TPTP Problem File: SLH0715^1.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_01584_070781__5949534_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1466 ( 679 unt; 186 typ; 0 def)
% Number of atoms : 3321 (1265 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9884 ( 254 ~; 15 |; 128 &;8174 @)
% ( 0 <=>;1313 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 1401 (1401 >; 0 *; 0 +; 0 <<)
% Number of symbols : 177 ( 174 usr; 28 con; 0-5 aty)
% Number of variables : 3448 ( 186 ^;3194 !; 68 ?;3448 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:46:51.565
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Set__Oset_It__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_J,type,
set_se5827506804761348711at_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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_na7233567106578532785at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_se3022870823424313865at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_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_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (174)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
comple3227554028126040196at_nat: set_se5827506804761348711at_nat > set_na7233567106578532785at_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
comple2605510978757769510at_nat: set_se3022870823424313865at_nat > set_nat_nat_nat_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
comple5448282615319421384at_nat: set_set_nat_nat > set_nat_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_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_001_Eo,type,
disjoi3706031484853323779_nat_o: ( ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_o ) > set_na7233567106578532785at_nat > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_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_001t__Nat__Onat,type,
disjoi2115914870343817253at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_nat ) > set_na7233567106578532785at_nat > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
disjoi6151367327612141793_nat_o: ( ( ( nat > nat ) > nat > nat ) > set_o ) > set_nat_nat_nat_nat > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_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,
disjoi4499352858376688327at_nat: ( ( ( nat > nat ) > nat > nat ) > set_nat ) > set_nat_nat_nat_nat > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
disjoi7375875898804509119_nat_o: ( ( nat > nat ) > set_o ) > set_nat_nat > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
disjoi831272138528337257at_nat: ( ( nat > nat ) > set_nat ) > set_nat_nat > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001t__Nat__Onat_001_Eo,type,
disjoi1808054049482533742_nat_o: ( nat > set_o ) > set_nat > $o ).
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_Disjoint__Sets_Odisjoint__family__on_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
disjoi7862385731094200888_nat_o: ( set_nat > set_o ) > set_set_nat > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
disjoi2115144663756723504at_nat: ( set_nat > set_nat ) > set_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_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_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_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_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_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_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_I_Eo_J,type,
minus_minus_set_o: set_o > set_o > set_o ).
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_Ojoin_001t__Nat__Onat,type,
hales_join_nat: ( nat > nat ) > ( nat > nat ) > nat > nat > nat > nat ).
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_001_Eo,type,
inf_inf_o: $o > $o > $o ).
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_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_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_I_Eo_J,type,
inf_inf_set_o: set_o > set_o > set_o ).
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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inf_in710756014367367485at_nat: set_set_nat_nat > set_set_nat_nat > set_set_nat_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_001_Eo,type,
sup_sup_o: $o > $o > $o ).
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_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_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_I_Eo_J,type,
sup_sup_set_o: set_o > set_o > set_o ).
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_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
sup_su4213647025997063966et_nat: set_set_set_nat > set_set_set_nat > set_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_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_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_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $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_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_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_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_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo7376149671870096959at_nat: set_set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
bot_bot_set_set_o: set_set_o ).
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_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_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_le5526148332077535835at_nat: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le747776305331315197at_nat: ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_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_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_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4954213926817602059at_nat: set_set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_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_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_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_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,
image_757773569841663411at_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_na7233567106578532785at_nat ) > set_na7233567106578532785at_nat > set_se5827506804761348711at_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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_2666519055618792072et_nat: ( ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_nat ) > set_na7233567106578532785at_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
image_8690456353314504180_nat_o: ( ( ( nat > nat ) > nat > nat ) > $o ) > set_nat_nat_nat_nat > set_o ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_8194121248528334964at_nat: ( ( ( nat > nat ) > nat > nat ) > nat ) > set_nat_nat_nat_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__Set__Oset_It__Nat__Onat_J,type,
image_1946857609996606506et_nat: ( ( ( nat > nat ) > nat > nat ) > set_nat ) > set_nat_nat_nat_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_3205354838064109189at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
image_nat_nat_o: ( ( nat > nat ) > $o ) > set_nat_nat > set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7432509271690132940et_nat: ( ( nat > nat ) > set_nat ) > set_nat_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_o_set_nat_nat: ( $o > set_nat_nat ) > set_o > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
image_o_set_nat: ( $o > set_nat ) > set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_o_set_set_nat: ( $o > set_set_nat ) > set_o > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_6393715451659844596at_nat: ( nat > ( nat > nat ) > nat > nat ) > set_nat > set_nat_nat_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7301343469591561292at_nat: ( nat > set_nat_nat ) > set_nat > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_8569768528772619084at_nat: ( set_nat > nat > nat ) > set_set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_Eo,type,
set_or7139685690850216873Than_o: $o > $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or9117062992132219044at_nat: set_nat_nat > set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_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_or9155507668907256820at_nat: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_na7233567106578532785at_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or3591701359631937174at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or9140604705432621368at_nat: ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_Eo,type,
set_ord_atMost_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or250740698829186286at_nat: set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001_Eo,type,
set_ord_lessThan_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_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_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_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
member4685516209270408648at_nat: set_na7233567106578532785at_nat > set_se5827506804761348711at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member7681264892014656106at_nat: set_nat_nat_nat_nat > set_se3022870823424313865at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_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_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_062____,type,
chi: ( 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_i____,type,
i: nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_m____,type,
m2: nat ).
thf(sy_v_n_H____,type,
n: nat ).
thf(sy_v_n____,type,
n2: nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_t,type,
t3: nat ).
thf(sy_v_y____,type,
y: nat > nat ).
