TPTP Problem File: SLH0613^1.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_01452_065385__5919896_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1452 ( 752 unt; 177 typ; 0 def)
% Number of atoms : 3079 (1225 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 8970 ( 284 ~; 35 |; 160 &;7434 @)
% ( 0 <=>;1057 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 5 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 1734 (1734 >; 0 *; 0 +; 0 <<)
% Number of symbols : 171 ( 168 usr; 28 con; 0-6 aty)
% Number of variables : 3124 ( 221 ^;2844 !; 59 ?;3124 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:46:46.137
%------------------------------------------------------------------------------
% 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 (168)
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_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_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_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_Ocube,type,
hales_cube: nat > nat > set_nat_nat ).
thf(sy_c_Hales__Jewett_Ois__line,type,
hales_is_line: ( 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_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_M_Eo_J,type,
bot_bo5587768346753192576_nat_o: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_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_M_Eo_J,type,
bot_bo622770940136227119_nat_o: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
bot_bo1568108970253895006_nat_o: ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bo6348804412059337741_nat_o: ( ( nat > nat ) > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
bot_bo1514271634159724301_nat_o: ( nat > nat > nat ) > $o ).
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__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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_3521005150465447523at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat ) > nat ) > set_na6626867396258451522at_nat > 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_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_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_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_001t__Nat__Onat_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,
image_3941236537129881699at_nat: ( nat > ( nat > nat ) > ( nat > nat ) > nat ) > set_nat > set_na6626867396258451522at_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6393715451659844596at_nat: ( nat > ( nat > nat ) > nat > nat ) > set_nat > set_nat_nat_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_5809701139083627781at_nat: ( nat > ( nat > nat ) > nat ) > set_nat > set_nat_nat_nat2 ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6919068903512877573at_nat: ( nat > nat > nat > nat ) > set_nat > set_nat_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_nat_nat_nat: ( 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__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_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_ma____,type,
ma: nat ).
thf(sy_v_n_H____,type,
n: nat ).
thf(sy_v_n____,type,
n2: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_s____,type,
s2: nat ).
thf(sy_v_t,type,
t3: nat ).
thf(sy_v_x____,type,
x: nat ).
% Relevant facts (1268)
thf(fact_0_a_I1_J,axiom,
ord_less_nat @ ma @ ( plus_plus_nat @ k @ one_one_nat ) ).
% a(1)
thf(fact_1_a_I2_J,axiom,
ord_less_nat @ na @ ( plus_plus_nat @ k @ one_one_nat ) ).
% a(2)
thf(fact_2_a_I4_J,axiom,
member_nat @ x @ ( bvar @ ma ) ).
% a(4)
thf(fact_3_a_I5_J,axiom,
member_nat @ x @ ( bvar @ na ) ).
% a(5)
thf(fact_4_empty__iff,axiom,
! [C: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ C @ bot_bo7445843802507891576at_nat ) ).
% empty_iff
thf(fact_5_empty__iff,axiom,
! [C: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ C @ bot_bo945813143650711160at_nat ) ).
% empty_iff
thf(fact_6_empty__iff,axiom,
! [C: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ C @ bot_bo3919185967433191911at_nat ) ).
% empty_iff
thf(fact_7_empty__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat] :
~ ( member4402528950554000163at_nat @ C @ bot_bo2074992577060541142at_nat ) ).
% empty_iff
thf(fact_8_empty__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( member8881365325514865170at_nat @ C @ bot_bo2676777031303994949at_nat ) ).
% empty_iff
thf(fact_9_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_10_all__not__in__conv,axiom,
! [A: set_nat_nat_nat] :
( ( ! [X: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ X @ A ) )
= ( A = bot_bo7445843802507891576at_nat ) ) ).
% all_not_in_conv
thf(fact_11_all__not__in__conv,axiom,
! [A: set_nat_nat_nat2] :
( ( ! [X: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ X @ A ) )
= ( A = bot_bo945813143650711160at_nat ) ) ).
% all_not_in_conv
thf(fact_12_all__not__in__conv,axiom,
! [A: set_nat_nat_nat_nat] :
( ( ! [X: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ X @ A ) )
= ( A = bot_bo3919185967433191911at_nat ) ) ).
% all_not_in_conv
thf(fact_13_all__not__in__conv,axiom,
! [A: set_na6626867396258451522at_nat] :
( ( ! [X: ( nat > nat ) > ( nat > nat ) > nat] :
~ ( member4402528950554000163at_nat @ X @ A ) )
= ( A = bot_bo2074992577060541142at_nat ) ) ).
% all_not_in_conv
thf(fact_14_all__not__in__conv,axiom,
! [A: set_na7233567106578532785at_nat] :
( ( ! [X: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( member8881365325514865170at_nat @ X @ A ) )
= ( A = bot_bo2676777031303994949at_nat ) ) ).
% all_not_in_conv
thf(fact_15_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_16_Collect__empty__eq,axiom,
! [P: ( nat > nat > nat ) > $o] :
( ( ( collect_nat_nat_nat2 @ P )
= bot_bo7445843802507891576at_nat )
= ( ! [X: nat > nat > nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_17_Collect__empty__eq,axiom,
! [P: ( ( nat > nat ) > nat ) > $o] :
( ( ( collect_nat_nat_nat @ P )
= bot_bo945813143650711160at_nat )
= ( ! [X: ( nat > nat ) > nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_18_Collect__empty__eq,axiom,
! [P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( ( collec3567154360959927026at_nat @ P )
= bot_bo3919185967433191911at_nat )
= ( ! [X: ( nat > nat ) > nat > nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_19_Collect__empty__eq,axiom,
! [P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ( ( collec2410089373097230945at_nat @ P )
= bot_bo2074992577060541142at_nat )
= ( ! [X: ( nat > nat ) > ( nat > nat ) > nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_20_Collect__empty__eq,axiom,
! [P: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o] :
( ( ( collec6535634078845029456at_nat @ P )
= bot_bo2676777031303994949at_nat )
= ( ! [X: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_21_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_22_Collect__empty__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( ! [X: nat > nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_23_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_24_empty__Collect__eq,axiom,
! [P: ( nat > nat > nat ) > $o] :
( ( bot_bo7445843802507891576at_nat
= ( collect_nat_nat_nat2 @ P ) )
= ( ! [X: nat > nat > nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_25_empty__Collect__eq,axiom,
! [P: ( ( nat > nat ) > nat ) > $o] :
( ( bot_bo945813143650711160at_nat
= ( collect_nat_nat_nat @ P ) )
= ( ! [X: ( nat > nat ) > nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_26_empty__Collect__eq,axiom,
! [P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( bot_bo3919185967433191911at_nat
= ( collec3567154360959927026at_nat @ P ) )
= ( ! [X: ( nat > nat ) > nat > nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_27_empty__Collect__eq,axiom,
! [P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ( bot_bo2074992577060541142at_nat
= ( collec2410089373097230945at_nat @ P ) )
= ( ! [X: ( nat > nat ) > ( nat > nat ) > nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_28_empty__Collect__eq,axiom,
! [P: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o] :
( ( bot_bo2676777031303994949at_nat
= ( collec6535634078845029456at_nat @ P ) )
= ( ! [X: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_29_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_30_empty__Collect__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P ) )
= ( ! [X: nat > nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_31_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_32_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_33_fact3,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_lessThan_nat @ k ) )
=> ( ( inf_inf_set_nat @ ( bl @ zero_zero_nat ) @ ( hales_set_incr @ n2 @ ( bs @ X2 ) ) )
= bot_bot_set_nat ) ) ).
% fact3
thf(fact_34_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_35_emptyE,axiom,
! [A2: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ A2 @ bot_bo7445843802507891576at_nat ) ).
% emptyE
thf(fact_36_emptyE,axiom,
! [A2: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ A2 @ bot_bo945813143650711160at_nat ) ).
% emptyE
thf(fact_37_emptyE,axiom,
! [A2: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ A2 @ bot_bo3919185967433191911at_nat ) ).
% emptyE
thf(fact_38_emptyE,axiom,
! [A2: ( nat > nat ) > ( nat > nat ) > nat] :
~ ( member4402528950554000163at_nat @ A2 @ bot_bo2074992577060541142at_nat ) ).
% emptyE
thf(fact_39_emptyE,axiom,
! [A2: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( member8881365325514865170at_nat @ A2 @ bot_bo2676777031303994949at_nat ) ).
% emptyE
thf(fact_40_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_41_equals0D,axiom,
! [A: set_nat_nat_nat,A2: nat > nat > nat] :
( ( A = bot_bo7445843802507891576at_nat )
=> ~ ( member_nat_nat_nat2 @ A2 @ A ) ) ).
% equals0D
thf(fact_42_equals0D,axiom,
! [A: set_nat_nat_nat2,A2: ( nat > nat ) > nat] :
( ( A = bot_bo945813143650711160at_nat )
=> ~ ( member_nat_nat_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_43_equals0D,axiom,
! [A: set_nat_nat_nat_nat,A2: ( nat > nat ) > nat > nat] :
( ( A = bot_bo3919185967433191911at_nat )
=> ~ ( member952132173341509300at_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_44_equals0D,axiom,
! [A: set_na6626867396258451522at_nat,A2: ( nat > nat ) > ( nat > nat ) > nat] :
( ( A = bot_bo2074992577060541142at_nat )
=> ~ ( member4402528950554000163at_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_45_equals0D,axiom,
! [A: set_na7233567106578532785at_nat,A2: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( A = bot_bo2676777031303994949at_nat )
=> ~ ( member8881365325514865170at_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_46_equals0I,axiom,
! [A: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_47_equals0I,axiom,
! [A: set_nat_nat_nat] :
( ! [Y: nat > nat > nat] :
~ ( member_nat_nat_nat2 @ Y @ A )
=> ( A = bot_bo7445843802507891576at_nat ) ) ).
% equals0I
thf(fact_48_equals0I,axiom,
! [A: set_nat_nat_nat2] :
( ! [Y: ( nat > nat ) > nat] :
~ ( member_nat_nat_nat @ Y @ A )
=> ( A = bot_bo945813143650711160at_nat ) ) ).
% equals0I
thf(fact_49_equals0I,axiom,
! [A: set_nat_nat_nat_nat] :
( ! [Y: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ Y @ A )
=> ( A = bot_bo3919185967433191911at_nat ) ) ).
% equals0I
thf(fact_50_equals0I,axiom,
! [A: set_na6626867396258451522at_nat] :
( ! [Y: ( nat > nat ) > ( nat > nat ) > nat] :
~ ( member4402528950554000163at_nat @ Y @ A )
=> ( A = bot_bo2074992577060541142at_nat ) ) ).
% equals0I
thf(fact_51_equals0I,axiom,
! [A: set_na7233567106578532785at_nat] :
( ! [Y: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( member8881365325514865170at_nat @ Y @ A )
=> ( A = bot_bo2676777031303994949at_nat ) ) ).
% equals0I
thf(fact_52_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_53_ex__in__conv,axiom,
! [A: set_nat_nat_nat] :
( ( ? [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ A ) )
= ( A != bot_bo7445843802507891576at_nat ) ) ).
% ex_in_conv
thf(fact_54_ex__in__conv,axiom,
! [A: set_nat_nat_nat2] :
( ( ? [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ A ) )
= ( A != bot_bo945813143650711160at_nat ) ) ).
% ex_in_conv
thf(fact_55_ex__in__conv,axiom,
! [A: set_nat_nat_nat_nat] :
( ( ? [X: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X @ A ) )
= ( A != bot_bo3919185967433191911at_nat ) ) ).
% ex_in_conv
thf(fact_56_ex__in__conv,axiom,
! [A: set_na6626867396258451522at_nat] :
( ( ? [X: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X @ A ) )
= ( A != bot_bo2074992577060541142at_nat ) ) ).
% ex_in_conv
thf(fact_57_ex__in__conv,axiom,
! [A: set_na7233567106578532785at_nat] :
( ( ? [X: ( nat > nat ) > ( nat > nat ) > nat > nat] : ( member8881365325514865170at_nat @ X @ A ) )
= ( A != bot_bo2676777031303994949at_nat ) ) ).
% ex_in_conv
thf(fact_58__C0_C,axiom,
na = zero_zero_nat ).
% "0"
thf(fact_59_a_I3_J,axiom,
ma != na ).
% a(3)
thf(fact_60_IntI,axiom,
! [C: nat > nat > nat,A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ A )
=> ( ( member_nat_nat_nat2 @ C @ B )
=> ( member_nat_nat_nat2 @ C @ ( inf_in5274420515160781174at_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_61_IntI,axiom,
! [C: ( nat > nat ) > nat,A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ A )
=> ( ( member_nat_nat_nat @ C @ B )
=> ( member_nat_nat_nat @ C @ ( inf_in7997761893158376566at_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_62_IntI,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ A )
=> ( ( member952132173341509300at_nat @ C @ B )
=> ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_63_IntI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ A )
=> ( ( member4402528950554000163at_nat @ C @ B )
=> ( member4402528950554000163at_nat @ C @ ( inf_in6213014276851238612at_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_64_IntI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ A )
=> ( ( member8881365325514865170at_nat @ C @ B )
=> ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_65_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_66_Int__iff,axiom,
! [C: nat > nat > nat,A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( inf_in5274420515160781174at_nat @ A @ B ) )
= ( ( member_nat_nat_nat2 @ C @ A )
& ( member_nat_nat_nat2 @ C @ B ) ) ) ).
% Int_iff
thf(fact_67_Int__iff,axiom,
! [C: ( nat > nat ) > nat,A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( inf_in7997761893158376566at_nat @ A @ B ) )
= ( ( member_nat_nat_nat @ C @ A )
& ( member_nat_nat_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_68_Int__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A @ B ) )
= ( ( member952132173341509300at_nat @ C @ A )
& ( member952132173341509300at_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_69_Int__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( inf_in6213014276851238612at_nat @ A @ B ) )
= ( ( member4402528950554000163at_nat @ C @ A )
& ( member4402528950554000163at_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_70_Int__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A @ B ) )
= ( ( member8881365325514865170at_nat @ C @ A )
& ( member8881365325514865170at_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_71_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_72_lessThan__eq__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ( set_ord_lessThan_nat @ X3 )
= ( set_ord_lessThan_nat @ Y2 ) )
= ( X3 = Y2 ) ) ).
% lessThan_eq_iff
thf(fact_73_lessThan__iff,axiom,
! [I: nat > nat > nat,K: nat > nat > nat] :
( ( member_nat_nat_nat2 @ I @ ( set_or3808701207811398603at_nat @ K ) )
= ( ord_less_nat_nat_nat2 @ I @ K ) ) ).
% lessThan_iff
thf(fact_74_lessThan__iff,axiom,
! [I: ( nat > nat ) > nat,K: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ I @ ( set_or2699333443382148811at_nat @ K ) )
= ( ord_less_nat_nat_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_75_lessThan__iff,axiom,
! [I: ( nat > nat ) > nat > nat,K: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I @ ( set_or7562748684798938298at_nat @ K ) )
= ( ord_le4629963735342356977at_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_76_lessThan__iff,axiom,
! [I: ( nat > nat ) > ( nat > nat ) > nat,K: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ I @ ( set_or6177432841829679145at_nat @ K ) )
= ( ord_le7877100967975825120at_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_77_lessThan__iff,axiom,
! [I: ( nat > nat ) > ( nat > nat ) > nat > nat,K: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ I @ ( set_or2490836252891414040at_nat @ K ) )
= ( ord_le338063099783794255at_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_78_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_79_assms_I2_J,axiom,
ord_less_eq_nat @ one_one_nat @ k ).
% assms(2)
thf(fact_80_fact1,axiom,
( ( inf_inf_set_nat @ ( hales_set_incr @ n2 @ ( bs @ k ) ) @ ( bl @ one_one_nat ) )
= bot_bot_set_nat ) ).
% fact1
thf(fact_81_fact4,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( 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 ) ) )
=> ( ( X2 != Xa )
=> ( ( inf_inf_set_nat @ ( hales_set_incr @ n2 @ ( bs @ X2 ) ) @ ( hales_set_incr @ n2 @ ( bs @ Xa ) ) )
= bot_bot_set_nat ) ) ) ) ).
% fact4
thf(fact_82_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_83_n__def,axiom,
( n2
= ( plus_plus_nat @ n @ d ) ) ).
% n_def
thf(fact_84_assms_I5_J,axiom,
ord_less_nat @ zero_zero_nat @ r ).
% assms(5)
thf(fact_85_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_86_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_87_Bvar__def,axiom,
( bvar
= ( ^ [I2: nat] : ( if_set_nat @ ( I2 = zero_zero_nat ) @ ( bl @ zero_zero_nat ) @ ( hales_set_incr @ n2 @ ( bs @ ( minus_minus_nat @ I2 @ one_one_nat ) ) ) ) ) ) ).
% Bvar_def
thf(fact_88_fact5,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) )
=> ( ( inf_inf_set_nat @ ( bvar @ X2 ) @ bstat )
= bot_bot_set_nat ) ) ).
