( ? [X6: nat > nat] : ( P @ X6 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [T4: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ T4 @ ( plus_plus_nat @ k @ one_one_nat ) @ m @ t3 @ r @ chi ) ).
%------------------------------------------------------------------------------