% Relevant facts (1274)
thf(fact_0__092_060open_062i_A_092_060in_062_ABT_A_Ik_A_L_A1_J_092_060close_062,axiom,
member_nat @ i @ ( bt @ ( plus_plus_nat @ k @ one_one_nat ) ) ).
% \<open>i \<in> BT (k + 1)\<close>
thf(fact_1_assms_I2_J,axiom,
ord_less_eq_nat @ one_one_nat @ k ).
% assms(2)
thf(fact_2_fact5,axiom,
! [X: nat] :
( ( member_nat @ X @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) )
=> ( ( inf_inf_set_nat @ ( bvar @ X ) @ bstat )
= bot_bot_set_nat ) ) ).
% fact5
thf(fact_3_Bstat__def,axiom,
( bstat
= ( sup_sup_set_nat @ ( hales_set_incr @ n2 @ ( bs @ k ) ) @ ( bl @ one_one_nat ) ) ) ).
% Bstat_def
thf(fact_4_fact1,axiom,
( ( inf_inf_set_nat @ ( hales_set_incr @ n2 @ ( bs @ k ) ) @ ( bl @ one_one_nat ) )
= bot_bot_set_nat ) ).
% fact1
thf(fact_5_fact4,axiom,
! [X: nat] :
( ( member_nat @ X @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) )
=> ( ( X != Xa )
=> ( ( inf_inf_set_nat @ ( hales_set_incr @ n2 @ ( bs @ X ) ) @ ( hales_set_incr @ n2 @ ( bs @ Xa ) ) )
= bot_bot_set_nat ) ) ) ) ).
% fact4
thf(fact_6_n__def,axiom,
( n2
= ( plus_plus_nat @ n @ d ) ) ).
% n_def
thf(fact_7_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_8_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_9__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_10_fact3,axiom,
! [X: nat] :
( ( member_nat @ X @ ( set_ord_lessThan_nat @ k ) )
=> ( ( inf_inf_set_nat @ ( bl @ zero_zero_nat ) @ ( hales_set_incr @ n2 @ ( bs @ X ) ) )
= bot_bot_set_nat ) ) ).
% fact3
thf(fact_11_F1,axiom,
~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ bt @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ) ) ).
% F1
thf(fact_12__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_13_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_14_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_15_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_16_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_17_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_18_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_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_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_29_image__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_30_image__eqI,axiom,
! [B: nat,F: set_nat > nat,X2: set_nat,A: set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A )
=> ( member_nat @ B @ ( image_set_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_31_image__eqI,axiom,
! [B: set_nat,F: nat > set_nat,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_32_image__eqI,axiom,
! [B: set_nat,F: set_nat > set_nat,X2: set_nat,A: set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_33_image__eqI,axiom,
! [B: nat > nat,F: nat > nat > nat,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_34_image__eqI,axiom,
! [B: nat,F: ( nat > nat ) > nat,X2: nat > nat,A: set_nat_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_35_image__eqI,axiom,
! [B: nat > nat,F: set_nat > nat > nat,X2: set_nat,A: set_set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A )
=> ( member_nat_nat @ B @ ( image_8569768528772619084at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_36_image__eqI,axiom,
! [B: set_nat,F: ( nat > nat ) > set_nat,X2: nat > nat,A: set_nat_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_set_nat @ B @ ( image_7432509271690132940et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_37_image__eqI,axiom,
! [B: nat > nat,F: ( nat > nat ) > nat > nat,X2: nat > nat,A: set_nat_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_38_image__eqI,axiom,
! [B: ( nat > nat ) > nat > nat,F: nat > ( nat > nat ) > nat > nat,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member952132173341509300at_nat @ B @ ( image_6393715451659844596at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_39_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_40_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_41_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X3: $o] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_42_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_43_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_44_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X3: $o] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_45_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_46_all__not__in__conv,axiom,
! [A: set_nat_nat] :
( ( ! [X3: nat > nat] :
~ ( member_nat_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_47_all__not__in__conv,axiom,
! [A: set_nat_nat_nat_nat] :
( ( ! [X3: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ X3 @ A ) )
= ( A = bot_bo3919185967433191911at_nat ) ) ).
% all_not_in_conv
thf(fact_48_all__not__in__conv,axiom,
! [A: set_na7233567106578532785at_nat] :
( ( ! [X3: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( member8881365325514865170at_nat @ X3 @ A ) )
= ( A = bot_bo2676777031303994949at_nat ) ) ).
% all_not_in_conv
thf(fact_49_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_50_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X3: $o] :
~ ( member_o @ X3 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_51_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_52_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_53_empty__iff,axiom,
! [C: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ C @ bot_bo3919185967433191911at_nat ) ).