% fact5
thf(fact_89_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_90_less__imp__neq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_91_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_92_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_93_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_94_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X4: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X4 )
=> ( P @ Y3 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_95_antisym__conv3,axiom,
! [Y2: nat,X3: nat] :
( ~ ( ord_less_nat @ Y2 @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_96_linorder__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_cases
thf(fact_97_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_98_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_99_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_100_mem__Collect__eq,axiom,
! [A2: nat > nat > nat,P: ( nat > nat > nat ) > $o] :
( ( member_nat_nat_nat2 @ A2 @ ( collect_nat_nat_nat2 @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_101_mem__Collect__eq,axiom,
! [A2: ( nat > nat ) > nat,P: ( ( nat > nat ) > nat ) > $o] :
( ( member_nat_nat_nat @ A2 @ ( collect_nat_nat_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_102_mem__Collect__eq,axiom,
! [A2: ( nat > nat ) > nat > nat,P: ( ( nat > nat ) > nat > nat ) > $o] :
( ( member952132173341509300at_nat @ A2 @ ( collec3567154360959927026at_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_103_mem__Collect__eq,axiom,
! [A2: ( nat > nat ) > ( nat > nat ) > nat,P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ( member4402528950554000163at_nat @ A2 @ ( collec2410089373097230945at_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_104_mem__Collect__eq,axiom,
! [A2: ( nat > nat ) > ( nat > nat ) > nat > nat,P: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o] :
( ( member8881365325514865170at_nat @ A2 @ ( collec6535634078845029456at_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_105_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_106_mem__Collect__eq,axiom,
! [A2: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A2 @ ( collect_nat_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_107_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_108_Collect__mem__eq,axiom,
! [A: set_nat_nat_nat] :
( ( collect_nat_nat_nat2
@ ^ [X: nat > nat > nat] : ( member_nat_nat_nat2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_109_Collect__mem__eq,axiom,
! [A: set_nat_nat_nat2] :
( ( collect_nat_nat_nat
@ ^ [X: ( nat > nat ) > nat] : ( member_nat_nat_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
! [A: set_nat_nat_nat_nat] :
( ( collec3567154360959927026at_nat
@ ^ [X: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_111_Collect__mem__eq,axiom,
! [A: set_na6626867396258451522at_nat] :
( ( collec2410089373097230945at_nat
@ ^ [X: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_112_Collect__mem__eq,axiom,
! [A: set_na7233567106578532785at_nat] :
( ( collec6535634078845029456at_nat
@ ^ [X: ( nat > nat ) > ( nat > nat ) > nat > nat] : ( member8881365325514865170at_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_113_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_116_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_117_Collect__cong,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X4: nat > nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat_nat @ P )
= ( collect_nat_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_118_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_119_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_120_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_121_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_122_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_123_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_124_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_125_IntE,axiom,
! [C: nat > nat > nat,A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( inf_in5274420515160781174at_nat @ A @ B ) )
=> ~ ( ( member_nat_nat_nat2 @ C @ A )
=> ~ ( member_nat_nat_nat2 @ C @ B ) ) ) ).
% IntE
thf(fact_126_IntE,axiom,
! [C: ( nat > nat ) > nat,A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( inf_in7997761893158376566at_nat @ A @ B ) )
=> ~ ( ( member_nat_nat_nat @ C @ A )
=> ~ ( member_nat_nat_nat @ C @ B ) ) ) ).
% IntE
thf(fact_127_IntE,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A @ B ) )
=> ~ ( ( member952132173341509300at_nat @ C @ A )
=> ~ ( member952132173341509300at_nat @ C @ B ) ) ) ).
% IntE
thf(fact_128_IntE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( inf_in6213014276851238612at_nat @ A @ B ) )
=> ~ ( ( member4402528950554000163at_nat @ C @ A )
=> ~ ( member4402528950554000163at_nat @ C @ B ) ) ) ).
% IntE
thf(fact_129_IntE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A @ B ) )
=> ~ ( ( member8881365325514865170at_nat @ C @ A )
=> ~ ( member8881365325514865170at_nat @ C @ B ) ) ) ).
% IntE
thf(fact_130_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_131_IntD1,axiom,
! [C: nat > nat > nat,A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( inf_in5274420515160781174at_nat @ A @ B ) )
=> ( member_nat_nat_nat2 @ C @ A ) ) ).
% IntD1
thf(fact_132_IntD1,axiom,
! [C: ( nat > nat ) > nat,A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( inf_in7997761893158376566at_nat @ A @ B ) )
=> ( member_nat_nat_nat @ C @ A ) ) ).
% IntD1
thf(fact_133_IntD1,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A @ B ) )
=> ( member952132173341509300at_nat @ C @ A ) ) ).
% IntD1
thf(fact_134_IntD1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( inf_in6213014276851238612at_nat @ A @ B ) )
=> ( member4402528950554000163at_nat @ C @ A ) ) ).
% IntD1
thf(fact_135_IntD1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A @ B ) )
=> ( member8881365325514865170at_nat @ C @ A ) ) ).
% IntD1
thf(fact_136_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_137_IntD2,axiom,
! [C: nat > nat > nat,A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( inf_in5274420515160781174at_nat @ A @ B ) )
=> ( member_nat_nat_nat2 @ C @ B ) ) ).
% IntD2
thf(fact_138_IntD2,axiom,
! [C: ( nat > nat ) > nat,A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( inf_in7997761893158376566at_nat @ A @ B ) )
=> ( member_nat_nat_nat @ C @ B ) ) ).
% IntD2
thf(fact_139_IntD2,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A @ B ) )
=> ( member952132173341509300at_nat @ C @ B ) ) ).
% IntD2
thf(fact_140_IntD2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( inf_in6213014276851238612at_nat @ A @ B ) )
=> ( member4402528950554000163at_nat @ C @ B ) ) ).
% IntD2
thf(fact_141_IntD2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A @ B ) )
=> ( member8881365325514865170at_nat @ C @ B ) ) ).
% IntD2
thf(fact_142_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_143_Int__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_144_Int__absorb,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_145_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_146_Int__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_147_Int__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_148_linorder__neqE,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_149_order__less__asym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% order_less_asym
thf(fact_150_linorder__neq__iff,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
= ( ( ord_less_nat @ X3 @ Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_151_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_152_order__less__trans,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X3 @ Z ) ) ) ).
% order_less_trans
thf(fact_153_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_154_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_155_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_156_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_157_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_158_order__less__not__sym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% order_less_not_sym
thf(fact_159_order__less__imp__triv,axiom,
! [X3: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_160_linorder__less__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ).
% linorder_less_linear
thf(fact_161_order__less__imp__not__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_162_order__less__imp__not__eq2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_163_order__less__imp__not__less,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_164_lessThan__strict__subset__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_165_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_166_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_167_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_168_bot__set__def,axiom,
( bot_bo7445843802507891576at_nat
= ( collect_nat_nat_nat2 @ bot_bo1514271634159724301_nat_o ) ) ).
% bot_set_def
thf(fact_169_bot__set__def,axiom,
( bot_bo945813143650711160at_nat
= ( collect_nat_nat_nat @ bot_bo6348804412059337741_nat_o ) ) ).
% bot_set_def
thf(fact_170_bot__set__def,axiom,
( bot_bo3919185967433191911at_nat
= ( collec3567154360959927026at_nat @ bot_bo1568108970253895006_nat_o ) ) ).
% bot_set_def
thf(fact_171_bot__set__def,axiom,
( bot_bo2074992577060541142at_nat
= ( collec2410089373097230945at_nat @ bot_bo622770940136227119_nat_o ) ) ).
% bot_set_def
thf(fact_172_bot__set__def,axiom,
( bot_bo2676777031303994949at_nat
= ( collec6535634078845029456at_nat @ bot_bo5587768346753192576_nat_o ) ) ).
% bot_set_def
thf(fact_173_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_174_disjoint__iff__not__equal,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_175_disjoint__iff__not__equal,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( ( inf_in5274420515160781174at_nat @ A @ B )
= bot_bo7445843802507891576at_nat )
= ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A )
=> ! [Y4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ Y4 @ B )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_176_disjoint__iff__not__equal,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( ( inf_in7997761893158376566at_nat @ A @ B )
= bot_bo945813143650711160at_nat )
= ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
=> ! [Y4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ Y4 @ B )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_177_disjoint__iff__not__equal,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( ( inf_in2949407623404935909at_nat @ A @ B )
= bot_bo3919185967433191911at_nat )
= ( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A )
=> ! [Y4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ Y4 @ B )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_178_disjoint__iff__not__equal,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( ( inf_in6213014276851238612at_nat @ A @ B )
= bot_bo2074992577060541142at_nat )
= ( ! [X: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X @ A )
=> ! [Y4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ Y4 @ B )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_179_disjoint__iff__not__equal,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( ( inf_in6008378084349164867at_nat @ A @ B )
= bot_bo2676777031303994949at_nat )
= ( ! [X: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X @ A )
=> ! [Y4: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ Y4 @ B )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_180_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_181_Int__empty__right,axiom,
! [A: set_nat_nat_nat] :
( ( inf_in5274420515160781174at_nat @ A @ bot_bo7445843802507891576at_nat )
= bot_bo7445843802507891576at_nat ) ).
% Int_empty_right
thf(fact_182_Int__empty__right,axiom,
! [A: set_nat_nat_nat2] :
( ( inf_in7997761893158376566at_nat @ A @ bot_bo945813143650711160at_nat )
= bot_bo945813143650711160at_nat ) ).
% Int_empty_right
thf(fact_183_Int__empty__right,axiom,
! [A: set_nat_nat_nat_nat] :
( ( inf_in2949407623404935909at_nat @ A @ bot_bo3919185967433191911at_nat )
= bot_bo3919185967433191911at_nat ) ).
% Int_empty_right
thf(fact_184_Int__empty__right,axiom,
! [A: set_na6626867396258451522at_nat] :
( ( inf_in6213014276851238612at_nat @ A @ bot_bo2074992577060541142at_nat )
= bot_bo2074992577060541142at_nat ) ).
% Int_empty_right
thf(fact_185_Int__empty__right,axiom,
! [A: set_na7233567106578532785at_nat] :
( ( inf_in6008378084349164867at_nat @ A @ bot_bo2676777031303994949at_nat )
= bot_bo2676777031303994949at_nat ) ).
% Int_empty_right
thf(fact_186_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_187_Int__empty__left,axiom,
! [B: set_nat_nat_nat] :
( ( inf_in5274420515160781174at_nat @ bot_bo7445843802507891576at_nat @ B )
= bot_bo7445843802507891576at_nat ) ).
% Int_empty_left
thf(fact_188_Int__empty__left,axiom,
! [B: set_nat_nat_nat2] :
( ( inf_in7997761893158376566at_nat @ bot_bo945813143650711160at_nat @ B )
= bot_bo945813143650711160at_nat ) ).
% Int_empty_left
thf(fact_189_Int__empty__left,axiom,
! [B: set_nat_nat_nat_nat] :
( ( inf_in2949407623404935909at_nat @ bot_bo3919185967433191911at_nat @ B )
= bot_bo3919185967433191911at_nat ) ).
% Int_empty_left
thf(fact_190_Int__empty__left,axiom,
! [B: set_na6626867396258451522at_nat] :
( ( inf_in6213014276851238612at_nat @ bot_bo2074992577060541142at_nat @ B )
= bot_bo2074992577060541142at_nat ) ).
% Int_empty_left
thf(fact_191_Int__empty__left,axiom,
! [B: set_na7233567106578532785at_nat] :
( ( inf_in6008378084349164867at_nat @ bot_bo2676777031303994949at_nat @ B )
= bot_bo2676777031303994949at_nat ) ).
% Int_empty_left
thf(fact_192_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_193_disjoint__iff,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( ( inf_in5274420515160781174at_nat @ A @ B )
= bot_bo7445843802507891576at_nat )
= ( ! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A )
=> ~ ( member_nat_nat_nat2 @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_194_disjoint__iff,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( ( inf_in7997761893158376566at_nat @ A @ B )
= bot_bo945813143650711160at_nat )
= ( ! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A )
=> ~ ( member_nat_nat_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_195_disjoint__iff,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( ( inf_in2949407623404935909at_nat @ A @ B )
= bot_bo3919185967433191911at_nat )
= ( ! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A )
=> ~ ( member952132173341509300at_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_196_disjoint__iff,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( ( inf_in6213014276851238612at_nat @ A @ B )
= bot_bo2074992577060541142at_nat )
= ( ! [X: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X @ A )
=> ~ ( member4402528950554000163at_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_197_disjoint__iff,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( ( inf_in6008378084349164867at_nat @ A @ B )
= bot_bo2676777031303994949at_nat )
= ( ! [X: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X @ A )
=> ~ ( member8881365325514865170at_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_198_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_nat @ X4 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_199_Int__emptyI,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ! [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A )
=> ~ ( member_nat_nat_nat2 @ X4 @ B ) )
=> ( ( inf_in5274420515160781174at_nat @ A @ B )
= bot_bo7445843802507891576at_nat ) ) ).
% Int_emptyI
thf(fact_200_Int__emptyI,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A )
=> ~ ( member_nat_nat_nat @ X4 @ B ) )
=> ( ( inf_in7997761893158376566at_nat @ A @ B )
= bot_bo945813143650711160at_nat ) ) ).
% Int_emptyI
thf(fact_201_Int__emptyI,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ! [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ A )
=> ~ ( member952132173341509300at_nat @ X4 @ B ) )
=> ( ( inf_in2949407623404935909at_nat @ A @ B )
= bot_bo3919185967433191911at_nat ) ) ).
% Int_emptyI
thf(fact_202_Int__emptyI,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ! [X4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X4 @ A )
=> ~ ( member4402528950554000163at_nat @ X4 @ B ) )
=> ( ( inf_in6213014276851238612at_nat @ A @ B )
= bot_bo2074992577060541142at_nat ) ) ).
% Int_emptyI
thf(fact_203_Int__emptyI,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ! [X4: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X4 @ A )
=> ~ ( member8881365325514865170at_nat @ X4 @ B ) )
=> ( ( inf_in6008378084349164867at_nat @ A @ B )
= bot_bo2676777031303994949at_nat ) ) ).
% Int_emptyI
thf(fact_204_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_205_bot_Onot__eq__extremum,axiom,
! [A2: set_nat_nat_nat] :
( ( A2 != bot_bo7445843802507891576at_nat )
= ( ord_le6871433888996735800at_nat @ bot_bo7445843802507891576at_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_206_bot_Onot__eq__extremum,axiom,
! [A2: set_nat_nat_nat2] :
( ( A2 != bot_bo945813143650711160at_nat )
= ( ord_le371403230139555384at_nat @ bot_bo945813143650711160at_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_207_bot_Onot__eq__extremum,axiom,
! [A2: set_nat_nat_nat_nat] :
( ( A2 != bot_bo3919185967433191911at_nat )
= ( ord_le6177938698872215975at_nat @ bot_bo3919185967433191911at_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_208_bot_Onot__eq__extremum,axiom,
! [A2: set_na6626867396258451522at_nat] :
( ( A2 != bot_bo2074992577060541142at_nat )
= ( ord_le2785809691299232406at_nat @ bot_bo2074992577060541142at_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_209_bot_Onot__eq__extremum,axiom,
! [A2: set_na7233567106578532785at_nat] :
( ( A2 != bot_bo2676777031303994949at_nat )
= ( ord_le7586516898478368261at_nat @ bot_bo2676777031303994949at_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_210_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_211_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_212_bot_Oextremum__strict,axiom,
! [A2: set_nat_nat_nat] :
~ ( ord_le6871433888996735800at_nat @ A2 @ bot_bo7445843802507891576at_nat ) ).
% bot.extremum_strict
thf(fact_213_bot_Oextremum__strict,axiom,
! [A2: set_nat_nat_nat2] :
~ ( ord_le371403230139555384at_nat @ A2 @ bot_bo945813143650711160at_nat ) ).
% bot.extremum_strict
thf(fact_214_bot_Oextremum__strict,axiom,
! [A2: set_nat_nat_nat_nat] :
~ ( ord_le6177938698872215975at_nat @ A2 @ bot_bo3919185967433191911at_nat ) ).
% bot.extremum_strict
thf(fact_215_bot_Oextremum__strict,axiom,
! [A2: set_na6626867396258451522at_nat] :
~ ( ord_le2785809691299232406at_nat @ A2 @ bot_bo2074992577060541142at_nat ) ).
% bot.extremum_strict
thf(fact_216_bot_Oextremum__strict,axiom,
! [A2: set_na7233567106578532785at_nat] :
~ ( ord_le7586516898478368261at_nat @ A2 @ bot_bo2676777031303994949at_nat ) ).
% bot.extremum_strict
thf(fact_217_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_218_Bstat__def,axiom,
( bstat
= ( sup_sup_set_nat @ ( hales_set_incr @ n2 @ ( bs @ k ) ) @ ( bl @ one_one_nat ) ) ) ).
% Bstat_def
thf(fact_219_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_220_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_221_add__less__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_222_add__less__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_223_less__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_224_less__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_225__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_226_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_227_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_228_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_229_order__refl,axiom,
! [X3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_230_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_231_dual__order_Orefl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_232_add__right__cancel,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_233_add__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_234_image__eqI,axiom,
! [B2: nat,F: nat > nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_235_image__eqI,axiom,
! [B2: set_nat,F: nat > set_nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_236_image__eqI,axiom,
! [B2: nat > nat,F: ( nat > nat ) > nat > nat,X3: nat > nat,A: set_nat_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_237_image__eqI,axiom,
! [B2: nat,F: ( nat > nat > nat ) > nat,X3: nat > nat > nat,A: set_nat_nat_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat_nat_nat2 @ X3 @ A )
=> ( member_nat @ B2 @ ( image_913610194320715013at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_238_image__eqI,axiom,
! [B2: nat,F: ( ( nat > nat ) > nat ) > nat,X3: ( nat > nat ) > nat,A: set_nat_nat_nat2] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat_nat_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_7809927846809980933at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_239_image__eqI,axiom,
! [B2: nat > nat > nat,F: nat > nat > nat > nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat_nat_nat2 @ B2 @ ( image_6919068903512877573at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_240_image__eqI,axiom,
! [B2: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat_nat_nat @ B2 @ ( image_5809701139083627781at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_241_image__eqI,axiom,
! [B2: nat,F: ( ( nat > nat ) > nat > nat ) > nat,X3: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member952132173341509300at_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_8194121248528334964at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_242_image__eqI,axiom,
! [B2: ( nat > nat ) > nat > nat,F: nat > ( nat > nat ) > nat > nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member952132173341509300at_nat @ B2 @ ( image_6393715451659844596at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_243_image__eqI,axiom,
! [B2: nat > nat > nat,F: ( nat > nat > nat ) > nat > nat > nat,X3: nat > nat > nat,A: set_nat_nat_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat_nat_nat2 @ X3 @ A )
=> ( member_nat_nat_nat2 @ B2 @ ( image_1896091034194584419at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_244_UnCI,axiom,
! [C: nat > nat > nat,B: set_nat_nat_nat,A: set_nat_nat_nat] :
( ( ~ ( member_nat_nat_nat2 @ C @ B )
=> ( member_nat_nat_nat2 @ C @ A ) )
=> ( member_nat_nat_nat2 @ C @ ( sup_su3334021163961628176at_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_245_UnCI,axiom,
! [C: ( nat > nat ) > nat,B: set_nat_nat_nat2,A: set_nat_nat_nat2] :
( ( ~ ( member_nat_nat_nat @ C @ B )
=> ( member_nat_nat_nat @ C @ A ) )
=> ( member_nat_nat_nat @ C @ ( sup_su6057362541959223568at_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_246_UnCI,axiom,
! [C: ( nat > nat ) > nat > nat,B: set_nat_nat_nat_nat,A: set_nat_nat_nat_nat] :
( ( ~ ( member952132173341509300at_nat @ C @ B )
=> ( member952132173341509300at_nat @ C @ A ) )
=> ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_247_UnCI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,B: set_na6626867396258451522at_nat,A: set_na6626867396258451522at_nat] :
( ( ~ ( member4402528950554000163at_nat @ C @ B )
=> ( member4402528950554000163at_nat @ C @ A ) )
=> ( member4402528950554000163at_nat @ C @ ( sup_su481250237928500590at_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_248_UnCI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,B: set_na7233567106578532785at_nat,A: set_na7233567106578532785at_nat] :
( ( ~ ( member8881365325514865170at_nat @ C @ B )
=> ( member8881365325514865170at_nat @ C @ A ) )
=> ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_249_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_250_UnCI,axiom,
! [C: set_nat,B: set_set_nat,A: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ A ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_251_Un__iff,axiom,
! [C: nat > nat > nat,A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( sup_su3334021163961628176at_nat @ A @ B ) )
= ( ( member_nat_nat_nat2 @ C @ A )
| ( member_nat_nat_nat2 @ C @ B ) ) ) ).
% Un_iff
thf(fact_252_Un__iff,axiom,
! [C: ( nat > nat ) > nat,A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( sup_su6057362541959223568at_nat @ A @ B ) )
= ( ( member_nat_nat_nat @ C @ A )
| ( member_nat_nat_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_253_Un__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A @ B ) )
= ( ( member952132173341509300at_nat @ C @ A )
| ( member952132173341509300at_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_254_Un__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( sup_su481250237928500590at_nat @ A @ B ) )
= ( ( member4402528950554000163at_nat @ C @ A )
| ( member4402528950554000163at_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_255_Un__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A @ B ) )
= ( ( member8881365325514865170at_nat @ C @ A )
| ( member8881365325514865170at_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_256_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_257_Un__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
| ( member_set_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_258__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_259_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_260_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_261_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_262_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_263_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_264_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y2 ) )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_265_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y2: nat] :
( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_266_add__cancel__right__right,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_267_add__cancel__right__left,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_268_add__cancel__left__right,axiom,
! [A2: nat,B2: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_269_add__cancel__left__left,axiom,
! [B2: nat,A2: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_270_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_271_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_272_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_273_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_274_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_275_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_276_add__diff__cancel__right_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_277_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_278_add__diff__cancel__left_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_279_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_280_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_281_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_282_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_283_image__empty,axiom,
! [F: nat > nat > nat > nat] :
( ( image_6919068903512877573at_nat @ F @ bot_bot_set_nat )
= bot_bo7445843802507891576at_nat ) ).
% image_empty
thf(fact_284_image__empty,axiom,
! [F: nat > ( nat > nat ) > nat] :
( ( image_5809701139083627781at_nat @ F @ bot_bot_set_nat )
= bot_bo945813143650711160at_nat ) ).
% image_empty
thf(fact_285_image__empty,axiom,
! [F: ( nat > nat > nat ) > nat] :
( ( image_913610194320715013at_nat @ F @ bot_bo7445843802507891576at_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_286_image__empty,axiom,
! [F: ( ( nat > nat ) > nat ) > nat] :
( ( image_7809927846809980933at_nat @ F @ bot_bo945813143650711160at_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_287_image__empty,axiom,
! [F: nat > ( nat > nat ) > nat > nat] :
( ( image_6393715451659844596at_nat @ F @ bot_bot_set_nat )
= bot_bo3919185967433191911at_nat ) ).
% image_empty
thf(fact_288_image__empty,axiom,
! [F: ( ( nat > nat ) > nat > nat ) > nat] :
( ( image_8194121248528334964at_nat @ F @ bot_bo3919185967433191911at_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_289_image__empty,axiom,
! [F: nat > ( nat > nat ) > ( nat > nat ) > nat] :
( ( image_3941236537129881699at_nat @ F @ bot_bot_set_nat )
= bot_bo2074992577060541142at_nat ) ).
% image_empty
thf(fact_290_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_291_empty__is__image,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_292_empty__is__image,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( image_3205354838064109189at_nat @ F @ A ) )
= ( A = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_293_empty__is__image,axiom,
! [F: ( nat > nat > nat ) > nat,A: set_nat_nat_nat] :
( ( bot_bot_set_nat
= ( image_913610194320715013at_nat @ F @ A ) )
= ( A = bot_bo7445843802507891576at_nat ) ) ).
% empty_is_image
thf(fact_294_empty__is__image,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( bot_bot_set_nat
= ( image_7809927846809980933at_nat @ F @ A ) )
= ( A = bot_bo945813143650711160at_nat ) ) ).
% empty_is_image
thf(fact_295_empty__is__image,axiom,
! [F: nat > nat > nat > nat,A: set_nat] :
( ( bot_bo7445843802507891576at_nat
= ( image_6919068903512877573at_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_296_empty__is__image,axiom,
! [F: nat > ( nat > nat ) > nat,A: set_nat] :
( ( bot_bo945813143650711160at_nat
= ( image_5809701139083627781at_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_297_empty__is__image,axiom,
! [F: ( ( nat > nat ) > nat > nat ) > nat,A: set_nat_nat_nat_nat] :
( ( bot_bot_set_nat
= ( image_8194121248528334964at_nat @ F @ A ) )
= ( A = bot_bo3919185967433191911at_nat ) ) ).
% empty_is_image
thf(fact_298_empty__is__image,axiom,
! [F: nat > ( nat > nat ) > nat > nat,A: set_nat] :
( ( bot_bo3919185967433191911at_nat
= ( image_6393715451659844596at_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_299_empty__is__image,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat,A: set_na6626867396258451522at_nat] :
( ( bot_bot_set_nat
= ( image_3521005150465447523at_nat @ F @ A ) )
= ( A = bot_bo2074992577060541142at_nat ) ) ).
% empty_is_image
thf(fact_300_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_301_image__is__empty,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( ( image_nat_set_nat @ F @ A )
= bot_bot_set_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_302_image__is__empty,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( ( image_3205354838064109189at_nat @ F @ A )
= bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_303_image__is__empty,axiom,
! [F: ( nat > nat > nat ) > nat,A: set_nat_nat_nat] :
( ( ( image_913610194320715013at_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bo7445843802507891576at_nat ) ) ).
% image_is_empty
thf(fact_304_image__is__empty,axiom,
! [F: ( ( nat > nat ) > nat ) > nat,A: set_nat_nat_nat2] :
( ( ( image_7809927846809980933at_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bo945813143650711160at_nat ) ) ).
% image_is_empty
thf(fact_305_image__is__empty,axiom,
! [F: nat > nat > nat > nat,A: set_nat] :
( ( ( image_6919068903512877573at_nat @ F @ A )
= bot_bo7445843802507891576at_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_306_image__is__empty,axiom,
! [F: nat > ( nat > nat ) > nat,A: set_nat] :
( ( ( image_5809701139083627781at_nat @ F @ A )
= bot_bo945813143650711160at_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_307_image__is__empty,axiom,
! [F: ( ( nat > nat ) > nat > nat ) > nat,A: set_nat_nat_nat_nat] :
( ( ( image_8194121248528334964at_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bo3919185967433191911at_nat ) ) ).
% image_is_empty
thf(fact_308_image__is__empty,axiom,
! [F: nat > ( nat > nat ) > nat > nat,A: set_nat] :
( ( ( image_6393715451659844596at_nat @ F @ A )
= bot_bo3919185967433191911at_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_309_image__is__empty,axiom,
! [F: ( ( nat > nat ) > ( nat > nat ) > nat ) > nat,A: set_na6626867396258451522at_nat] :
( ( ( image_3521005150465447523at_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bo2074992577060541142at_nat ) ) ).
% image_is_empty
thf(fact_310_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_311_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_312_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_313_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_314_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_315_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_316_lessThan__subset__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X3 ) @ ( set_ord_lessThan_nat @ Y2 ) )
= ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_317_Un__empty,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 ) ) ) ).
% Un_empty
thf(fact_318_Un__empty,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 ) ) ) ).
% Un_empty
thf(fact_319_Un__empty,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( ( sup_su3334021163961628176at_nat @ A @ B )
= bot_bo7445843802507891576at_nat )
= ( ( A = bot_bo7445843802507891576at_nat )
& ( B = bot_bo7445843802507891576at_nat ) ) ) ).
% Un_empty
thf(fact_320_Un__empty,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( ( sup_su6057362541959223568at_nat @ A @ B )
= bot_bo945813143650711160at_nat )
= ( ( A = bot_bo945813143650711160at_nat )
& ( B = bot_bo945813143650711160at_nat ) ) ) ).
% Un_empty
thf(fact_321_Un__empty,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( ( sup_su3836648520750444671at_nat @ A @ B )
= bot_bo3919185967433191911at_nat )
= ( ( A = bot_bo3919185967433191911at_nat )
& ( B = bot_bo3919185967433191911at_nat ) ) ) ).
% Un_empty
thf(fact_322_Un__empty,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( ( sup_su481250237928500590at_nat @ A @ B )
= bot_bo2074992577060541142at_nat )
= ( ( A = bot_bo2074992577060541142at_nat )
& ( B = bot_bo2074992577060541142at_nat ) ) ) ).
% Un_empty
thf(fact_323_Un__empty,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( ( sup_su8594648213498475741at_nat @ A @ B )
= bot_bo2676777031303994949at_nat )
= ( ( A = bot_bo2676777031303994949at_nat )
& ( B = bot_bo2676777031303994949at_nat ) ) ) ).
% Un_empty
thf(fact_324_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_325_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_326_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_327_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_328_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_329_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_330_Un__Int__eq_I1_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_331_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_332_Un__Int__eq_I2_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_333_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_334_Un__Int__eq_I3_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( inf_inf_set_set_nat @ S @ ( sup_sup_set_set_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_335_Un__Int__eq_I4_J,axiom,
! [T: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T @ ( sup_sup_set_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_336_Un__Int__eq_I4_J,axiom,
! [T: set_set_nat,S: set_set_nat] :
( ( inf_inf_set_set_nat @ T @ ( sup_sup_set_set_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_337_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_338_Int__Un__eq_I1_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_339_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_340_Int__Un__eq_I2_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_341_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_342_Int__Un__eq_I3_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( sup_sup_set_set_nat @ S @ ( inf_inf_set_set_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_343_Int__Un__eq_I4_J,axiom,
! [T: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T @ ( inf_inf_set_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_344_Int__Un__eq_I4_J,axiom,
! [T: set_set_nat,S: set_set_nat] :
( ( sup_sup_set_set_nat @ T @ ( inf_inf_set_set_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_345_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_346_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_347_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_348_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_349_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_350_diff__add__zero,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_351_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_352_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_353_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_354_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_355_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_356_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_357_calculation,axiom,
~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ bt @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ) ) ).
% calculation
thf(fact_358_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_359_le__cases3,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_360_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_361_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
& ( ord_le9059583361652607317at_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_362_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_363_ord__eq__le__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( A2 = B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_364_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_365_ord__le__eq__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_366_order__antisym,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_367_order__antisym,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% order_antisym
thf(fact_368_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_369_order_Otrans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_370_order__trans,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_371_order__trans,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ Z )
=> ( ord_le9059583361652607317at_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_372_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_373_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_374_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B5 @ A5 )
& ( ord_le9059583361652607317at_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_375_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_376_dual__order_Oantisym,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_377_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_378_dual__order_Otrans,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C @ B2 )
=> ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_379_diff__add,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% diff_add
thf(fact_380_le__add__diff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_381_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_382_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_383_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_384_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_385_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_386_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_387_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_388_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A2 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_389_UnE,axiom,
! [C: nat > nat > nat,A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ ( sup_su3334021163961628176at_nat @ A @ B ) )
=> ( ~ ( member_nat_nat_nat2 @ C @ A )
=> ( member_nat_nat_nat2 @ C @ B ) ) ) ).
% UnE
thf(fact_390_UnE,axiom,
! [C: ( nat > nat ) > nat,A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ ( sup_su6057362541959223568at_nat @ A @ B ) )
=> ( ~ ( member_nat_nat_nat @ C @ A )
=> ( member_nat_nat_nat @ C @ B ) ) ) ).
% UnE
thf(fact_391_UnE,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A @ B ) )
=> ( ~ ( member952132173341509300at_nat @ C @ A )
=> ( member952132173341509300at_nat @ C @ B ) ) ) ).
% UnE
thf(fact_392_UnE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ ( sup_su481250237928500590at_nat @ A @ B ) )
=> ( ~ ( member4402528950554000163at_nat @ C @ A )
=> ( member4402528950554000163at_nat @ C @ B ) ) ) ).
% UnE
thf(fact_393_UnE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A @ B ) )
=> ( ~ ( member8881365325514865170at_nat @ C @ A )
=> ( member8881365325514865170at_nat @ C @ B ) ) ) ).
% UnE
thf(fact_394_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_395_UnE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( ~ ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% UnE
thf(fact_396_UnI1,axiom,
! [C: nat > nat > nat,A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ A )
=> ( member_nat_nat_nat2 @ C @ ( sup_su3334021163961628176at_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_397_UnI1,axiom,
! [C: ( nat > nat ) > nat,A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ A )
=> ( member_nat_nat_nat @ C @ ( sup_su6057362541959223568at_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_398_UnI1,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ A )
=> ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_399_UnI1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ A )
=> ( member4402528950554000163at_nat @ C @ ( sup_su481250237928500590at_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_400_UnI1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ A )
=> ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_401_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_402_UnI1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_403_UnI2,axiom,
! [C: nat > nat > nat,B: set_nat_nat_nat,A: set_nat_nat_nat] :
( ( member_nat_nat_nat2 @ C @ B )
=> ( member_nat_nat_nat2 @ C @ ( sup_su3334021163961628176at_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_404_UnI2,axiom,
! [C: ( nat > nat ) > nat,B: set_nat_nat_nat2,A: set_nat_nat_nat2] :
( ( member_nat_nat_nat @ C @ B )
=> ( member_nat_nat_nat @ C @ ( sup_su6057362541959223568at_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_405_UnI2,axiom,
! [C: ( nat > nat ) > nat > nat,B: set_nat_nat_nat_nat,A: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ B )
=> ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_406_UnI2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat,B: set_na6626867396258451522at_nat,A: set_na6626867396258451522at_nat] :
( ( member4402528950554000163at_nat @ C @ B )
=> ( member4402528950554000163at_nat @ C @ ( sup_su481250237928500590at_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_407_UnI2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,B: set_na7233567106578532785at_nat,A: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ B )
=> ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_408_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_409_UnI2,axiom,
! [C: set_nat,B: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_410_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) )
| ? [X: nat] :
( ( member_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_411_bex__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ? [X: set_nat] :
( ( member_set_nat @ X @ ( sup_sup_set_set_nat @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ X ) )
| ? [X: set_nat] :
( ( member_set_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_412_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_413_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat @ X3 @ A )
=> ( member_set_nat @ ( F @ X3 ) @ ( image_nat_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_414_imageI,axiom,
! [X3: nat > nat,A: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat @ ( F @ X3 ) @ ( image_3205354838064109189at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_415_imageI,axiom,
! [X3: nat > nat > nat,A: set_nat_nat_nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_913610194320715013at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_416_imageI,axiom,
! [X3: ( nat > nat ) > nat,A: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_7809927846809980933at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_417_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > nat > nat > nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat_nat_nat2 @ ( F @ X3 ) @ ( image_6919068903512877573at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_418_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat_nat_nat @ ( F @ X3 ) @ ( image_5809701139083627781at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_419_imageI,axiom,
! [X3: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,F: ( ( nat > nat ) > nat > nat ) > nat] :
( ( member952132173341509300at_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_8194121248528334964at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_420_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > ( nat > nat ) > nat > nat] :
( ( member_nat @ X3 @ A )
=> ( member952132173341509300at_nat @ ( F @ X3 ) @ ( image_6393715451659844596at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_421_imageI,axiom,
! [X3: nat > nat > nat,A: set_nat_nat_nat,F: ( nat > nat > nat ) > nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A )
=> ( member_nat_nat_nat2 @ ( F @ X3 ) @ ( image_1896091034194584419at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_422_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( P @ X ) )
& ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_423_ball__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ! [X: set_nat] :
( ( member_set_nat @ X @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( P @ X ) )
& ! [X: set_nat] :
( ( member_set_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_424_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_425_Un__assoc,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_426_image__Un,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( image_3205354838064109189at_nat @ F @ ( sup_sup_set_nat_nat @ A @ B ) )
= ( sup_sup_set_nat_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ ( image_3205354838064109189at_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_427_image__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_428_image__Un,axiom,
! [F: nat > set_nat,A: set_nat,B: set_nat] :
( ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_429_image__Un,axiom,
! [F: set_nat > nat,A: set_set_nat,B: set_set_nat] :
( ( image_set_nat_nat @ F @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_set_nat_nat @ F @ A ) @ ( image_set_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_430_image__Un,axiom,
! [F: set_nat > set_nat,A: set_set_nat,B: set_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_set_nat @ ( image_7916887816326733075et_nat @ F @ A ) @ ( image_7916887816326733075et_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_431_psubsetD,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le6871433888996735800at_nat @ A @ B )
=> ( ( member_nat_nat_nat2 @ C @ A )
=> ( member_nat_nat_nat2 @ C @ B ) ) ) ).