% empty_iff
thf(fact_54_empty__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( member8881365325514865170at_nat @ C @ bot_bo2676777031303994949at_nat ) ).
% empty_iff
thf(fact_55_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_56_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_57_Int__iff,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) )
= ( ( member_set_nat @ C @ A )
& ( member_set_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_58_Int__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B2 ) )
= ( ( member_nat_nat @ C @ A )
& ( member_nat_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_59_Int__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A @ B2 ) )
= ( ( member952132173341509300at_nat @ C @ A )
& ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_60_Int__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A @ B2 ) )
= ( ( member8881365325514865170at_nat @ C @ A )
& ( member8881365325514865170at_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_61_Int__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_62_IntI,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_63_IntI,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( ( member_nat_nat @ C @ B2 )
=> ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_64_IntI,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ A )
=> ( ( member952132173341509300at_nat @ C @ B2 )
=> ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_65_IntI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ A )
=> ( ( member8881365325514865170at_nat @ C @ B2 )
=> ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_66_IntI,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_67_Un__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B2 ) )
= ( ( member_nat_nat @ C @ A )
| ( member_nat_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_68_Un__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A @ B2 ) )
= ( ( member952132173341509300at_nat @ C @ A )
| ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_69_Un__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A @ B2 ) )
= ( ( member8881365325514865170at_nat @ C @ A )
| ( member8881365325514865170at_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_70_Un__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_71_Un__iff,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B2 ) )
= ( ( member_set_nat @ C @ A )
| ( member_set_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_72_UnCI,axiom,
! [C: nat > nat,B2: set_nat_nat,A: set_nat_nat] :
( ( ~ ( member_nat_nat @ C @ B2 )
=> ( member_nat_nat @ C @ A ) )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B2 ) ) ) ).
% UnCI
thf(fact_73_UnCI,axiom,
! [C: ( nat > nat ) > nat > nat,B2: set_nat_nat_nat_nat,A: set_nat_nat_nat_nat] :
( ( ~ ( member952132173341509300at_nat @ C @ B2 )
=> ( member952132173341509300at_nat @ C @ A ) )
=> ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A @ B2 ) ) ) ).
% UnCI
thf(fact_74_UnCI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,B2: set_na7233567106578532785at_nat,A: set_na7233567106578532785at_nat] :
( ( ~ ( member8881365325514865170at_nat @ C @ B2 )
=> ( member8881365325514865170at_nat @ C @ A ) )
=> ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A @ B2 ) ) ) ).
% UnCI
thf(fact_75_UnCI,axiom,
! [C: nat,B2: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnCI
thf(fact_76_UnCI,axiom,
! [C: set_nat,B2: set_set_nat,A: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ A ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B2 ) ) ) ).
% UnCI
thf(fact_77_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_78_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_79_image__is__empty,axiom,
! [F: $o > nat,A: set_o] :
( ( ( image_o_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_80_image__is__empty,axiom,
! [F: nat > $o,A: set_nat] :
( ( ( image_nat_o @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_81_image__is__empty,axiom,
! [F: $o > $o,A: set_o] :
( ( ( image_o_o @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_82_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_83_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_84_empty__is__image,axiom,
! [F: $o > nat,A: set_o] :
( ( bot_bot_set_nat
= ( image_o_nat @ F @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_85_empty__is__image,axiom,
! [F: nat > $o,A: set_nat] :
( ( bot_bot_set_o
= ( image_nat_o @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_86_empty__is__image,axiom,
! [F: $o > $o,A: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_87_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_88_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_89_image__empty,axiom,
! [F: nat > $o] :
( ( image_nat_o @ F @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% image_empty
thf(fact_90_image__empty,axiom,
! [F: $o > nat] :
( ( image_o_nat @ F @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_91_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_92_Un__empty,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A @ B2 )
= bot_bot_set_set_nat )
= ( ( A = bot_bot_set_set_nat )
& ( B2 = bot_bot_set_set_nat ) ) ) ).
% Un_empty
thf(fact_93_Un__empty,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_94_Un__empty,axiom,
! [A: set_o,B2: set_o] :
( ( ( sup_sup_set_o @ A @ B2 )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_95_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_96_Bvar__def,axiom,
( bvar
= ( ^ [I: nat] : ( if_set_nat @ ( I = zero_zero_nat ) @ ( bl @ zero_zero_nat ) @ ( hales_set_incr @ n2 @ ( bs @ ( minus_minus_nat @ I @ one_one_nat ) ) ) ) ) ) ).