% psubsetD
thf(fact_432_psubsetD,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le371403230139555384at_nat @ A @ B )
=> ( ( member_nat_nat_nat @ C @ A )
=> ( member_nat_nat_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_433_psubsetD,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat,C: ( nat > nat ) > nat > nat] :
( ( ord_le6177938698872215975at_nat @ A @ B )
=> ( ( member952132173341509300at_nat @ C @ A )
=> ( member952132173341509300at_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_434_psubsetD,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat,C: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le2785809691299232406at_nat @ A @ B )
=> ( ( member4402528950554000163at_nat @ C @ A )
=> ( member4402528950554000163at_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_435_psubsetD,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat,C: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( ord_le7586516898478368261at_nat @ A @ B )
=> ( ( member8881365325514865170at_nat @ C @ A )
=> ( member8881365325514865170at_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_436_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_437_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_438_Un__absorb,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_439_image__iff,axiom,
! [Z: set_nat,F: nat > set_nat,A: set_nat] :
( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_440_image__iff,axiom,
! [Z: nat > nat,F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ Z @ ( image_3205354838064109189at_nat @ F @ A ) )
= ( ? [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_441_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_442_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_443_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_444_bex__imageD,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_nat_set_nat @ F @ A ) )
& ( P @ X2 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_445_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
& ( P @ X2 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_446_bex__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,P: ( nat > nat ) > $o] :
( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( image_3205354838064109189at_nat @ F @ A ) )
& ( P @ X2 ) )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_447_image__cong,axiom,
! [M3: set_nat_nat,N3: set_nat_nat,F: ( nat > nat ) > nat > nat,G: ( nat > nat ) > nat > nat] :
( ( M3 = N3 )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ N3 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_3205354838064109189at_nat @ F @ M3 )
= ( image_3205354838064109189at_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_448_image__cong,axiom,
! [M3: set_nat,N3: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M3 = N3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N3 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_set_nat @ F @ M3 )
= ( image_nat_set_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_449_image__cong,axiom,
! [M3: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
( ( M3 = N3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N3 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_nat @ F @ M3 )
= ( image_nat_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_450_ball__imageD,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_nat_set_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_451_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_452_ball__imageD,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,P: ( nat > nat ) > $o] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( image_3205354838064109189at_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_453_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_454_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_455_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_456_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_457_antisym,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_458_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_459_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: set_nat,F: nat > set_nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_460_rev__image__eqI,axiom,
! [X3: nat > nat,A: set_nat_nat,B2: nat > nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_461_rev__image__eqI,axiom,
! [X3: nat > nat > nat,A: set_nat_nat_nat,B2: nat,F: ( nat > nat > nat ) > nat] :
( ( member_nat_nat_nat2 @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_913610194320715013at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_462_rev__image__eqI,axiom,
! [X3: ( nat > nat ) > nat,A: set_nat_nat_nat2,B2: nat,F: ( ( nat > nat ) > nat ) > nat] :
( ( member_nat_nat_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_7809927846809980933at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_463_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: nat > nat > nat,F: nat > nat > nat > nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat_nat_nat2 @ B2 @ ( image_6919068903512877573at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_464_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: ( nat > nat ) > nat,F: nat > ( nat > nat ) > nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat_nat_nat @ B2 @ ( image_5809701139083627781at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_465_rev__image__eqI,axiom,
! [X3: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: nat,F: ( ( nat > nat ) > nat > nat ) > nat] :
( ( member952132173341509300at_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_8194121248528334964at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_466_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: ( nat > nat ) > nat > nat,F: nat > ( nat > nat ) > nat > nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member952132173341509300at_nat @ B2 @ ( image_6393715451659844596at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_467_rev__image__eqI,axiom,
! [X3: nat > nat > nat,A: set_nat_nat_nat,B2: nat > nat > nat,F: ( nat > nat > nat ) > nat > nat > nat] :
( ( member_nat_nat_nat2 @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat_nat_nat2 @ B2 @ ( image_1896091034194584419at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_468_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_469_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_470_Un__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_471_Un__left__absorb,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 ) ) ).
% Un_left_absorb
thf(fact_472_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_473_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_474_Un__left__commute,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) )
= ( sup_sup_set_set_nat @ B @ ( sup_sup_set_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_475_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_476_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_477_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B5 )
& ( ord_le9059583361652607317at_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_478_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_479_order__subst1,axiom,
! [A2: nat,F: set_nat_nat > nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_480_order__subst1,axiom,
! [A2: set_nat_nat,F: nat > set_nat_nat,B2: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_481_order__subst1,axiom,
! [A2: set_nat_nat,F: set_nat_nat > set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_482_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_483_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_484_order__subst2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_485_order__subst2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_486_order__eq__refl,axiom,
! [X3: nat,Y2: nat] :
( ( X3 = Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_487_order__eq__refl,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat] :
( ( X3 = Y2 )
=> ( ord_le9059583361652607317at_nat @ X3 @ Y2 ) ) ).
% order_eq_refl
thf(fact_488_linorder__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_linear
thf(fact_489_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_490_ord__eq__le__subst,axiom,
! [A2: set_nat_nat,F: nat > set_nat_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_491_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat_nat > nat,B2: set_nat_nat,C: set_nat_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_492_ord__eq__le__subst,axiom,
! [A2: set_nat_nat,F: set_nat_nat > set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_493_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_494_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_495_ord__le__eq__subst,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_496_ord__le__eq__subst,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_le9059583361652607317at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_497_linorder__le__cases,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_le_cases
thf(fact_498_order__antisym__conv,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_499_order__antisym__conv,axiom,
! [Y2: set_nat_nat,X3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X3 )
=> ( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_500_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M3: nat] :
( ( P @ X3 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_501_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_502_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_503_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_504_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_505_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_506_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_507_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_508_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_509_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_510_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_511_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_512_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_513_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_514_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_515_diff__right__commute,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_516_diff__diff__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_517_add__implies__diff,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_518_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_519_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_520_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_521_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_522_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_523_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
? [C3: nat] :
( B5
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_524_add__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_525_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C4: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C4 ) ) ) ).
% less_eqE
thf(fact_526_add__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_527_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_528_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_529_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_530_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_531_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_532_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_533_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_534_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_535_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_536_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_537_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_538_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_539_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_540_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_541_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_542_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_543_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_544_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_545_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_546_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_547_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_548_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_549_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_550_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_551_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_552_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_553_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_554_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_555_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_556_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_557_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_558_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_559_Un__empty__left,axiom,
! [B: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_560_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_561_Un__empty__left,axiom,
! [B: set_nat_nat_nat] :
( ( sup_su3334021163961628176at_nat @ bot_bo7445843802507891576at_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_562_Un__empty__left,axiom,
! [B: set_nat_nat_nat2] :
( ( sup_su6057362541959223568at_nat @ bot_bo945813143650711160at_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_563_Un__empty__left,axiom,
! [B: set_nat_nat_nat_nat] :
( ( sup_su3836648520750444671at_nat @ bot_bo3919185967433191911at_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_564_Un__empty__left,axiom,
! [B: set_na6626867396258451522at_nat] :
( ( sup_su481250237928500590at_nat @ bot_bo2074992577060541142at_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_565_Un__empty__left,axiom,
! [B: set_na7233567106578532785at_nat] :
( ( sup_su8594648213498475741at_nat @ bot_bo2676777031303994949at_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_566_Un__empty__right,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ bot_bot_set_set_nat )
= A ) ).
% Un_empty_right
thf(fact_567_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_568_Un__empty__right,axiom,
! [A: set_nat_nat_nat] :
( ( sup_su3334021163961628176at_nat @ A @ bot_bo7445843802507891576at_nat )
= A ) ).
% Un_empty_right
thf(fact_569_Un__empty__right,axiom,
! [A: set_nat_nat_nat2] :
( ( sup_su6057362541959223568at_nat @ A @ bot_bo945813143650711160at_nat )
= A ) ).
% Un_empty_right
thf(fact_570_Un__empty__right,axiom,
! [A: set_nat_nat_nat_nat] :
( ( sup_su3836648520750444671at_nat @ A @ bot_bo3919185967433191911at_nat )
= A ) ).
% Un_empty_right
thf(fact_571_Un__empty__right,axiom,
! [A: set_na6626867396258451522at_nat] :
( ( sup_su481250237928500590at_nat @ A @ bot_bo2074992577060541142at_nat )
= A ) ).
% Un_empty_right
thf(fact_572_Un__empty__right,axiom,
! [A: set_na7233567106578532785at_nat] :
( ( sup_su8594648213498475741at_nat @ A @ bot_bo2676777031303994949at_nat )
= A ) ).
% Un_empty_right
thf(fact_573_Un__Int__distrib2,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ C2 ) @ A )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ A ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_574_Un__Int__distrib2,axiom,
! [B: set_set_nat,C2: set_set_nat,A: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B @ C2 ) @ A )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B @ A ) @ ( sup_sup_set_set_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_575_Int__Un__distrib2,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ A ) @ ( inf_inf_set_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_576_Int__Un__distrib2,axiom,
! [B: set_set_nat,C2: set_set_nat,A: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B @ C2 ) @ A )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B @ A ) @ ( inf_inf_set_set_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_577_Un__Int__distrib,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_578_Un__Int__distrib,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( inf_inf_set_set_nat @ B @ C2 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_579_Int__Un__distrib,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_580_Int__Un__distrib,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_581_Un__Int__crazy,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ B @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ B @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_582_Un__Int__crazy,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ B @ C2 ) ) @ ( inf_inf_set_set_nat @ C2 @ A ) )
= ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ B @ C2 ) ) @ ( sup_sup_set_set_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_583_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_584_order__le__imp__less__or__eq,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ( ord_less_set_nat_nat @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_585_linorder__le__less__linear,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
| ( ord_less_nat @ Y2 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_586_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_587_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_set_nat_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_588_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_589_order__less__le__subst1,axiom,
! [A2: set_nat_nat,F: nat > set_nat_nat,B2: nat,C: nat] :
( ( ord_less_set_nat_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_590_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat_nat > nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_591_order__less__le__subst1,axiom,
! [A2: set_nat_nat,F: set_nat_nat > set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_592_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_593_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_594_order__le__less__subst2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_595_order__le__less__subst2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_596_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_597_order__le__less__subst1,axiom,
! [A2: set_nat_nat,F: nat > set_nat_nat,B2: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_set_nat_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_598_order__less__le__trans,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_nat @ X3 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_599_order__less__le__trans,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_less_set_nat_nat @ X3 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ Z )
=> ( ord_less_set_nat_nat @ X3 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_600_order__le__less__trans,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X3 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_601_order__le__less__trans,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ( ord_less_set_nat_nat @ Y2 @ Z )
=> ( ord_less_set_nat_nat @ X3 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_602_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_603_order__neq__le__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2 != B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_604_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_605_order__le__neq__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_606_order__less__imp__le,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_607_order__less__imp__le,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ X3 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_608_linorder__not__less,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% linorder_not_less
thf(fact_609_linorder__not__le,axiom,
! [X3: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X3 ) ) ).
% linorder_not_le
thf(fact_610_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_611_order__less__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_612_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_613_order__le__less,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_less_set_nat_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_614_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_615_dual__order_Ostrict__implies__order,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_616_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_617_order_Ostrict__implies__order,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_618_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_619_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [B5: set_nat_nat,A5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B5 @ A5 )
& ~ ( ord_le9059583361652607317at_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_620_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_621_dual__order_Ostrict__trans2,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C @ B2 )
=> ( ord_less_set_nat_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_622_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_623_dual__order_Ostrict__trans1,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ord_less_set_nat_nat @ C @ B2 )
=> ( ord_less_set_nat_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_624_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_625_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [B5: set_nat_nat,A5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_626_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_627_dual__order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B5: set_nat_nat,A5: set_nat_nat] :
( ( ord_less_set_nat_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_628_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_629_order_Ostrict__iff__not,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B5 )
& ~ ( ord_le9059583361652607317at_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_630_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_631_order_Ostrict__trans2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ord_less_set_nat_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_632_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_633_order_Ostrict__trans1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat_nat @ B2 @ C )
=> ( ord_less_set_nat_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_634_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_635_order_Ostrict__iff__order,axiom,
( ord_less_set_nat_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_636_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_637_order_Oorder__iff__strict,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ( ord_less_set_nat_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_638_not__le__imp__less,axiom,
! [Y2: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ord_less_nat @ X3 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_639_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_640_less__le__not__le,axiom,
( ord_less_set_nat_nat
= ( ^ [X: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y4 )
& ~ ( ord_le9059583361652607317at_nat @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_641_antisym__conv2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_642_antisym__conv2,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ( ~ ( ord_less_set_nat_nat @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_643_antisym__conv1,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_644_antisym__conv1,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat] :
( ~ ( ord_less_set_nat_nat @ X3 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_645_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_646_nless__le,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ~ ( ord_less_set_nat_nat @ A2 @ B2 ) )
= ( ~ ( ord_le9059583361652607317at_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_647_leI,axiom,
! [X3: nat,Y2: nat] :
( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ).
% leI
thf(fact_648_leD,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y2 ) ) ).
% leD
thf(fact_649_leD,axiom,
! [Y2: set_nat_nat,X3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X3 )
=> ~ ( ord_less_set_nat_nat @ X3 @ Y2 ) ) ).
% leD
thf(fact_650_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_651_not__psubset__empty,axiom,
! [A: set_nat_nat_nat] :
~ ( ord_le6871433888996735800at_nat @ A @ bot_bo7445843802507891576at_nat ) ).
% not_psubset_empty
thf(fact_652_not__psubset__empty,axiom,
! [A: set_nat_nat_nat2] :
~ ( ord_le371403230139555384at_nat @ A @ bot_bo945813143650711160at_nat ) ).
% not_psubset_empty
thf(fact_653_not__psubset__empty,axiom,
! [A: set_nat_nat_nat_nat] :
~ ( ord_le6177938698872215975at_nat @ A @ bot_bo3919185967433191911at_nat ) ).
% not_psubset_empty
thf(fact_654_not__psubset__empty,axiom,
! [A: set_na6626867396258451522at_nat] :
~ ( ord_le2785809691299232406at_nat @ A @ bot_bo2074992577060541142at_nat ) ).
% not_psubset_empty
thf(fact_655_not__psubset__empty,axiom,
! [A: set_na7233567106578532785at_nat] :
~ ( ord_le7586516898478368261at_nat @ A @ bot_bo2676777031303994949at_nat ) ).
% not_psubset_empty
thf(fact_656_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_657_bot_Oextremum,axiom,
! [A2: set_nat_nat_nat] : ( ord_le3211623285424100676at_nat @ bot_bo7445843802507891576at_nat @ A2 ) ).
% bot.extremum
thf(fact_658_bot_Oextremum,axiom,
! [A2: set_nat_nat_nat2] : ( ord_le5934964663421696068at_nat @ bot_bo945813143650711160at_nat @ A2 ) ).
% bot.extremum
thf(fact_659_bot_Oextremum,axiom,
! [A2: set_nat_nat_nat_nat] : ( ord_le5260717879541182899at_nat @ bot_bo3919185967433191911at_nat @ A2 ) ).
% bot.extremum
thf(fact_660_bot_Oextremum,axiom,
! [A2: set_na6626867396258451522at_nat] : ( ord_le973658574027395234at_nat @ bot_bo2074992577060541142at_nat @ A2 ) ).
% bot.extremum
thf(fact_661_bot_Oextremum,axiom,
! [A2: set_na7233567106578532785at_nat] : ( ord_le8099187209609443857at_nat @ bot_bo2676777031303994949at_nat @ A2 ) ).
% bot.extremum
thf(fact_662_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_663_bot_Oextremum,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% bot.extremum
thf(fact_664_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_665_bot_Oextremum__unique,axiom,
! [A2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ bot_bo7445843802507891576at_nat )
= ( A2 = bot_bo7445843802507891576at_nat ) ) ).
% bot.extremum_unique
thf(fact_666_bot_Oextremum__unique,axiom,
! [A2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ A2 @ bot_bo945813143650711160at_nat )
= ( A2 = bot_bo945813143650711160at_nat ) ) ).
% bot.extremum_unique
thf(fact_667_bot_Oextremum__unique,axiom,
! [A2: set_nat_nat_nat_nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ bot_bo3919185967433191911at_nat )
= ( A2 = bot_bo3919185967433191911at_nat ) ) ).
% bot.extremum_unique
thf(fact_668_bot_Oextremum__unique,axiom,
! [A2: set_na6626867396258451522at_nat] :
( ( ord_le973658574027395234at_nat @ A2 @ bot_bo2074992577060541142at_nat )
= ( A2 = bot_bo2074992577060541142at_nat ) ) ).
% bot.extremum_unique
thf(fact_669_bot_Oextremum__unique,axiom,
! [A2: set_na7233567106578532785at_nat] :
( ( ord_le8099187209609443857at_nat @ A2 @ bot_bo2676777031303994949at_nat )
= ( A2 = bot_bo2676777031303994949at_nat ) ) ).
% bot.extremum_unique
thf(fact_670_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_671_bot_Oextremum__unique,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% bot.extremum_unique
thf(fact_672_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_673_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A2 @ bot_bo7445843802507891576at_nat )
=> ( A2 = bot_bo7445843802507891576at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_674_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ A2 @ bot_bo945813143650711160at_nat )
=> ( A2 = bot_bo945813143650711160at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_675_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat_nat_nat_nat] :
( ( ord_le5260717879541182899at_nat @ A2 @ bot_bo3919185967433191911at_nat )
=> ( A2 = bot_bo3919185967433191911at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_676_bot_Oextremum__uniqueI,axiom,
! [A2: set_na6626867396258451522at_nat] :
( ( ord_le973658574027395234at_nat @ A2 @ bot_bo2074992577060541142at_nat )
=> ( A2 = bot_bo2074992577060541142at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_677_bot_Oextremum__uniqueI,axiom,
! [A2: set_na7233567106578532785at_nat] :
( ( ord_le8099187209609443857at_nat @ A2 @ bot_bo2676777031303994949at_nat )
=> ( A2 = bot_bo2676777031303994949at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_678_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_679_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_680_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_681_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_682_add__nonpos__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_683_add__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_684_add__increasing2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_685_add__decreasing2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_686_add__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_687_add__decreasing,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_688_add__less__le__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_689_add__le__less__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_690_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_691_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_692_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_693_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_694_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_695_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_696_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_697_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_698_add__strict__increasing2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_699_add__strict__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_700_add__pos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_701_add__nonpos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_702_add__nonneg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_703_add__neg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_704_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_705_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ( ( ord_less_nat @ A2 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_706_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_707_add__right__imp__eq,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_708_add__left__imp__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_709_add_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_710_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B5: nat] : ( plus_plus_nat @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_711_add_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_712_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_713_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_714_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_715_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_716_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_717_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_718_linorder__neqE__nat,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_719_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_720_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_721_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_722_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_723_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_724_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_725_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_726_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_727_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_728_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_729_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_730_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_731_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_732_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_733_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_734_add__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_735_add__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_736_add__strict__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_737_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_738_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_739_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_740_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_741_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_742_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_743_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_744_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_745_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_746_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_747_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_748_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_749_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_750_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_751_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_752_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_753_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_754_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_755_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_756_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_757_pos__add__strict,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_758_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ! [C4: nat] :
( ( B2
= ( plus_plus_nat @ A2 @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_759_add__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_760_add__neg__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_761_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_762_le__add__diff__inverse,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_763_le__add__diff__inverse2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_764_sup__inf__absorb,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y2 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_765_sup__inf__absorb,axiom,
! [X3: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( inf_inf_set_set_nat @ X3 @ Y2 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_766_inf__sup__absorb,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y2 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_767_inf__sup__absorb,axiom,
! [X3: set_set_nat,Y2: set_set_nat] :
( ( inf_inf_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ X3 @ Y2 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_768_sup__bot__left,axiom,
! [X3: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_769_sup__bot__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_770_sup__bot__left,axiom,
! [X3: set_nat_nat_nat] :
( ( sup_su3334021163961628176at_nat @ bot_bo7445843802507891576at_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_771_sup__bot__left,axiom,
! [X3: set_nat_nat_nat2] :
( ( sup_su6057362541959223568at_nat @ bot_bo945813143650711160at_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_772_sup__bot__left,axiom,
! [X3: set_nat_nat_nat_nat] :
( ( sup_su3836648520750444671at_nat @ bot_bo3919185967433191911at_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_773_sup__bot__left,axiom,
! [X3: set_na6626867396258451522at_nat] :
( ( sup_su481250237928500590at_nat @ bot_bo2074992577060541142at_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_774_sup__bot__left,axiom,
! [X3: set_na7233567106578532785at_nat] :
( ( sup_su8594648213498475741at_nat @ bot_bo2676777031303994949at_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_775_sup__bot__right,axiom,
! [X3: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ bot_bot_set_set_nat )
= X3 ) ).