% Bvar_def
thf(fact_97_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_98_rev__image__eqI,axiom,
! [X2: set_nat,A: set_set_nat,B: nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_set_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_99_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_100_rev__image__eqI,axiom,
! [X2: set_nat,A: set_set_nat,B: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_101_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat @ B @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_102_rev__image__eqI,axiom,
! [X2: nat > nat,A: set_nat_nat,B: nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_103_rev__image__eqI,axiom,
! [X2: set_nat,A: set_set_nat,B: nat > nat,F: set_nat > nat > nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat @ B @ ( image_8569768528772619084at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_104_rev__image__eqI,axiom,
! [X2: nat > nat,A: set_nat_nat,B: set_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_nat @ B @ ( image_7432509271690132940et_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_105_rev__image__eqI,axiom,
! [X2: nat > nat,A: set_nat_nat,B: nat > nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat_nat @ B @ ( image_3205354838064109189at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_106_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B: ( nat > nat ) > nat > nat,F: nat > ( nat > nat ) > nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B
= ( F @ X2 ) )
=> ( member952132173341509300at_nat @ B @ ( image_6393715451659844596at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_107_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 ) )
=> ! [X: nat] :
( ( member_nat @ X @ A )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_108_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X: nat] :
( ( member_nat @ X @ A )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_109_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M = N )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_set_nat @ F @ M )
= ( image_nat_set_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_110_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_111_bex__imageD,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ? [X: set_nat] :
( ( member_set_nat @ X @ ( image_nat_set_nat @ F @ A ) )
& ( P @ X ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_112_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
& ( P @ X ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_113_image__iff,axiom,
! [Z: set_nat,F: nat > set_nat,A: set_nat] :
( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_114_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_115_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_116_imageI,axiom,
! [X2: set_nat,A: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_set_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_117_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_118_imageI,axiom,
! [X2: set_nat,A: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ).
% imageI
thf(fact_119_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_nat_nat_nat2 @ F @ A ) ) ) ).
% imageI
thf(fact_120_imageI,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_121_imageI,axiom,
! [X2: set_nat,A: set_set_nat,F: set_nat > nat > nat] :
( ( member_set_nat @ X2 @ A )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_8569768528772619084at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_122_imageI,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_7432509271690132940et_nat @ F @ A ) ) ) ).
% imageI
thf(fact_123_imageI,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_3205354838064109189at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_124_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > ( nat > nat ) > nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member952132173341509300at_nat @ ( F @ X2 ) @ ( image_6393715451659844596at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_125_image__Un,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_126_image__Un,axiom,
! [F: nat > set_nat,A: set_nat,B2: set_nat] :
( ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_127_image__Un,axiom,
! [F: set_nat > nat,A: set_set_nat,B2: set_set_nat] :
( ( image_set_nat_nat @ F @ ( sup_sup_set_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ ( image_set_nat_nat @ F @ A ) @ ( image_set_nat_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_128_image__Un,axiom,
! [F: set_nat > set_nat,A: set_set_nat,B2: set_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( sup_sup_set_set_nat @ A @ B2 ) )
= ( sup_sup_set_set_nat @ ( image_7916887816326733075et_nat @ F @ A ) @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_129_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_130_ex__in__conv,axiom,
! [A: set_nat_nat] :
( ( ? [X3: nat > nat] : ( member_nat_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_131_ex__in__conv,axiom,
! [A: set_nat_nat_nat_nat] :
( ( ? [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A ) )
= ( A != bot_bo3919185967433191911at_nat ) ) ).
% ex_in_conv
thf(fact_132_ex__in__conv,axiom,
! [A: set_na7233567106578532785at_nat] :
( ( ? [X3: ( nat > nat ) > ( nat > nat ) > nat > nat] : ( member8881365325514865170at_nat @ X3 @ A ) )
= ( A != bot_bo2676777031303994949at_nat ) ) ).
% ex_in_conv
thf(fact_133_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_134_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X3: $o] : ( member_o @ X3 @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_135_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y: set_nat] :
~ ( member_set_nat @ Y @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_136_equals0I,axiom,
! [A: set_nat_nat] :
( ! [Y: nat > nat] :
~ ( member_nat_nat @ Y @ A )
=> ( A = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_137_equals0I,axiom,
! [A: set_nat_nat_nat_nat] :
( ! [Y: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ Y @ A )
=> ( A = bot_bo3919185967433191911at_nat ) ) ).
% equals0I
thf(fact_138_equals0I,axiom,
! [A: set_na7233567106578532785at_nat] :
( ! [Y: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( member8881365325514865170at_nat @ Y @ A )
=> ( A = bot_bo2676777031303994949at_nat ) ) ).
% equals0I
thf(fact_139_equals0I,axiom,
! [A: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_140_equals0I,axiom,
! [A: set_o] :
( ! [Y: $o] :
~ ( member_o @ Y @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_141_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_142_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_143_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_144_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_145_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_146_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_147_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_148_Collect__mem__eq,axiom,
! [A: set_nat_nat_nat_nat] :
( ( collec3567154360959927026at_nat
@ ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_149_Collect__mem__eq,axiom,
! [A: set_na7233567106578532785at_nat] :
( ( collec6535634078845029456at_nat
@ ^ [X3: ( nat > nat ) > ( nat > nat ) > nat > nat] : ( member8881365325514865170at_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_150_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_151_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_152_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_153_equals0D,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( A = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_154_equals0D,axiom,
! [A: set_nat_nat_nat_nat,A2: ( nat > nat ) > nat > nat] :
( ( A = bot_bo3919185967433191911at_nat )
=> ~ ( member952132173341509300at_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_155_equals0D,axiom,
! [A: set_na7233567106578532785at_nat,A2: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( A = bot_bo2676777031303994949at_nat )
=> ~ ( member8881365325514865170at_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_156_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_157_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_158_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_159_emptyE,axiom,
! [A2: nat > nat] :
~ ( member_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_160_emptyE,axiom,
! [A2: ( nat > nat ) > nat > nat] :
~ ( member952132173341509300at_nat @ A2 @ bot_bo3919185967433191911at_nat ) ).