% sup_bot_right
thf(fact_776_sup__bot__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% sup_bot_right
thf(fact_777_sup__bot__right,axiom,
! [X3: set_nat_nat_nat] :
( ( sup_su3334021163961628176at_nat @ X3 @ bot_bo7445843802507891576at_nat )
= X3 ) ).
% sup_bot_right
thf(fact_778_sup__bot__right,axiom,
! [X3: set_nat_nat_nat2] :
( ( sup_su6057362541959223568at_nat @ X3 @ bot_bo945813143650711160at_nat )
= X3 ) ).
% sup_bot_right
thf(fact_779_sup__bot__right,axiom,
! [X3: set_nat_nat_nat_nat] :
( ( sup_su3836648520750444671at_nat @ X3 @ bot_bo3919185967433191911at_nat )
= X3 ) ).
% sup_bot_right
thf(fact_780_sup__bot__right,axiom,
! [X3: set_na6626867396258451522at_nat] :
( ( sup_su481250237928500590at_nat @ X3 @ bot_bo2074992577060541142at_nat )
= X3 ) ).
% sup_bot_right
thf(fact_781_sup__bot__right,axiom,
! [X3: set_na7233567106578532785at_nat] :
( ( sup_su8594648213498475741at_nat @ X3 @ bot_bo2676777031303994949at_nat )
= X3 ) ).
% sup_bot_right
thf(fact_782_bot__eq__sup__iff,axiom,
! [X3: set_set_nat,Y2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ X3 @ Y2 ) )
= ( ( X3 = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_783_bot__eq__sup__iff,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X3 @ Y2 ) )
= ( ( X3 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_784_bot__eq__sup__iff,axiom,
! [X3: set_nat_nat_nat,Y2: set_nat_nat_nat] :
( ( bot_bo7445843802507891576at_nat
= ( sup_su3334021163961628176at_nat @ X3 @ Y2 ) )
= ( ( X3 = bot_bo7445843802507891576at_nat )
& ( Y2 = bot_bo7445843802507891576at_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_785_bot__eq__sup__iff,axiom,
! [X3: set_nat_nat_nat2,Y2: set_nat_nat_nat2] :
( ( bot_bo945813143650711160at_nat
= ( sup_su6057362541959223568at_nat @ X3 @ Y2 ) )
= ( ( X3 = bot_bo945813143650711160at_nat )
& ( Y2 = bot_bo945813143650711160at_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_786_bot__eq__sup__iff,axiom,
! [X3: set_nat_nat_nat_nat,Y2: set_nat_nat_nat_nat] :
( ( bot_bo3919185967433191911at_nat
= ( sup_su3836648520750444671at_nat @ X3 @ Y2 ) )
= ( ( X3 = bot_bo3919185967433191911at_nat )
& ( Y2 = bot_bo3919185967433191911at_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_787_bot__eq__sup__iff,axiom,
! [X3: set_na6626867396258451522at_nat,Y2: set_na6626867396258451522at_nat] :
( ( bot_bo2074992577060541142at_nat
= ( sup_su481250237928500590at_nat @ X3 @ Y2 ) )
= ( ( X3 = bot_bo2074992577060541142at_nat )
& ( Y2 = bot_bo2074992577060541142at_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_788_bot__eq__sup__iff,axiom,
! [X3: set_na7233567106578532785at_nat,Y2: set_na7233567106578532785at_nat] :
( ( bot_bo2676777031303994949at_nat
= ( sup_su8594648213498475741at_nat @ X3 @ Y2 ) )
= ( ( X3 = bot_bo2676777031303994949at_nat )
& ( Y2 = bot_bo2676777031303994949at_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_789_sup__eq__bot__iff,axiom,
! [X3: set_set_nat,Y2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ X3 @ Y2 )
= bot_bot_set_set_nat )
= ( ( X3 = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_790_sup__eq__bot__iff,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X3 @ Y2 )
= bot_bot_set_nat )
= ( ( X3 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_791_sup__eq__bot__iff,axiom,
! [X3: set_nat_nat_nat,Y2: set_nat_nat_nat] :
( ( ( sup_su3334021163961628176at_nat @ X3 @ Y2 )
= bot_bo7445843802507891576at_nat )
= ( ( X3 = bot_bo7445843802507891576at_nat )
& ( Y2 = bot_bo7445843802507891576at_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_792_sup__eq__bot__iff,axiom,
! [X3: set_nat_nat_nat2,Y2: set_nat_nat_nat2] :
( ( ( sup_su6057362541959223568at_nat @ X3 @ Y2 )
= bot_bo945813143650711160at_nat )
= ( ( X3 = bot_bo945813143650711160at_nat )
& ( Y2 = bot_bo945813143650711160at_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_793_sup__eq__bot__iff,axiom,
! [X3: set_nat_nat_nat_nat,Y2: set_nat_nat_nat_nat] :
( ( ( sup_su3836648520750444671at_nat @ X3 @ Y2 )
= bot_bo3919185967433191911at_nat )
= ( ( X3 = bot_bo3919185967433191911at_nat )
& ( Y2 = bot_bo3919185967433191911at_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_794_sup__eq__bot__iff,axiom,
! [X3: set_na6626867396258451522at_nat,Y2: set_na6626867396258451522at_nat] :
( ( ( sup_su481250237928500590at_nat @ X3 @ Y2 )
= bot_bo2074992577060541142at_nat )
= ( ( X3 = bot_bo2074992577060541142at_nat )
& ( Y2 = bot_bo2074992577060541142at_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_795_sup__eq__bot__iff,axiom,
! [X3: set_na7233567106578532785at_nat,Y2: set_na7233567106578532785at_nat] :
( ( ( sup_su8594648213498475741at_nat @ X3 @ Y2 )
= bot_bo2676777031303994949at_nat )
= ( ( X3 = bot_bo2676777031303994949at_nat )
& ( Y2 = bot_bo2676777031303994949at_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_796_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ( A2 = bot_bot_set_set_nat )
& ( B2 = bot_bot_set_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_797_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_798_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( ( sup_su3334021163961628176at_nat @ A2 @ B2 )
= bot_bo7445843802507891576at_nat )
= ( ( A2 = bot_bo7445843802507891576at_nat )
& ( B2 = bot_bo7445843802507891576at_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_799_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( ( sup_su6057362541959223568at_nat @ A2 @ B2 )
= bot_bo945813143650711160at_nat )
= ( ( A2 = bot_bo945813143650711160at_nat )
& ( B2 = bot_bo945813143650711160at_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_800_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( ( sup_su3836648520750444671at_nat @ A2 @ B2 )
= bot_bo3919185967433191911at_nat )
= ( ( A2 = bot_bo3919185967433191911at_nat )
& ( B2 = bot_bo3919185967433191911at_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_801_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( ( sup_su481250237928500590at_nat @ A2 @ B2 )
= bot_bo2074992577060541142at_nat )
= ( ( A2 = bot_bo2074992577060541142at_nat )
& ( B2 = bot_bo2074992577060541142at_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_802_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( ( sup_su8594648213498475741at_nat @ A2 @ B2 )
= bot_bo2676777031303994949at_nat )
= ( ( A2 = bot_bo2676777031303994949at_nat )
& ( B2 = bot_bo2676777031303994949at_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_803_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_804_Diff__cancel,axiom,
! [A: set_nat_nat_nat] :
( ( minus_7721066311745899709at_nat @ A @ A )
= bot_bo7445843802507891576at_nat ) ).
% Diff_cancel
thf(fact_805_Diff__cancel,axiom,
! [A: set_nat_nat_nat2] :
( ( minus_1221035652888719293at_nat @ A @ A )
= bot_bo945813143650711160at_nat ) ).
% Diff_cancel
thf(fact_806_Diff__cancel,axiom,
! [A: set_nat_nat_nat_nat] :
( ( minus_4646100876039749548at_nat @ A @ A )
= bot_bo3919185967433191911at_nat ) ).
% Diff_cancel
thf(fact_807_Diff__cancel,axiom,
! [A: set_na6626867396258451522at_nat] :
( ( minus_5225787954611647771at_nat @ A @ A )
= bot_bo2074992577060541142at_nat ) ).
% Diff_cancel
thf(fact_808_Diff__cancel,axiom,
! [A: set_na7233567106578532785at_nat] :
( ( minus_9165053394918225162at_nat @ A @ A )
= bot_bo2676777031303994949at_nat ) ).
% Diff_cancel
thf(fact_809_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_810_empty__Diff,axiom,
! [A: set_nat_nat_nat] :
( ( minus_7721066311745899709at_nat @ bot_bo7445843802507891576at_nat @ A )
= bot_bo7445843802507891576at_nat ) ).
% empty_Diff
thf(fact_811_empty__Diff,axiom,
! [A: set_nat_nat_nat2] :
( ( minus_1221035652888719293at_nat @ bot_bo945813143650711160at_nat @ A )
= bot_bo945813143650711160at_nat ) ).
% empty_Diff
thf(fact_812_empty__Diff,axiom,
! [A: set_nat_nat_nat_nat] :
( ( minus_4646100876039749548at_nat @ bot_bo3919185967433191911at_nat @ A )
= bot_bo3919185967433191911at_nat ) ).
% empty_Diff
thf(fact_813_empty__Diff,axiom,
! [A: set_na6626867396258451522at_nat] :
( ( minus_5225787954611647771at_nat @ bot_bo2074992577060541142at_nat @ A )
= bot_bo2074992577060541142at_nat ) ).
% empty_Diff
thf(fact_814_empty__Diff,axiom,
! [A: set_na7233567106578532785at_nat] :
( ( minus_9165053394918225162at_nat @ bot_bo2676777031303994949at_nat @ A )
= bot_bo2676777031303994949at_nat ) ).
% empty_Diff
thf(fact_815_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_816_Diff__empty,axiom,
! [A: set_nat_nat_nat] :
( ( minus_7721066311745899709at_nat @ A @ bot_bo7445843802507891576at_nat )
= A ) ).
% Diff_empty
thf(fact_817_Diff__empty,axiom,
! [A: set_nat_nat_nat2] :
( ( minus_1221035652888719293at_nat @ A @ bot_bo945813143650711160at_nat )
= A ) ).
% Diff_empty
thf(fact_818_Diff__empty,axiom,
! [A: set_nat_nat_nat_nat] :
( ( minus_4646100876039749548at_nat @ A @ bot_bo3919185967433191911at_nat )
= A ) ).
% Diff_empty
thf(fact_819_Diff__empty,axiom,
! [A: set_na6626867396258451522at_nat] :
( ( minus_5225787954611647771at_nat @ A @ bot_bo2074992577060541142at_nat )
= A ) ).
% Diff_empty
thf(fact_820_Diff__empty,axiom,
! [A: set_na7233567106578532785at_nat] :
( ( minus_9165053394918225162at_nat @ A @ bot_bo2676777031303994949at_nat )
= A ) ).
% Diff_empty
thf(fact_821_subsetI,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ! [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A )
=> ( member_nat_nat_nat2 @ X4 @ B ) )
=> ( ord_le3211623285424100676at_nat @ A @ B ) ) ).
% subsetI
thf(fact_822_subsetI,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A )
=> ( member_nat_nat_nat @ X4 @ B ) )
=> ( ord_le5934964663421696068at_nat @ A @ B ) ) ).
% subsetI
thf(fact_823_subsetI,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ! [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ A )
=> ( member952132173341509300at_nat @ X4 @ B ) )
=> ( ord_le5260717879541182899at_nat @ A @ B ) ) ).
% subsetI
thf(fact_824_subsetI,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ! [X4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X4 @ A )
=> ( member4402528950554000163at_nat @ X4 @ B ) )
=> ( ord_le973658574027395234at_nat @ A @ B ) ) ).
% subsetI
thf(fact_825_subsetI,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ! [X4: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X4 @ A )
=> ( member8881365325514865170at_nat @ X4 @ B ) )
=> ( ord_le8099187209609443857at_nat @ A @ B ) ) ).
% subsetI
thf(fact_826_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_827_subsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( member_nat_nat @ X4 @ B ) )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% subsetI
thf(fact_828_subset__antisym,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_829_inf_Oidem,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_830_inf__idem,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_831_inf_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_832_inf__left__idem,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y2 ) )
= ( inf_inf_set_nat @ X3 @ Y2 ) ) ).
% inf_left_idem
thf(fact_833_inf_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_834_inf__right__idem,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ Y2 )
= ( inf_inf_set_nat @ X3 @ Y2 ) ) ).
% inf_right_idem
thf(fact_835_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_836_sup_Oidem,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_837_sup__idem,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_838_sup__idem,axiom,
! [X3: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_839_sup_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_840_sup_Oleft__idem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_841_sup__left__idem,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y2 ) )
= ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_842_sup__left__idem,axiom,
! [X3: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ X3 @ Y2 ) )
= ( sup_sup_set_set_nat @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_843_sup_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_844_sup_Oright__idem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_845_Un__Diff__cancel2,axiom,
! [B: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A ) @ A )
= ( sup_sup_set_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_846_Un__Diff__cancel2,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ B @ A ) @ A )
= ( sup_sup_set_set_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_847_Un__Diff__cancel,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_848_Un__Diff__cancel,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( minus_2163939370556025621et_nat @ B @ A ) )
= ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_849_le__inf__iff,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z ) )
= ( ( ord_less_eq_set_nat @ X3 @ Y2 )
& ( ord_less_eq_set_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_850_le__inf__iff,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y2 @ Z ) )
= ( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_851_le__inf__iff,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ ( inf_inf_set_nat_nat @ Y2 @ Z ) )
= ( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
& ( ord_le9059583361652607317at_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_852_inf_Obounded__iff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_853_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_854_inf_Obounded__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) )
= ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
& ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_855_le__sup__iff,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ Z )
= ( ( ord_less_eq_set_nat @ X3 @ Z )
& ( ord_less_eq_set_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_856_le__sup__iff,axiom,
! [X3: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X3 @ Y2 ) @ Z )
= ( ( ord_le6893508408891458716et_nat @ X3 @ Z )
& ( ord_le6893508408891458716et_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_857_le__sup__iff,axiom,
! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y2 ) @ Z )
= ( ( ord_less_eq_nat @ X3 @ Z )
& ( ord_less_eq_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_858_le__sup__iff,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X3 @ Y2 ) @ Z )
= ( ( ord_le9059583361652607317at_nat @ X3 @ Z )
& ( ord_le9059583361652607317at_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_859_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_860_sup_Obounded__iff,axiom,
! [B2: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C ) @ A2 )
= ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
& ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_861_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_862_sup_Obounded__iff,axiom,
! [B2: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C ) @ A2 )
= ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
& ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_863_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_864_Diff__eq__empty__iff,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( ( minus_7721066311745899709at_nat @ A @ B )
= bot_bo7445843802507891576at_nat )
= ( ord_le3211623285424100676at_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_865_Diff__eq__empty__iff,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( ( minus_1221035652888719293at_nat @ A @ B )
= bot_bo945813143650711160at_nat )
= ( ord_le5934964663421696068at_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_866_Diff__eq__empty__iff,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( ( minus_4646100876039749548at_nat @ A @ B )
= bot_bo3919185967433191911at_nat )
= ( ord_le5260717879541182899at_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_867_Diff__eq__empty__iff,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( ( minus_5225787954611647771at_nat @ A @ B )
= bot_bo2074992577060541142at_nat )
= ( ord_le973658574027395234at_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_868_Diff__eq__empty__iff,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( ( minus_9165053394918225162at_nat @ A @ B )
= bot_bo2676777031303994949at_nat )
= ( ord_le8099187209609443857at_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_869_Diff__eq__empty__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( minus_8121590178497047118at_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_870_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_871_empty__subsetI,axiom,
! [A: set_nat_nat_nat] : ( ord_le3211623285424100676at_nat @ bot_bo7445843802507891576at_nat @ A ) ).
% empty_subsetI
thf(fact_872_empty__subsetI,axiom,
! [A: set_nat_nat_nat2] : ( ord_le5934964663421696068at_nat @ bot_bo945813143650711160at_nat @ A ) ).
% empty_subsetI
thf(fact_873_empty__subsetI,axiom,
! [A: set_nat_nat_nat_nat] : ( ord_le5260717879541182899at_nat @ bot_bo3919185967433191911at_nat @ A ) ).
% empty_subsetI
thf(fact_874_empty__subsetI,axiom,
! [A: set_na6626867396258451522at_nat] : ( ord_le973658574027395234at_nat @ bot_bo2074992577060541142at_nat @ A ) ).
% empty_subsetI
thf(fact_875_empty__subsetI,axiom,
! [A: set_na7233567106578532785at_nat] : ( ord_le8099187209609443857at_nat @ bot_bo2676777031303994949at_nat @ A ) ).
% empty_subsetI
thf(fact_876_empty__subsetI,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% empty_subsetI
thf(fact_877_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_878_subset__empty,axiom,
! [A: set_nat_nat_nat] :
( ( ord_le3211623285424100676at_nat @ A @ bot_bo7445843802507891576at_nat )
= ( A = bot_bo7445843802507891576at_nat ) ) ).
% subset_empty
thf(fact_879_subset__empty,axiom,
! [A: set_nat_nat_nat2] :
( ( ord_le5934964663421696068at_nat @ A @ bot_bo945813143650711160at_nat )
= ( A = bot_bo945813143650711160at_nat ) ) ).
% subset_empty
thf(fact_880_subset__empty,axiom,
! [A: set_nat_nat_nat_nat] :
( ( ord_le5260717879541182899at_nat @ A @ bot_bo3919185967433191911at_nat )
= ( A = bot_bo3919185967433191911at_nat ) ) ).
% subset_empty
thf(fact_881_subset__empty,axiom,
! [A: set_na6626867396258451522at_nat] :
( ( ord_le973658574027395234at_nat @ A @ bot_bo2074992577060541142at_nat )
= ( A = bot_bo2074992577060541142at_nat ) ) ).
% subset_empty
thf(fact_882_subset__empty,axiom,
! [A: set_na7233567106578532785at_nat] :
( ( ord_le8099187209609443857at_nat @ A @ bot_bo2676777031303994949at_nat )
= ( A = bot_bo2676777031303994949at_nat ) ) ).
% subset_empty
thf(fact_883_subset__empty,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% subset_empty
thf(fact_884_inf__bot__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_885_inf__bot__right,axiom,
! [X3: set_nat_nat_nat] :
( ( inf_in5274420515160781174at_nat @ X3 @ bot_bo7445843802507891576at_nat )
= bot_bo7445843802507891576at_nat ) ).
% inf_bot_right
thf(fact_886_inf__bot__right,axiom,
! [X3: set_nat_nat_nat2] :
( ( inf_in7997761893158376566at_nat @ X3 @ bot_bo945813143650711160at_nat )
= bot_bo945813143650711160at_nat ) ).
% inf_bot_right
thf(fact_887_inf__bot__right,axiom,
! [X3: set_nat_nat_nat_nat] :
( ( inf_in2949407623404935909at_nat @ X3 @ bot_bo3919185967433191911at_nat )
= bot_bo3919185967433191911at_nat ) ).
% inf_bot_right
thf(fact_888_inf__bot__right,axiom,
! [X3: set_na6626867396258451522at_nat] :
( ( inf_in6213014276851238612at_nat @ X3 @ bot_bo2074992577060541142at_nat )
= bot_bo2074992577060541142at_nat ) ).
% inf_bot_right
thf(fact_889_inf__bot__right,axiom,
! [X3: set_na7233567106578532785at_nat] :
( ( inf_in6008378084349164867at_nat @ X3 @ bot_bo2676777031303994949at_nat )
= bot_bo2676777031303994949at_nat ) ).
% inf_bot_right
thf(fact_890_inf__bot__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_891_inf__bot__left,axiom,
! [X3: set_nat_nat_nat] :
( ( inf_in5274420515160781174at_nat @ bot_bo7445843802507891576at_nat @ X3 )
= bot_bo7445843802507891576at_nat ) ).
% inf_bot_left
thf(fact_892_inf__bot__left,axiom,
! [X3: set_nat_nat_nat2] :
( ( inf_in7997761893158376566at_nat @ bot_bo945813143650711160at_nat @ X3 )
= bot_bo945813143650711160at_nat ) ).
% inf_bot_left
thf(fact_893_inf__bot__left,axiom,
! [X3: set_nat_nat_nat_nat] :
( ( inf_in2949407623404935909at_nat @ bot_bo3919185967433191911at_nat @ X3 )
= bot_bo3919185967433191911at_nat ) ).
% inf_bot_left
thf(fact_894_inf__bot__left,axiom,
! [X3: set_na6626867396258451522at_nat] :
( ( inf_in6213014276851238612at_nat @ bot_bo2074992577060541142at_nat @ X3 )
= bot_bo2074992577060541142at_nat ) ).
% inf_bot_left
thf(fact_895_inf__bot__left,axiom,
! [X3: set_na7233567106578532785at_nat] :
( ( inf_in6008378084349164867at_nat @ bot_bo2676777031303994949at_nat @ X3 )
= bot_bo2676777031303994949at_nat ) ).
% inf_bot_left
thf(fact_896_sup__bot_Oright__neutral,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ bot_bot_set_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_897_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_898_sup__bot_Oright__neutral,axiom,
! [A2: set_nat_nat_nat] :
( ( sup_su3334021163961628176at_nat @ A2 @ bot_bo7445843802507891576at_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_899_sup__bot_Oright__neutral,axiom,
! [A2: set_nat_nat_nat2] :
( ( sup_su6057362541959223568at_nat @ A2 @ bot_bo945813143650711160at_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_900_sup__bot_Oright__neutral,axiom,
! [A2: set_nat_nat_nat_nat] :
( ( sup_su3836648520750444671at_nat @ A2 @ bot_bo3919185967433191911at_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_901_sup__bot_Oright__neutral,axiom,
! [A2: set_na6626867396258451522at_nat] :
( ( sup_su481250237928500590at_nat @ A2 @ bot_bo2074992577060541142at_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_902_sup__bot_Oright__neutral,axiom,
! [A2: set_na7233567106578532785at_nat] :
( ( sup_su8594648213498475741at_nat @ A2 @ bot_bo2676777031303994949at_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_903_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_set_nat )
& ( B2 = bot_bot_set_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_904_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_905_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat_nat_nat,B2: set_nat_nat_nat] :
( ( bot_bo7445843802507891576at_nat
= ( sup_su3334021163961628176at_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo7445843802507891576at_nat )
& ( B2 = bot_bo7445843802507891576at_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_906_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat_nat_nat2,B2: set_nat_nat_nat2] :
( ( bot_bo945813143650711160at_nat
= ( sup_su6057362541959223568at_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo945813143650711160at_nat )
& ( B2 = bot_bo945813143650711160at_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_907_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( bot_bo3919185967433191911at_nat
= ( sup_su3836648520750444671at_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo3919185967433191911at_nat )
& ( B2 = bot_bo3919185967433191911at_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_908_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_na6626867396258451522at_nat,B2: set_na6626867396258451522at_nat] :
( ( bot_bo2074992577060541142at_nat
= ( sup_su481250237928500590at_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo2074992577060541142at_nat )
& ( B2 = bot_bo2074992577060541142at_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_909_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( bot_bo2676777031303994949at_nat
= ( sup_su8594648213498475741at_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo2676777031303994949at_nat )
& ( B2 = bot_bo2676777031303994949at_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_910_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_911_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_912_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat_nat_nat] :
( ( sup_su3334021163961628176at_nat @ bot_bo7445843802507891576at_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_913_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat_nat_nat2] :
( ( sup_su6057362541959223568at_nat @ bot_bo945813143650711160at_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_914_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat_nat_nat_nat] :
( ( sup_su3836648520750444671at_nat @ bot_bo3919185967433191911at_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_915_sup__bot_Oleft__neutral,axiom,
! [A2: set_na6626867396258451522at_nat] :
( ( sup_su481250237928500590at_nat @ bot_bo2074992577060541142at_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_916_sup__bot_Oleft__neutral,axiom,
! [A2: set_na7233567106578532785at_nat] :
( ( sup_su8594648213498475741at_nat @ bot_bo2676777031303994949at_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_917_Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_918_Diff__disjoint,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( inf_in5274420515160781174at_nat @ A @ ( minus_7721066311745899709at_nat @ B @ A ) )
= bot_bo7445843802507891576at_nat ) ).
% Diff_disjoint
thf(fact_919_Diff__disjoint,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( inf_in7997761893158376566at_nat @ A @ ( minus_1221035652888719293at_nat @ B @ A ) )
= bot_bo945813143650711160at_nat ) ).
% Diff_disjoint
thf(fact_920_Diff__disjoint,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( inf_in2949407623404935909at_nat @ A @ ( minus_4646100876039749548at_nat @ B @ A ) )
= bot_bo3919185967433191911at_nat ) ).
% Diff_disjoint
thf(fact_921_Diff__disjoint,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( inf_in6213014276851238612at_nat @ A @ ( minus_5225787954611647771at_nat @ B @ A ) )
= bot_bo2074992577060541142at_nat ) ).
% Diff_disjoint
thf(fact_922_Diff__disjoint,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( inf_in6008378084349164867at_nat @ A @ ( minus_9165053394918225162at_nat @ B @ A ) )
= bot_bo2676777031303994949at_nat ) ).
% Diff_disjoint
thf(fact_923_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_924_Int__subset__iff,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( ord_le9059583361652607317at_nat @ C2 @ A )
& ( ord_le9059583361652607317at_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_925_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_926_Un__subset__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 )
= ( ( ord_le6893508408891458716et_nat @ A @ C2 )
& ( ord_le6893508408891458716et_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_927_Un__subset__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C2 )
= ( ( ord_le9059583361652607317at_nat @ A @ C2 )
& ( ord_le9059583361652607317at_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_928_psubsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_929_in__mono,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat,X3: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A @ B )
=> ( ( member_nat_nat_nat2 @ X3 @ A )
=> ( member_nat_nat_nat2 @ X3 @ B ) ) ) ).
% in_mono
thf(fact_930_in__mono,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2,X3: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A @ B )
=> ( ( member_nat_nat_nat @ X3 @ A )
=> ( member_nat_nat_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_931_in__mono,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat,X3: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A @ B )
=> ( ( member952132173341509300at_nat @ X3 @ A )
=> ( member952132173341509300at_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_932_in__mono,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat,X3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le973658574027395234at_nat @ A @ B )
=> ( ( member4402528950554000163at_nat @ X3 @ A )
=> ( member4402528950554000163at_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_933_in__mono,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat,X3: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( ord_le8099187209609443857at_nat @ A @ B )
=> ( ( member8881365325514865170at_nat @ X3 @ A )
=> ( member8881365325514865170at_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_934_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_935_in__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,X3: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_936_subsetD,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat,C: nat > nat > nat] :
( ( ord_le3211623285424100676at_nat @ A @ B )
=> ( ( member_nat_nat_nat2 @ C @ A )
=> ( member_nat_nat_nat2 @ C @ B ) ) ) ).
% subsetD
thf(fact_937_subsetD,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2,C: ( nat > nat ) > nat] :
( ( ord_le5934964663421696068at_nat @ A @ B )
=> ( ( member_nat_nat_nat @ C @ A )
=> ( member_nat_nat_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_938_subsetD,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat,C: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A @ B )
=> ( ( member952132173341509300at_nat @ C @ A )
=> ( member952132173341509300at_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_939_subsetD,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat,C: ( nat > nat ) > ( nat > nat ) > nat] :
( ( ord_le973658574027395234at_nat @ A @ B )
=> ( ( member4402528950554000163at_nat @ C @ A )
=> ( member4402528950554000163at_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_940_subsetD,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat,C: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( ord_le8099187209609443857at_nat @ A @ B )
=> ( ( member8881365325514865170at_nat @ C @ A )
=> ( member8881365325514865170at_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_941_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_942_subsetD,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_943_Diff__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,D3: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ D3 @ B )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ C2 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_944_equalityE,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ~ ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ~ ( ord_le9059583361652607317at_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_945_subset__eq,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A4: set_nat_nat_nat,B4: set_nat_nat_nat] :
! [X: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X @ A4 )
=> ( member_nat_nat_nat2 @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_946_subset__eq,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A4: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
! [X: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X @ A4 )
=> ( member_nat_nat_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_947_subset__eq,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A4: set_nat_nat_nat_nat,B4: set_nat_nat_nat_nat] :
! [X: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X @ A4 )
=> ( member952132173341509300at_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_948_subset__eq,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A4: set_na6626867396258451522at_nat,B4: set_na6626867396258451522at_nat] :
! [X: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X @ A4 )
=> ( member4402528950554000163at_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_949_subset__eq,axiom,
( ord_le8099187209609443857at_nat
= ( ^ [A4: set_na7233567106578532785at_nat,B4: set_na7233567106578532785at_nat] :
! [X: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X @ A4 )
=> ( member8881365325514865170at_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_950_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_951_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [X: nat > nat] :
( ( member_nat_nat @ X @ A4 )
=> ( member_nat_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_952_equalityD1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_953_equalityD2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_954_subset__iff,axiom,
( ord_le3211623285424100676at_nat
= ( ^ [A4: set_nat_nat_nat,B4: set_nat_nat_nat] :
! [T3: nat > nat > nat] :
( ( member_nat_nat_nat2 @ T3 @ A4 )
=> ( member_nat_nat_nat2 @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_955_subset__iff,axiom,
( ord_le5934964663421696068at_nat
= ( ^ [A4: set_nat_nat_nat2,B4: set_nat_nat_nat2] :
! [T3: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ T3 @ A4 )
=> ( member_nat_nat_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_956_subset__iff,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A4: set_nat_nat_nat_nat,B4: set_nat_nat_nat_nat] :
! [T3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ T3 @ A4 )
=> ( member952132173341509300at_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_957_subset__iff,axiom,
( ord_le973658574027395234at_nat
= ( ^ [A4: set_na6626867396258451522at_nat,B4: set_na6626867396258451522at_nat] :
! [T3: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ T3 @ A4 )
=> ( member4402528950554000163at_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_958_subset__iff,axiom,
( ord_le8099187209609443857at_nat
= ( ^ [A4: set_na7233567106578532785at_nat,B4: set_na7233567106578532785at_nat] :
! [T3: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ T3 @ A4 )
=> ( member8881365325514865170at_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_959_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A4 )
=> ( member_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_960_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [T3: nat > nat] :
( ( member_nat_nat @ T3 @ A4 )
=> ( member_nat_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_961_Diff__subset,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_962_double__diff,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ( minus_8121590178497047118at_nat @ B @ ( minus_8121590178497047118at_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_963_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% subset_refl
thf(fact_964_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_965_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_966_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X4: nat > nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_967_subset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_le9059583361652607317at_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_968_set__eq__subset,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_969_Diff__partition,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_970_Diff__partition,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( sup_sup_set_set_nat @ A @ ( minus_2163939370556025621et_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_971_Diff__partition,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( sup_sup_set_nat_nat @ A @ ( minus_8121590178497047118at_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_972_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X: set_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_973_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_974_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_975_Diff__subset__conv,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_976_Diff__subset__conv,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ C2 )
= ( ord_le6893508408891458716et_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_977_Diff__subset__conv,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ C2 )
= ( ord_le9059583361652607317at_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_978_image__diff__subset,axiom,
! [F: nat > set_nat,A: set_nat,B: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_979_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_980_image__diff__subset,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ ( image_3205354838064109189at_nat @ F @ B ) ) @ ( image_3205354838064109189at_nat @ F @ ( minus_8121590178497047118at_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_981_Int__Diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_982_Diff__Int2,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ ( inf_inf_set_nat @ B @ C2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_983_Diff__Diff__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( minus_minus_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_984_Diff__Int__distrib,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A ) @ ( inf_inf_set_nat @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_985_Diff__Int__distrib2,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_986_Un__Diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C2 ) @ ( minus_minus_set_nat @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_987_Un__Diff,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ C2 ) @ ( minus_2163939370556025621et_nat @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_988_subset__image__iff,axiom,
! [B: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_989_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_990_subset__image__iff,axiom,
! [B: set_nat_nat,F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ ( image_3205354838064109189at_nat @ F @ A ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A )
& ( B
= ( image_3205354838064109189at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_991_image__subset__iff,axiom,
! [F: nat > set_nat,A: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_set_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_992_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_993_image__subset__iff,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ B )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A )
=> ( member_nat_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_994_subset__imageE,axiom,
! [B: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
=> ( B
!= ( image_nat_set_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_995_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
=> ( B
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_996_subset__imageE,axiom,
! [B: set_nat_nat,F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ ( image_3205354838064109189at_nat @ F @ A ) )
=> ~ ! [C5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C5 @ A )
=> ( B
!= ( image_3205354838064109189at_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_997_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_998_image__subsetI,axiom,
! [A: set_nat,F: nat > set_nat,B: set_set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_999_image__subsetI,axiom,
! [A: set_nat,F: nat > nat > nat,B: set_nat_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1000_image__subsetI,axiom,
! [A: set_nat_nat_nat,F: ( nat > nat > nat ) > nat,B: set_nat] :
( ! [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_913610194320715013at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1001_image__subsetI,axiom,
! [A: set_nat_nat_nat2,F: ( ( nat > nat ) > nat ) > nat,B: set_nat] :
( ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_7809927846809980933at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1002_image__subsetI,axiom,
! [A: set_nat,F: nat > nat > nat > nat,B: set_nat_nat_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat_nat_nat2 @ ( F @ X4 ) @ B ) )
=> ( ord_le3211623285424100676at_nat @ ( image_6919068903512877573at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1003_image__subsetI,axiom,
! [A: set_nat,F: nat > ( nat > nat ) > nat,B: set_nat_nat_nat2] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat_nat_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le5934964663421696068at_nat @ ( image_5809701139083627781at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1004_image__subsetI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat > nat,B: set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( member_nat_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1005_image__subsetI,axiom,
! [A: set_nat_nat_nat_nat,F: ( ( nat > nat ) > nat > nat ) > nat,B: set_nat] :
( ! [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_8194121248528334964at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1006_image__subsetI,axiom,
! [A: set_nat,F: nat > ( nat > nat ) > nat > nat,B: set_nat_nat_nat_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member952132173341509300at_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le5260717879541182899at_nat @ ( image_6393715451659844596at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1007_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_1008_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_1009_image__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ ( image_3205354838064109189at_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_1010_psubset__imp__ex__mem,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( ord_le6871433888996735800at_nat @ A @ B )
=> ? [B3: nat > nat > nat] : ( member_nat_nat_nat2 @ B3 @ ( minus_7721066311745899709at_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1011_psubset__imp__ex__mem,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( ord_le371403230139555384at_nat @ A @ B )
=> ? [B3: ( nat > nat ) > nat] : ( member_nat_nat_nat @ B3 @ ( minus_1221035652888719293at_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1012_psubset__imp__ex__mem,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( ord_le6177938698872215975at_nat @ A @ B )
=> ? [B3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ B3 @ ( minus_4646100876039749548at_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1013_psubset__imp__ex__mem,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( ord_le2785809691299232406at_nat @ A @ B )
=> ? [B3: ( nat > nat ) > ( nat > nat ) > nat] : ( member4402528950554000163at_nat @ B3 @ ( minus_5225787954611647771at_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1014_psubset__imp__ex__mem,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( ord_le7586516898478368261at_nat @ A @ B )