% emptyE
thf(fact_161_emptyE,axiom,
! [A2: ( nat > nat ) > ( nat > nat ) > nat > nat] :
~ ( member8881365325514865170at_nat @ A2 @ bot_bo2676777031303994949at_nat ) ).
% emptyE
thf(fact_162_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_163_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_164_Int__left__commute,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_165_Int__left__absorb,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A @ B2 ) ) ).
% Int_left_absorb
thf(fact_166_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_167_Int__absorb,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_168_Int__assoc,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_169_IntD2,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) )
=> ( member_set_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_170_IntD2,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B2 ) )
=> ( member_nat_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_171_IntD2,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A @ B2 ) )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_172_IntD2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A @ B2 ) )
=> ( member8881365325514865170at_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_173_IntD2,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
=> ( member_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_174_IntD1,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) )
=> ( member_set_nat @ C @ A ) ) ).
% IntD1
thf(fact_175_IntD1,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B2 ) )
=> ( member_nat_nat @ C @ A ) ) ).
% IntD1
thf(fact_176_IntD1,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A @ B2 ) )
=> ( member952132173341509300at_nat @ C @ A ) ) ).
% IntD1
thf(fact_177_IntD1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A @ B2 ) )
=> ( member8881365325514865170at_nat @ C @ A ) ) ).
% IntD1
thf(fact_178_IntD1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_179_IntE,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ~ ( member_set_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_180_IntE,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B2 ) )
=> ~ ( ( member_nat_nat @ C @ A )
=> ~ ( member_nat_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_181_IntE,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( inf_in2949407623404935909at_nat @ A @ B2 ) )
=> ~ ( ( member952132173341509300at_nat @ C @ A )
=> ~ ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_182_IntE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( inf_in6008378084349164867at_nat @ A @ B2 ) )
=> ~ ( ( member8881365325514865170at_nat @ C @ A )
=> ~ ( member8881365325514865170at_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_183_IntE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_184_Un__left__commute,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_185_Un__left__commute,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_set_nat @ B2 @ ( sup_sup_set_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_186_Un__left__absorb,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% Un_left_absorb
thf(fact_187_Un__left__absorb,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ A @ B2 ) )
= ( sup_sup_set_set_nat @ A @ B2 ) ) ).
% Un_left_absorb
thf(fact_188_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_189_Un__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] : ( sup_sup_set_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_190_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_191_Un__absorb,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_192_Un__assoc,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_193_Un__assoc,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B2 ) @ C2 )
= ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_194_ball__Un,axiom,
! [A: set_nat,B2: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A @ B2 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_195_ball__Un,axiom,
! [A: set_set_nat,B2: set_set_nat,P: set_nat > $o] :
( ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ A @ B2 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_196_bex__Un,axiom,
! [A: set_nat,B2: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A @ B2 ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_197_bex__Un,axiom,
! [A: set_set_nat,B2: set_set_nat,P: set_nat > $o] :
( ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ A @ B2 ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_198_UnI2,axiom,
! [C: nat > nat,B2: set_nat_nat,A: set_nat_nat] :
( ( member_nat_nat @ C @ B2 )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B2 ) ) ) ).
% UnI2
thf(fact_199_UnI2,axiom,
! [C: ( nat > nat ) > nat > nat,B2: set_nat_nat_nat_nat,A: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ B2 )
=> ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A @ B2 ) ) ) ).
% UnI2
thf(fact_200_UnI2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,B2: set_na7233567106578532785at_nat,A: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ B2 )
=> ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A @ B2 ) ) ) ).
% UnI2
thf(fact_201_UnI2,axiom,
! [C: nat,B2: set_nat,A: set_nat] :
( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnI2
thf(fact_202_UnI2,axiom,
! [C: set_nat,B2: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B2 ) ) ) ).
% UnI2
thf(fact_203_UnI1,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B2 ) ) ) ).
% UnI1
thf(fact_204_UnI1,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ A )
=> ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A @ B2 ) ) ) ).
% UnI1
thf(fact_205_UnI1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ A )
=> ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A @ B2 ) ) ) ).
% UnI1
thf(fact_206_UnI1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% UnI1
thf(fact_207_UnI1,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B2 ) ) ) ).
% UnI1
thf(fact_208_UnE,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B2 ) )
=> ( ~ ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_209_UnE,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( sup_su3836648520750444671at_nat @ A @ B2 ) )
=> ( ~ ( member952132173341509300at_nat @ C @ A )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_210_UnE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( sup_su8594648213498475741at_nat @ A @ B2 ) )
=> ( ~ ( member8881365325514865170at_nat @ C @ A )
=> ( member8881365325514865170at_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_211_UnE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_212_UnE,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B2 ) )
=> ( ~ ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_213_disjoint__iff__not__equal,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
=> ( X3 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_214_disjoint__iff__not__equal,axiom,
! [A: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A @ B2 )
= bot_bot_set_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ B2 )
=> ( X3 = (~ Y2) ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_215_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_216_Int__empty__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_217_Int__empty__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_218_Int__empty__left,axiom,
! [B2: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B2 )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_219_disjoint__iff,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A @ B2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ~ ( member_set_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_220_disjoint__iff,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A @ B2 )
= bot_bot_set_nat_nat )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ~ ( member_nat_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_221_disjoint__iff,axiom,
! [A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( ( inf_in2949407623404935909at_nat @ A @ B2 )
= bot_bo3919185967433191911at_nat )
= ( ! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A )
=> ~ ( member952132173341509300at_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_222_disjoint__iff,axiom,
! [A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( ( inf_in6008378084349164867at_nat @ A @ B2 )
= bot_bo2676777031303994949at_nat )
= ( ! [X3: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X3 @ A )
=> ~ ( member8881365325514865170at_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_223_disjoint__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_224_disjoint__iff,axiom,
! [A: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A @ B2 )
= bot_bot_set_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member_o @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_225_Int__emptyI,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ~ ( member_set_nat @ X4 @ B2 ) )
=> ( ( inf_inf_set_set_nat @ A @ B2 )
= bot_bot_set_set_nat ) ) ).
% Int_emptyI
thf(fact_226_Int__emptyI,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ~ ( member_nat_nat @ X4 @ B2 ) )
=> ( ( inf_inf_set_nat_nat @ A @ B2 )
= bot_bot_set_nat_nat ) ) ).
% Int_emptyI
thf(fact_227_Int__emptyI,axiom,
! [A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ! [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ A )
=> ~ ( member952132173341509300at_nat @ X4 @ B2 ) )
=> ( ( inf_in2949407623404935909at_nat @ A @ B2 )
= bot_bo3919185967433191911at_nat ) ) ).
% Int_emptyI
thf(fact_228_Int__emptyI,axiom,
! [A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ! [X4: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X4 @ A )
=> ~ ( member8881365325514865170at_nat @ X4 @ B2 ) )
=> ( ( inf_in6008378084349164867at_nat @ A @ B2 )
= bot_bo2676777031303994949at_nat ) ) ).
% Int_emptyI
thf(fact_229_Int__emptyI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_nat @ X4 @ B2 ) )
=> ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_230_Int__emptyI,axiom,
! [A: set_o,B2: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ~ ( member_o @ X4 @ B2 ) )
=> ( ( inf_inf_set_o @ A @ B2 )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_231_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_232_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_233_Un__empty__right,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% Un_empty_right
thf(fact_234_Un__empty__left,axiom,
! [B2: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_235_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_236_Un__empty__left,axiom,
! [B2: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_237_Un__Int__distrib2,axiom,
! [B2: set_nat,C2: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ C2 ) @ A )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ A ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_238_Un__Int__distrib2,axiom,
! [B2: set_set_nat,C2: set_set_nat,A: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B2 @ C2 ) @ A )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B2 @ A ) @ ( sup_sup_set_set_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_239_Int__Un__distrib2,axiom,
! [B2: set_nat,C2: set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ A ) @ ( inf_inf_set_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_240_Int__Un__distrib2,axiom,
! [B2: set_set_nat,C2: set_set_nat,A: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B2 @ A ) @ ( inf_inf_set_set_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_241_Un__Int__distrib,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_242_Un__Int__distrib,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( inf_inf_set_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A @ B2 ) @ ( sup_sup_set_set_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_243_Int__Un__distrib,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_244_Int__Un__distrib,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B2 ) @ ( inf_inf_set_set_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_245_Un__Int__crazy,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ B2 @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_246_Un__Int__crazy,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B2 ) @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) @ ( inf_inf_set_set_nat @ C2 @ A ) )
= ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A @ B2 ) @ ( sup_sup_set_set_nat @ B2 @ C2 ) ) @ ( sup_sup_set_set_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_247_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_248_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_249_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_250_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_251_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_252__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_253_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_254_lessThan__subset__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_255_sup__inf__absorb,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ X2 @ Y3 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_256_sup__inf__absorb,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ X2 @ Y3 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_257_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_258_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_259_subsetI,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ X4 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_260_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_261_subsetI,axiom,
! [A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ! [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ A )
=> ( member952132173341509300at_nat @ X4 @ B2 ) )
=> ( ord_le5260717879541182899at_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_262_subsetI,axiom,
! [A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ! [X4: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X4 @ A )
=> ( member8881365325514865170at_nat @ X4 @ B2 ) )
=> ( ord_le8099187209609443857at_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_263_subsetI,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( member_nat_nat @ X4 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_264_subset__antisym,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_265_inf_Oidem,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_266_inf__idem,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_267_inf_Oleft__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_268_inf__left__idem,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ X2 @ Y3 ) )
= ( inf_inf_set_nat @ X2 @ Y3 ) ) ).
% inf_left_idem
thf(fact_269_inf_Oright__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B )
= ( inf_inf_set_nat @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_270_inf__right__idem,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 )
= ( inf_inf_set_nat @ X2 @ Y3 ) ) ).
% inf_right_idem
thf(fact_271_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_272_sup_Oidem,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_273_sup__idem,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_274_sup__idem,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_275_sup_Oleft__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_276_sup_Oleft__idem,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B ) )
= ( sup_sup_set_set_nat @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_277_sup__left__idem,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) )
= ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% sup_left_idem
thf(fact_278_sup__left__idem,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y3 ) )
= ( sup_sup_set_set_nat @ X2 @ Y3 ) ) ).