=> ? [B3: ( nat > nat ) > ( nat > nat ) > nat > nat] : ( member8881365325514865170at_nat @ B3 @ ( minus_9165053394918225162at_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1015_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1016_Int__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_1017_Int__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B: set_nat_nat,D3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ D3 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ C2 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_1018_Int__lower1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_1019_Int__lower1,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_1020_Int__lower2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_1021_Int__lower2,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_1022_Int__absorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1023_Int__absorb1,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( inf_inf_set_nat_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1024_Int__absorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_1025_Int__absorb2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( inf_inf_set_nat_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_1026_Int__greatest,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_1027_Int__greatest,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A )
=> ( ( ord_le9059583361652607317at_nat @ C2 @ B )
=> ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_1028_Int__Collect__mono,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat,P: ( nat > nat > nat ) > $o,Q: ( nat > nat > nat ) > $o] :
( ( ord_le3211623285424100676at_nat @ A @ B )
=> ( ! [X4: nat > nat > nat] :
( ( member_nat_nat_nat2 @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le3211623285424100676at_nat @ ( inf_in5274420515160781174at_nat @ A @ ( collect_nat_nat_nat2 @ P ) ) @ ( inf_in5274420515160781174at_nat @ B @ ( collect_nat_nat_nat2 @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1029_Int__Collect__mono,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2,P: ( ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > nat ) > $o] :
( ( ord_le5934964663421696068at_nat @ A @ B )
=> ( ! [X4: ( nat > nat ) > nat] :
( ( member_nat_nat_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le5934964663421696068at_nat @ ( inf_in7997761893158376566at_nat @ A @ ( collect_nat_nat_nat @ P ) ) @ ( inf_in7997761893158376566at_nat @ B @ ( collect_nat_nat_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1030_Int__Collect__mono,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat,P: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
( ( ord_le5260717879541182899at_nat @ A @ B )
=> ( ! [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le5260717879541182899at_nat @ ( inf_in2949407623404935909at_nat @ A @ ( collec3567154360959927026at_nat @ P ) ) @ ( inf_in2949407623404935909at_nat @ B @ ( collec3567154360959927026at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1031_Int__Collect__mono,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat,P: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o,Q: ( ( nat > nat ) > ( nat > nat ) > nat ) > $o] :
( ( ord_le973658574027395234at_nat @ A @ B )
=> ( ! [X4: ( nat > nat ) > ( nat > nat ) > nat] :
( ( member4402528950554000163at_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le973658574027395234at_nat @ ( inf_in6213014276851238612at_nat @ A @ ( collec2410089373097230945at_nat @ P ) ) @ ( inf_in6213014276851238612at_nat @ B @ ( collec2410089373097230945at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1032_Int__Collect__mono,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat,P: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o] :
( ( ord_le8099187209609443857at_nat @ A @ B )
=> ( ! [X4: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le8099187209609443857at_nat @ ( inf_in6008378084349164867at_nat @ A @ ( collec6535634078845029456at_nat @ P ) ) @ ( inf_in6008378084349164867at_nat @ B @ ( collec6535634078845029456at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1033_Int__Collect__mono,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B @ ( collect_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1034_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1035_Int__Collect__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ ( collect_nat_nat @ P ) ) @ ( inf_inf_set_nat_nat @ B @ ( collect_nat_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1036_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_1037_subset__Un__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_1038_subset__Un__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_1039_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ A )
=> ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ B )
=> ( C2
!= ( sup_sup_set_nat @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1040_subset__UnE,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) )
=> ~ ! [A6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A6 @ A )
=> ! [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ B )
=> ( C2
!= ( sup_sup_set_set_nat @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1041_subset__UnE,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ~ ! [A6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A6 @ A )
=> ! [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B6 @ B )
=> ( C2
!= ( sup_sup_set_nat_nat @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1042_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_1043_Un__absorb2,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( sup_sup_set_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_1044_Un__absorb2,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( sup_sup_set_nat_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_1045_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1046_Un__absorb1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( sup_sup_set_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1047_Un__absorb1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( sup_sup_set_nat_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1048_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_1049_Un__upper2,axiom,
! [B: set_set_nat,A: set_set_nat] : ( ord_le6893508408891458716et_nat @ B @ ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_1050_Un__upper2,axiom,
! [B: set_nat_nat,A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B @ ( sup_sup_set_nat_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_1051_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_1052_Un__upper1,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_1053_Un__upper1,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ ( sup_sup_set_nat_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_1054_Un__least,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_1055_Un__least,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_1056_Un__least,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_1057_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_1058_Un__mono,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat,D3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ D3 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_1059_Un__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B: set_nat_nat,D3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ D3 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_1060_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_1061_subset__psubset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ B @ C2 )
=> ( ord_less_set_nat_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1062_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_1063_psubset__subset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_less_set_nat_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1064_psubset__imp__subset,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_1065_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_1066_psubsetE,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ~ ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_1067_Int__Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_1068_Int__Diff__disjoint,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( inf_in5274420515160781174at_nat @ ( inf_in5274420515160781174at_nat @ A @ B ) @ ( minus_7721066311745899709at_nat @ A @ B ) )
= bot_bo7445843802507891576at_nat ) ).
% Int_Diff_disjoint
thf(fact_1069_Int__Diff__disjoint,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( inf_in7997761893158376566at_nat @ ( inf_in7997761893158376566at_nat @ A @ B ) @ ( minus_1221035652888719293at_nat @ A @ B ) )
= bot_bo945813143650711160at_nat ) ).
% Int_Diff_disjoint
thf(fact_1070_Int__Diff__disjoint,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( inf_in2949407623404935909at_nat @ ( inf_in2949407623404935909at_nat @ A @ B ) @ ( minus_4646100876039749548at_nat @ A @ B ) )
= bot_bo3919185967433191911at_nat ) ).
% Int_Diff_disjoint
thf(fact_1071_Int__Diff__disjoint,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( inf_in6213014276851238612at_nat @ ( inf_in6213014276851238612at_nat @ A @ B ) @ ( minus_5225787954611647771at_nat @ A @ B ) )
= bot_bo2074992577060541142at_nat ) ).
% Int_Diff_disjoint
thf(fact_1072_Int__Diff__disjoint,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( inf_in6008378084349164867at_nat @ ( inf_in6008378084349164867at_nat @ A @ B ) @ ( minus_9165053394918225162at_nat @ A @ B ) )
= bot_bo2676777031303994949at_nat ) ).
% Int_Diff_disjoint
thf(fact_1073_Diff__triv,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1074_Diff__triv,axiom,
! [A: set_nat_nat_nat,B: set_nat_nat_nat] :
( ( ( inf_in5274420515160781174at_nat @ A @ B )
= bot_bo7445843802507891576at_nat )
=> ( ( minus_7721066311745899709at_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1075_Diff__triv,axiom,
! [A: set_nat_nat_nat2,B: set_nat_nat_nat2] :
( ( ( inf_in7997761893158376566at_nat @ A @ B )
= bot_bo945813143650711160at_nat )
=> ( ( minus_1221035652888719293at_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1076_Diff__triv,axiom,
! [A: set_nat_nat_nat_nat,B: set_nat_nat_nat_nat] :
( ( ( inf_in2949407623404935909at_nat @ A @ B )
= bot_bo3919185967433191911at_nat )
=> ( ( minus_4646100876039749548at_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1077_Diff__triv,axiom,
! [A: set_na6626867396258451522at_nat,B: set_na6626867396258451522at_nat] :
( ( ( inf_in6213014276851238612at_nat @ A @ B )
= bot_bo2074992577060541142at_nat )
=> ( ( minus_5225787954611647771at_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1078_Diff__triv,axiom,
! [A: set_na7233567106578532785at_nat,B: set_na7233567106578532785at_nat] :
( ( ( inf_in6008378084349164867at_nat @ A @ B )
= bot_bo2676777031303994949at_nat )
=> ( ( minus_9165053394918225162at_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1079_Diff__Un,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1080_Diff__Un,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) )
= ( inf_inf_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1081_Diff__Int,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1082_Diff__Int,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( inf_inf_set_set_nat @ B @ C2 ) )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1083_Int__Diff__Un,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1084_Int__Diff__Un,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1085_Un__Diff__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1086_Un__Diff__Int,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( inf_inf_set_set_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1087_image__Int__subset,axiom,
! [F: nat > set_nat,A: set_nat,B: set_nat] : ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1088_image__Int__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1089_image__Int__subset,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ ( inf_inf_set_nat_nat @ A @ B ) ) @ ( inf_inf_set_nat_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ ( image_3205354838064109189at_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1090_image__Int__subset,axiom,
! [F: nat > nat > nat,A: set_nat,B: set_nat] : ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_nat_nat @ ( image_nat_nat_nat @ F @ A ) @ ( image_nat_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1091_Un__Int__assoc__eq,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1092_Un__Int__assoc__eq,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) )
= ( ord_le6893508408891458716et_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1093_Un__Int__assoc__eq,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) ) )
= ( ord_le9059583361652607317at_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1094_set__incr__altdef,axiom,
( hales_set_incr
= ( ^ [N2: nat] : ( image_nat_nat @ ( plus_plus_nat @ N2 ) ) ) ) ).
% set_incr_altdef
thf(fact_1095_inf__sup__aci_I4_J,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y2 ) )
= ( inf_inf_set_nat @ X3 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_1096_inf__sup__aci_I3_J,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z ) )
= ( inf_inf_set_nat @ Y2 @ ( inf_inf_set_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_1097_inf__sup__aci_I2_J,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ Z )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_1098_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_1099_inf_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_1100_inf__assoc,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ Z )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_1101_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( inf_inf_set_nat @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_1102_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X ) ) ) ).
% inf_commute
thf(fact_1103_inf_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A2 @ C ) )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_1104_inf__left__commute,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z ) )
= ( inf_inf_set_nat @ Y2 @ ( inf_inf_set_nat @ X3 @ Z ) ) ) ).
% inf_left_commute
thf(fact_1105_inf__sup__aci_I8_J,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y2 ) )
= ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_1106_inf__sup__aci_I8_J,axiom,
! [X3: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ X3 @ Y2 ) )
= ( sup_sup_set_set_nat @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_1107_inf__sup__aci_I7_J,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_1108_inf__sup__aci_I7_J,axiom,
! [X3: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_1109_inf__sup__aci_I6_J,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_1110_inf__sup__aci_I6_J,axiom,
! [X3: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_1111_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_1112_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_nat
= ( ^ [X: set_set_nat,Y4: set_set_nat] : ( sup_sup_set_set_nat @ Y4 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_1113_sup_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_1114_sup_Oassoc,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_1115_sup__assoc,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_1116_sup__assoc,axiom,
! [X3: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X3 @ Y2 ) @ Z )
= ( sup_sup_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_1117_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_1118_sup_Ocommute,axiom,
( sup_sup_set_set_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] : ( sup_sup_set_set_nat @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_1119_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X ) ) ) ).
% sup_commute
thf(fact_1120_sup__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [X: set_set_nat,Y4: set_set_nat] : ( sup_sup_set_set_nat @ Y4 @ X ) ) ) ).
% sup_commute
thf(fact_1121_sup_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_1122_sup_Oleft__commute,axiom,
! [B2: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ B2 @ ( sup_sup_set_set_nat @ A2 @ C ) )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_1123_sup__left__commute,axiom,
! [X3: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_1124_sup__left__commute,axiom,
! [X3: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_1125_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1126_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1127_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1128_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1129_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1130_inf__sup__ord_I2_J,axiom,
! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_1131_inf__sup__ord_I2_J,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_1132_inf__sup__ord_I2_J,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_1133_inf__sup__ord_I1_J,axiom,
! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_1134_inf__sup__ord_I1_J,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_1135_inf__sup__ord_I1_J,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_1136_inf__le1,axiom,
! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_1137_inf__le1,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_1138_inf__le1,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_1139_inf__le2,axiom,
! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_1140_inf__le2,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_1141_inf__le2,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_1142_le__infE,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_set_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_1143_le__infE,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_1144_le__infE,axiom,
! [X3: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ X3 @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_1145_le__infI,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ B2 )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1146_le__infI,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1147_le__infI,axiom,
! [X3: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X3 @ B2 )
=> ( ord_le9059583361652607317at_nat @ X3 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1148_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1149_inf__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1150_inf__mono,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B2: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ ( inf_inf_set_nat_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1151_le__infI1,axiom,
! [A2: set_nat,X3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_1152_le__infI1,axiom,
! [A2: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_1153_le__infI1,axiom,
! [A2: set_nat_nat,X3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X3 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_1154_le__infI2,axiom,
! [B2: set_nat,X3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_1155_le__infI2,axiom,
! [B2: nat,X3: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_1156_le__infI2,axiom,
! [B2: set_nat_nat,X3: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ X3 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_1157_inf_OorderE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1158_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1159_inf_OorderE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1160_inf_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_1161_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_1162_inf_OorderI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2
= ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_1163_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y2: set_nat] :
( ! [X4: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F @ X4 @ Y ) @ X4 )
=> ( ! [X4: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F @ X4 @ Y ) @ Y )
=> ( ! [X4: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ( ord_less_eq_set_nat @ X4 @ Z3 )
=> ( ord_less_eq_set_nat @ X4 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_1164_inf__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y2: nat] :
( ! [X4: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y ) @ X4 )
=> ( ! [X4: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y ) @ Y )
=> ( ! [X4: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_1165_inf__unique,axiom,
! [F: set_nat_nat > set_nat_nat > set_nat_nat,X3: set_nat_nat,Y2: set_nat_nat] :
( ! [X4: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F @ X4 @ Y ) @ X4 )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F @ X4 @ Y ) @ Y )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat,Z3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ X4 @ Z3 )
=> ( ord_le9059583361652607317at_nat @ X4 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat_nat @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_1166_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1167_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( inf_inf_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1168_le__iff__inf,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X: set_nat_nat,Y4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1169_inf_Oabsorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_1170_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_1171_inf_Oabsorb1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_1172_inf_Oabsorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1173_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1174_inf_Oabsorb2,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1175_inf__absorb1,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ( inf_inf_set_nat @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_1176_inf__absorb1,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( inf_inf_nat @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_1177_inf__absorb1,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ( inf_inf_set_nat_nat @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_1178_inf__absorb2,axiom,
! [Y2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( ( inf_inf_set_nat @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_1179_inf__absorb2,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_1180_inf__absorb2,axiom,
! [Y2: set_nat_nat,X3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X3 )
=> ( ( inf_inf_set_nat_nat @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_1181_inf_OboundedE,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_1182_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_1183_inf_OboundedE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_1184_inf_OboundedI,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1185_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1186_inf_OboundedI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1187_inf__greatest,axiom,
! [X3: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ X3 @ Z )
=> ( ord_le9059583361652607317at_nat @ X3 @ ( inf_inf_set_nat_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1188__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_1189_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_1190_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1191_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_1192_is__line__elim__t__1,axiom,
! [L2: nat > nat > nat,N: nat,T2: nat] :
( ( hales_is_line @ L2 @ N @ T2 )
=> ( ( T2 = 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 )
& ! [X2: nat] :
( ( member_nat @ X2 @ B_1 )
=> ! [Xa: nat] :
( ( ord_less_nat @ Xa @ T2 )
=> ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ T2 )
=> ( ( L2 @ Xa @ X2 )
= ( L2 @ Y3 @ X2 ) ) ) ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B_0 )
=> ! [S3: nat] :
( ( ord_less_nat @ S3 @ T2 )
=> ( ( L2 @ S3 @ X2 )
= S3 ) ) ) ) ) ) ).
% is_line_elim_t_1
thf(fact_1193_d__def,axiom,
( d
= ( minus_minus_nat @ m @ ( plus_plus_nat @ n @ m2 ) ) ) ).
% d_def
thf(fact_1194_M_H__prop,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ n @ m2 ) @ m ).
% M'_prop
thf(fact_1195__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_1196_line__points__in__cube__unfolded,axiom,
! [L2: nat > nat > nat,N: nat,T2: nat,S2: nat,J: nat] :
( ( hales_is_line @ L2 @ N @ T2 )
=> ( ( ord_less_nat @ S2 @ T2 )
=> ( ( ord_less_nat @ J @ N )
=> ( member_nat @ ( L2 @ S2 @ J ) @ ( set_ord_lessThan_nat @ T2 ) ) ) ) ) ).
% line_points_in_cube_unfolded
thf(fact_1197_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ( P @ M ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_1198_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less_nat @ M @ N )
& ( P @ M ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1199_fT__def,axiom,
( fT
= ( ^ [X: nat] : ( if_nat @ ( member_nat @ X @ ( bl @ one_one_nat ) ) @ ( fL @ X ) @ ( if_nat @ ( member_nat @ X @ ( hales_set_incr @ n2 @ ( bs @ k ) ) ) @ ( fS @ ( minus_minus_nat @ X @ n2 ) ) @ undefined_nat ) ) ) ) ).
% fT_def
thf(fact_1200_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_1201_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_1202_BfL__props_I1_J,axiom,
disjoi6798895846410478970at_nat @ bl @ ( set_ord_atMost_nat @ one_one_nat ) ).
% BfL_props(1)
thf(fact_1203_BfS__props_I1_J,axiom,
disjoi6798895846410478970at_nat @ bs @ ( set_ord_atMost_nat @ k ) ).
% BfS_props(1)
thf(fact_1204_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1205_fact2,axiom,
( ( inf_inf_set_nat @ ( bl @ zero_zero_nat )
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( hales_set_incr @ n2 @ ( bs @ I2 ) )
@ ( set_ord_lessThan_nat @ k ) ) ) )
= bot_bot_set_nat ) ).
% fact2
thf(fact_1206_set__incr__def,axiom,
( hales_set_incr
= ( ^ [N2: nat] :
( image_nat_nat
@ ^ [A5: nat] : ( plus_plus_nat @ A5 @ N2 ) ) ) ) ).