% sup_left_idem
thf(fact_279_sup_Oright__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ B )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_280_sup_Oright__idem,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ B )
= ( sup_sup_set_set_nat @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_281_lessThan__eq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ( set_ord_lessThan_nat @ X2 )
= ( set_ord_lessThan_nat @ Y3 ) )
= ( X2 = Y3 ) ) ).
% lessThan_eq_iff
thf(fact_282_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_283_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_284_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_285_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_286_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_287_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_288_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_289_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_290_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_291_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_292_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_293_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_294_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_295_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_296_le__inf__iff,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) )
= ( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_297_le__inf__iff,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) )
= ( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_298_le__inf__iff,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ Y3 @ Z ) )
= ( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
& ( ord_le9059583361652607317at_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_299_inf_Obounded__iff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_300_inf_Obounded__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_301_inf_Obounded__iff,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C ) )
= ( ( ord_le9059583361652607317at_nat @ A2 @ B )
& ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_302_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_303_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_304_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_305_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_306_le__sup__iff,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ Z )
= ( ( ord_less_eq_set_nat @ X2 @ Z )
& ( ord_less_eq_set_nat @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_307_le__sup__iff,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X2 @ Y3 ) @ Z )
= ( ( ord_le6893508408891458716et_nat @ X2 @ Z )
& ( ord_le6893508408891458716et_nat @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_308_le__sup__iff,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ Z )
= ( ( ord_less_eq_nat @ X2 @ Z )
& ( ord_less_eq_nat @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_309_le__sup__iff,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X2 @ Y3 ) @ Z )
= ( ( ord_le9059583361652607317at_nat @ X2 @ Z )
& ( ord_le9059583361652607317at_nat @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_310_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_311_sup_Obounded__iff,axiom,
! [B: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B @ C ) @ A2 )
= ( ( ord_le6893508408891458716et_nat @ B @ A2 )
& ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_312_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_313_sup_Obounded__iff,axiom,
! [B: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B @ C ) @ A2 )
= ( ( ord_le9059583361652607317at_nat @ B @ A2 )
& ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_314_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_315_subset__empty,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_316_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_317_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_318_empty__subsetI,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% empty_subsetI
thf(fact_319_empty__subsetI,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% empty_subsetI
thf(fact_320_inf__bot__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_321_inf__bot__left,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X2 )
= bot_bot_set_o ) ).
% inf_bot_left
thf(fact_322_inf__bot__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_323_inf__bot__right,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ X2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% inf_bot_right
thf(fact_324_sup__bot__left,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_325_sup__bot__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_326_sup__bot__left,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_327_sup__bot__right,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ bot_bot_set_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_328_sup__bot__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_329_sup__bot__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
= X2 ) ).
% sup_bot_right
thf(fact_330_bot__eq__sup__iff,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ X2 @ Y3 ) )
= ( ( X2 = bot_bot_set_set_nat )
& ( Y3 = bot_bot_set_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_331_bot__eq__sup__iff,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X2 @ Y3 ) )
= ( ( X2 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_332_bot__eq__sup__iff,axiom,
! [X2: set_o,Y3: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ X2 @ Y3 ) )
= ( ( X2 = bot_bot_set_o )
& ( Y3 = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_333_sup__eq__bot__iff,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ( ( sup_sup_set_set_nat @ X2 @ Y3 )
= bot_bot_set_set_nat )
= ( ( X2 = bot_bot_set_set_nat )
& ( Y3 = bot_bot_set_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_334_sup__eq__bot__iff,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y3 )
= bot_bot_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_335_sup__eq__bot__iff,axiom,
! [X2: set_o,Y3: set_o] :
( ( ( sup_sup_set_o @ X2 @ Y3 )
= bot_bot_set_o )
= ( ( X2 = bot_bot_set_o )
& ( Y3 = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_336_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A2 @ B )
= bot_bot_set_set_nat )
= ( ( A2 = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_337_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_338_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ( A2 = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_339_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_340_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_341_sup__bot_Oleft__neutral,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_342_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ A2 @ B ) )
= ( ( A2 = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_343_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B ) )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_344_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_o,B: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ A2 @ B ) )
= ( ( A2 = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_345_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_346_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_347_sup__bot_Oright__neutral,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_348_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_349_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_350_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_351_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_352_inf__sup__absorb,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_353_inf__sup__absorb,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y3 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_354_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_355_Int__subset__iff,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B2 ) )
= ( ( ord_le9059583361652607317at_nat @ C2 @ A )
& ( ord_le9059583361652607317at_nat @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_356_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_357_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_358_Un__subset__iff,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_359_Un__subset__iff,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B2 ) @ C2 )
= ( ( ord_le6893508408891458716et_nat @ A @ C2 )
& ( ord_le6893508408891458716et_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_360_Un__subset__iff,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B2 ) @ C2 )
= ( ( ord_le9059583361652607317at_nat @ A @ C2 )
& ( ord_le9059583361652607317at_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_361_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_362_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_363_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_364_atLeastLessThan__empty,axiom,
! [B: $o,A2: $o] :
( ( ord_less_eq_o @ B @ A2 )
=> ( ( set_or7139685690850216873Than_o @ A2 @ B )
= bot_bot_set_o ) ) ).