% set_incr_def
thf(fact_1207__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] :
? [I2: nat] :
( ( Uu
= ( hales_set_incr @ n2 @ ( bs @ I2 ) ) )
& ( member_nat @ I2 @ ( 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_1208__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] :
? [I2: nat] :
( ( Uu
= ( hales_set_incr @ n2 @ ( bs @ I2 ) ) )
& ( member_nat @ I2 @ ( set_ord_lessThan_nat @ k ) ) ) ) ) ).
% \<open>{} \<notin> {set_incr n__ (BS i) |i. i \<in> {..<k}}\<close>
thf(fact_1209_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_1210_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ t3 ).
% assms(1)
thf(fact_1211__092_060open_0620_A_060_As_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ s2 ).
% \<open>0 < s\<close>
thf(fact_1212_assms_I4_J,axiom,
! [K4: nat,R: nat] :
( ( ord_less_eq_nat @ K4 @ k )
=> ( hales_lhj @ R @ t3 @ K4 ) ) ).
% assms(4)
thf(fact_1213_s__def,axiom,
( s2
= ( power_power_nat @ r @ ( power_power_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) @ m2 ) ) ) ).
% s_def
thf(fact_1214_BfL__props_I4_J,axiom,
( member_nat_nat @ fL
@ ( piE_nat_nat @ ( bl @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% BfL_props(4)
thf(fact_1215_BfS__props_I4_J,axiom,
( member_nat_nat @ fS
@ ( piE_nat_nat @ ( bs @ k )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% BfS_props(4)
thf(fact_1216_BfS__props_I6_J,axiom,
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( bs @ k ) )
=> ( ( s @ X2 @ Xa )
= ( fS @ Xa ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ k )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( bs @ J3 ) )
=> ( ( s @ X2 @ Xa )
= ( X2 @ J3 ) ) ) ) ) ) ).
% BfS_props(6)
thf(fact_1217_BfL__props_I6_J,axiom,
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( bl @ one_one_nat ) )
=> ( ( l @ X2 @ Xa )
= ( fL @ Xa ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ one_one_nat )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( bl @ J3 ) )
=> ( ( l @ X2 @ Xa )
= ( X2 @ J3 ) ) ) ) ) ) ).
% BfL_props(6)
thf(fact_1218_T__def,axiom,
( t2
= ( restri4446420529079022766at_nat
@ ^ [X: nat > nat] :
( t @ ( restrict_nat_nat @ X @ ( set_ord_lessThan_nat @ one_one_nat ) )
@ ( restrict_nat_nat
@ ^ [Y4: nat] : ( X @ ( plus_plus_nat @ Y4 @ 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_1219__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_1220__092_060chi_062L__def,axiom,
( chi_L
= ( restri6011711336257459485at_nat
@ ^ [X: nat > nat] :
( restrict_nat_nat_nat
@ ^ [Y4: nat > nat] : ( chi @ ( hales_join_nat @ X @ Y4 @ 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_1221_Tset__def,axiom,
( tset
= ( collect_nat_nat
@ ^ [Uu: nat > nat] :
? [I2: nat,S4: nat > nat] :
( ( Uu
= ( hales_join_nat @ ( l_line @ I2 ) @ S4 @ n2 @ m2 ) )
& ( member_nat @ I2 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
& ( member_nat_nat @ S4 @ ( image_3205354838064109189at_nat @ s @ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ) ) ) ).
% Tset_def
thf(fact_1222_cube__def,axiom,
( hales_cube
= ( ^ [N2: nat,T3: nat] :
( piE_nat_nat @ ( set_ord_lessThan_nat @ N2 )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T3 ) ) ) ) ).
% cube_def
thf(fact_1223_join__cubes,axiom,
! [F: nat > nat,N: nat,T2: nat,G: nat > nat,M2: nat] :
( ( member_nat_nat @ F @ ( hales_cube @ N @ ( plus_plus_nat @ T2 @ one_one_nat ) ) )
=> ( ( member_nat_nat @ G @ ( hales_cube @ M2 @ ( plus_plus_nat @ T2 @ one_one_nat ) ) )
=> ( member_nat_nat @ ( hales_join_nat @ F @ G @ N @ M2 ) @ ( hales_cube @ ( plus_plus_nat @ N @ M2 ) @ ( plus_plus_nat @ T2 @ one_one_nat ) ) ) ) ) ).
% join_cubes
thf(fact_1224_cube__props_I1_J,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ T2 ) )
& ( ( X4 @ zero_zero_nat )
= S2 ) ) ) ).
% cube_props(1)
thf(fact_1225_line__points__in__cube,axiom,
! [L2: nat > nat > nat,N: nat,T2: nat,S2: nat] :
( ( hales_is_line @ L2 @ N @ T2 )
=> ( ( ord_less_nat @ S2 @ T2 )
=> ( member_nat_nat @ ( L2 @ S2 ) @ ( hales_cube @ N @ T2 ) ) ) ) ).
% line_points_in_cube
thf(fact_1226_cube__subset,axiom,
! [N: nat,T2: nat] : ( ord_le9059583361652607317at_nat @ ( hales_cube @ N @ T2 ) @ ( hales_cube @ N @ ( plus_plus_nat @ T2 @ one_one_nat ) ) ) ).
% cube_subset
thf(fact_1227_cube__restrict,axiom,
! [J: nat,N: nat,Y2: nat > nat,T2: nat] :
( ( ord_less_nat @ J @ N )
=> ( ( member_nat_nat @ Y2 @ ( hales_cube @ N @ T2 ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ Y2 @ ( set_ord_lessThan_nat @ J ) ) @ ( hales_cube @ J @ T2 ) ) ) ) ).
% cube_restrict
thf(fact_1228_split__cube_I1_J,axiom,
! [X3: nat > nat,K: nat,T2: nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T2 ) )
=> ( member_nat_nat @ ( restrict_nat_nat @ X3 @ ( set_ord_lessThan_nat @ one_one_nat ) ) @ ( hales_cube @ one_one_nat @ T2 ) ) ) ).
% split_cube(1)
thf(fact_1229_split__cube_I2_J,axiom,
! [X3: nat > nat,K: nat,T2: nat] :
( ( member_nat_nat @ X3 @ ( hales_cube @ ( plus_plus_nat @ K @ one_one_nat ) @ T2 ) )
=> ( member_nat_nat
@ ( restrict_nat_nat
@ ^ [Y4: nat] : ( X3 @ ( plus_plus_nat @ Y4 @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ K ) )
@ ( hales_cube @ K @ T2 ) ) ) ).
% split_cube(2)
thf(fact_1230_A,axiom,
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( 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 @ X2 @ Xa @ n2 @ m2 ) ) @ ( set_ord_lessThan_nat @ r ) ) ) ) ).
% A
thf(fact_1231_BfS__props_I5_J,axiom,
( member952132173341509300at_nat @ s
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% BfS_props(5)
thf(fact_1232_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 ) )
@ ^ [I2: nat > nat] : ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% BfL_props(5)
thf(fact_1233__092_060chi_062__prop,axiom,
( member_nat_nat_nat @ chi
@ ( piE_nat_nat_nat @ ( hales_cube @ m @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% \<chi>_prop
thf(fact_1234_L__line__base__prop,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( member_nat_nat @ ( l_line @ X2 ) @ ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% L_line_base_prop
thf(fact_1235_T_H__def,axiom,
( t
= ( restri1704181820465610764at_nat
@ ^ [X: nat > nat] :
( restri4446420529079022766at_nat
@ ^ [Y4: nat > nat] : ( hales_join_nat @ ( l_line @ ( X @ zero_zero_nat ) ) @ ( s @ Y4 ) @ 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_1236_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_1237__092_060chi_062L__prop,axiom,
( member4402528950554000163at_nat @ chi_L
@ ( piE_na7569501297962130601at_nat @ ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I2: 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_1238_T_H__prop,axiom,
( member8881365325514865170at_nat @ t
@ ( piE_na5223350113562215832at_nat @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I2: 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_1239_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 ) )
@ ^ [I2: nat > nat] : ( hales_cube @ ( plus_plus_nat @ n2 @ m2 ) @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% T_prop
thf(fact_1240_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 ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ s2 ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ one_one_nat @ N5 @ t3 @ s2 @ Chi ) ) ) ) ).
% n'_props
thf(fact_1241__092_060chi_062S__def,axiom,
( chi_S
= ( restrict_nat_nat_nat
@ ^ [Y4: nat > nat] : ( chi @ ( hales_join_nat @ ( l_line @ zero_zero_nat ) @ Y4 @ n2 @ m2 ) )
@ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% \<chi>S_def
thf(fact_1242__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 )
@ ^ [I2: 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 ) )
@ ^ [I2: nat > nat] : ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) )
=> ~ ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( BL @ one_one_nat ) )
=> ( ( l @ X2 @ Xa )
= ( FL @ Xa ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ one_one_nat )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( BL @ J3 ) )
=> ( ( l @ X2 @ Xa )
= ( X2 @ 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_1243_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 ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ k @ M6 @ t3 @ r @ Chi ) ) ) ) ).
% m_props
thf(fact_1244__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 )
@ ^ [I2: 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 ) )
@ ^ [I2: nat > nat] : ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) )
=> ~ ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( BS @ k ) )
=> ( ( s @ X2 @ Xa )
= ( FS @ Xa ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ k )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( BS @ J3 ) )
=> ( ( s @ X2 @ Xa )
= ( X2 @ 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_1245__092_060chi_062L__s__def,axiom,
( chi_L_s
= ( restrict_nat_nat_nat
@ ^ [X: nat > nat] : ( phi @ ( chi_L @ X ) )
@ ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% \<chi>L_s_def
thf(fact_1246_L__line__def,axiom,
( l_line
= ( restrict_nat_nat_nat2
@ ^ [S4: 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 )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% L_line_def
thf(fact_1247_is__line__def,axiom,
( hales_is_line
= ( ^ [L3: nat > nat > nat,N2: nat,T3: nat] :
( ( member_nat_nat_nat2 @ L3
@ ( piE_nat_nat_nat2 @ ( set_ord_lessThan_nat @ T3 )
@ ^ [I2: nat] : ( hales_cube @ N2 @ T3 ) ) )
& ! [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
=> ( ! [X: nat] :
( ( ord_less_nat @ X @ T3 )
=> ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ T3 )
=> ( ( L3 @ X @ J4 )
= ( L3 @ Y4 @ J4 ) ) ) )
| ! [S4: nat] :
( ( ord_less_nat @ S4 @ T3 )
=> ( ( L3 @ S4 @ J4 )
= S4 ) ) ) )
& ? [J4: nat] :
( ( ord_less_nat @ J4 @ N2 )
& ! [S4: nat] :
( ( ord_less_nat @ S4 @ T3 )
=> ( ( L3 @ S4 @ J4 )
= S4 ) ) ) ) ) ) ).
% is_line_def
thf(fact_1248_dim1__subspace__elims_I4_J,axiom,
! [B: nat > set_nat,N: nat,F: nat > nat,T2: nat,S: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T2 ) ) )
=> ( ( member952132173341509300at_nat @ S
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ one_one_nat @ T2 )
@ ^ [I2: nat > nat] : ( hales_cube @ N @ T2 ) ) )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ T2 ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B @ one_one_nat ) )
=> ( ( S @ X4 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B @ J2 ) )
=> ( ( S @ X4 @ Xa2 )
= ( X4 @ J2 ) ) ) ) ) )
=> ( ( B @ zero_zero_nat )
!= bot_bot_set_nat ) ) ) ) ) ) ) ).
% dim1_subspace_elims(4)
thf(fact_1249_dim1__subspace__elims_I3_J,axiom,
! [B: nat > set_nat,N: nat,F: nat > nat,T2: nat,S: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T2 ) ) )
=> ( ( member952132173341509300at_nat @ S
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ one_one_nat @ T2 )
@ ^ [I2: nat > nat] : ( hales_cube @ N @ T2 ) ) )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ T2 ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B @ one_one_nat ) )
=> ( ( S @ X4 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B @ J2 ) )
=> ( ( S @ X4 @ Xa2 )
= ( X4 @ J2 ) ) ) ) ) )
=> ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( hales_cube @ one_one_nat @ T2 ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( B @ one_one_nat ) )
=> ( ( S @ X2 @ Xa )
= ( F @ Xa ) ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( B @ zero_zero_nat ) )
=> ( ( S @ X2 @ Xa )
= ( X2 @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ).
% dim1_subspace_elims(3)
thf(fact_1250_dim1__subspace__elims_I2_J,axiom,
! [B: nat > set_nat,N: nat,F: nat > nat,T2: nat,S: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T2 ) ) )
=> ( ( member952132173341509300at_nat @ S
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ one_one_nat @ T2 )
@ ^ [I2: nat > nat] : ( hales_cube @ N @ T2 ) ) )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ T2 ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B @ one_one_nat ) )
=> ( ( S @ X4 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B @ J2 ) )
=> ( ( S @ X4 @ Xa2 )
= ( X4 @ J2 ) ) ) ) ) )
=> ( ( inf_inf_set_nat @ ( B @ zero_zero_nat ) @ ( B @ one_one_nat ) )
= bot_bot_set_nat ) ) ) ) ) ) ) ).
% dim1_subspace_elims(2)
thf(fact_1251_dim1__subspace__elims_I1_J,axiom,
! [B: nat > set_nat,N: nat,F: nat > nat,T2: nat,S: ( nat > nat ) > nat > nat] :
( ( disjoi6798895846410478970at_nat @ B @ ( set_ord_atMost_nat @ one_one_nat ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ B @ ( set_ord_lessThan_nat @ one_one_nat ) ) )
=> ( ( member_nat_nat @ F
@ ( piE_nat_nat @ ( B @ one_one_nat )
@ ^ [I2: nat] : ( set_ord_lessThan_nat @ T2 ) ) )
=> ( ( member952132173341509300at_nat @ S
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ one_one_nat @ T2 )
@ ^ [I2: nat > nat] : ( hales_cube @ N @ T2 ) ) )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ ( hales_cube @ one_one_nat @ T2 ) )
=> ( ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B @ one_one_nat ) )
=> ( ( S @ X4 @ Xa2 )
= ( F @ Xa2 ) ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ one_one_nat )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( B @ J2 ) )
=> ( ( S @ X4 @ Xa2 )
= ( X4 @ J2 ) ) ) ) ) )
=> ( ( sup_sup_set_nat @ ( B @ zero_zero_nat ) @ ( B @ one_one_nat ) )
= ( set_ord_lessThan_nat @ N ) ) ) ) ) ) ) ) ).
% dim1_subspace_elims(1)
thf(fact_1252__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 ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ s2 ) ) ) ).
% \<open>\<chi>L_s \<in> cube n__ (t + 1) \<rightarrow>\<^sub>E {..<s}\<close>
thf(fact_1253__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 ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) ) ).
% \<open>\<chi>S \<in> cube m__ (t + 1) \<rightarrow>\<^sub>E {..<r}\<close>
thf(fact_1254_L__prop,axiom,
hales_4261547300027266985ce_nat @ l @ one_one_nat @ n2 @ t3 @ s2 @ chi_L_s ).
% L_prop
thf(fact_1255_S__prop,axiom,
hales_4261547300027266985ce_nat @ s @ k @ m2 @ t3 @ r @ chi_S ).
% S_prop
thf(fact_1256__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,
~ ! [L4: ( nat > nat ) > nat > nat] :
~ ( hales_4261547300027266985ce_nat @ L4 @ 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_1257__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,
~ ! [S5: ( nat > nat ) > nat > nat] :
~ ( hales_4261547300027266985ce_nat @ S5 @ 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_1258__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 ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ s2 ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ 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_1259__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 ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ 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_1260_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 ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ s2 ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] :
( ( hales_4261547300027266985ce_nat @ S5 @ one_one_nat @ n2 @ t3 @ s2 @ Chi2 )
& ( hales_is_line
@ ( restrict_nat_nat_nat2
@ ^ [S4: nat] :
( S5
@ ( 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 )
= S4 ) ) ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
@ n2
@ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% line_subspace_s
thf(fact_1261_cube__props_I4_J,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( member_nat_nat
@ ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T2 ) )
& ( ( P4 @ zero_zero_nat )
= S2 ) ) )
@ ( hales_cube @ one_one_nat @ T2 ) ) ) ).
% cube_props(4)
thf(fact_1262_cube__props_I2_J,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( ( fChoice_nat_nat
@ ^ [P4: nat > nat] :
( ( member_nat_nat @ P4 @ ( hales_cube @ one_one_nat @ T2 ) )
& ( ( P4 @ zero_zero_nat )
= S2 ) )
@ zero_zero_nat )
= S2 ) ) ).
% cube_props(2)
thf(fact_1263_dim0__layered__subspace__ex,axiom,
! [Chi2: ( nat > nat ) > nat,N: nat,T2: nat,R: nat] :
( ( member_nat_nat_nat @ Chi2
@ ( piE_nat_nat_nat @ ( hales_cube @ N @ ( plus_plus_nat @ T2 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R ) ) )
=> ? [S5: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S5 @ zero_zero_nat @ N @ T2 @ R @ Chi2 ) ) ).
% dim0_layered_subspace_ex
thf(fact_1264_lhj__def,axiom,
( hales_lhj
= ( ^ [R2: nat,T3: nat,K2: 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 @ T3 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ R2 ) ) )
=> ? [S6: ( nat > nat ) > nat > nat] : ( hales_4261547300027266985ce_nat @ S6 @ K2 @ N8 @ T3 @ R2 @ Chi3 ) ) ) ) ) ) ).
% lhj_def
thf(fact_1265__092_060phi_062__prop,axiom,
( bij_be1059735840858801910at_nat @ phi
@ ( piE_nat_nat_nat @ ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I2: nat > nat] : ( set_ord_lessThan_nat @ r ) )
@ ( set_ord_lessThan_nat @ s2 ) ) ).
% \<phi>_prop
thf(fact_1266_nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1267__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 ) )
@ ^ [I2: 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>
% Helper facts (6)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y2: nat] :
( ( if_nat @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y2: nat] :
( ( if_nat @ $true @ X3 @ Y2 )
= X3 ) ).
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,
! [X3: set_nat,Y2: set_nat] :
( ( if_set_nat @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( if_set_nat @ $true @ X3 @ Y2 )
= X3 ) ).
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,
member_nat @ x @ bot_bot_set_nat ).
%------------------------------------------------------------------------------