% atLeastLessThan_empty
thf(fact_365_atLeastLessThan__empty,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( ( set_or9117062992132219044at_nat @ A2 @ B )
= bot_bo7376149671870096959at_nat ) ) ).
% atLeastLessThan_empty
thf(fact_366_atLeastLessThan__empty,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( set_or4665077453230672383an_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_367_ivl__subset,axiom,
! [I2: nat,J: nat,M2: nat,N2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ J @ I2 )
| ( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_eq_nat @ J @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_368_image__add__atLeastLessThan,axiom,
! [K: nat,I2: nat,J: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ K ) @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
= ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% image_add_atLeastLessThan
thf(fact_369_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_370_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_371_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_372_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_373_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_374_in__mono,axiom,
! [A: set_set_nat,B2: set_set_nat,X2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_375_in__mono,axiom,
! [A: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_376_in__mono,axiom,
! [A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat,X2: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A @ B2 )
=> ( ( member952132173341509300at_nat @ X2 @ A )
=> ( member952132173341509300at_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_377_in__mono,axiom,
! [A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat,X2: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( ord_le8099187209609443857at_nat @ A @ B2 )
=> ( ( member8881365325514865170at_nat @ X2 @ A )
=> ( member8881365325514865170at_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_378_in__mono,axiom,
! [A: set_nat_nat,B2: set_nat_nat,X2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_379_subsetD,axiom,
! [A: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_380_subsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_381_subsetD,axiom,
! [A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat,C: ( nat > nat ) > nat > nat] :
( ( ord_le5260717879541182899at_nat @ A @ B2 )
=> ( ( member952132173341509300at_nat @ C @ A )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_382_subsetD,axiom,
! [A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat,C: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( ord_le8099187209609443857at_nat @ A @ B2 )
=> ( ( member8881365325514865170at_nat @ C @ A )
=> ( member8881365325514865170at_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_383_subsetD,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_384_equalityE,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( A = B2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ A ) ) ) ).
% equalityE
thf(fact_385_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_386_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_387_subset__eq,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
! [X3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X3 @ A3 )
=> ( member952132173341509300at_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_388_subset__eq,axiom,
( ord_le8099187209609443857at_nat
= ( ^ [A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
! [X3: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ X3 @ A3 )
=> ( member8881365325514865170at_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_389_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
=> ( member_nat_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_390_equalityD1,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( A = B2 )
=> ( ord_le9059583361652607317at_nat @ A @ B2 ) ) ).
% equalityD1
thf(fact_391_equalityD2,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( A = B2 )
=> ( ord_le9059583361652607317at_nat @ B2 @ A ) ) ).
% equalityD2
thf(fact_392_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [T2: set_nat] :
( ( member_set_nat @ T2 @ A3 )
=> ( member_set_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_393_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_394_subset__iff,axiom,
( ord_le5260717879541182899at_nat
= ( ^ [A3: set_nat_nat_nat_nat,B3: set_nat_nat_nat_nat] :
! [T2: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ T2 @ A3 )
=> ( member952132173341509300at_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_395_subset__iff,axiom,
( ord_le8099187209609443857at_nat
= ( ^ [A3: set_na7233567106578532785at_nat,B3: set_na7233567106578532785at_nat] :
! [T2: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ T2 @ A3 )
=> ( member8881365325514865170at_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_396_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [T2: nat > nat] :
( ( member_nat_nat @ T2 @ A3 )
=> ( member_nat_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_397_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% subset_refl
thf(fact_398_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_399_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_400_subset__trans,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
=> ( ord_le9059583361652607317at_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_401_set__eq__subset,axiom,
( ( ^ [Y4: set_nat_nat,Z2: set_nat_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_402_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_403_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_404_diff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_405_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_406_atLeastLessThan__subset__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A2 @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_407_add__implies__diff,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_408_diff__diff__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_409_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_410_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_411_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_412_diff__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_413_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_414_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_415_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_416_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_417_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_418_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_419_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_420_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_421_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_422_subset__image__iff,axiom,
! [B2: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_423_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_424_subset__image__iff,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A )
& ( B2
= ( image_3205354838064109189at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_425_image__subset__iff,axiom,
! [F: nat > set_nat,A: set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_set_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_426_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_427_subset__imageE,axiom,
! [B2: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B2
!= ( image_nat_set_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_428_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B2
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_429_subset__imageE,axiom,
! [B2: set_nat_nat,F: ( nat > nat ) > nat > nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_3205354838064109189at_nat @ F @ A ) )
=> ~ ! [C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C3 @ A )
=> ( B2
!= ( image_3205354838064109189at_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_430_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_431_image__subsetI,axiom,
! [A: set_set_nat,F: set_nat > nat,B2: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_432_image__subsetI,axiom,
! [A: set_nat,F: nat > set_nat,B2: set_set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_433_image__subsetI,axiom,
! [A: set_set_nat,F: set_nat > set_nat,B2: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_434_image__subsetI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat,B2: set_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_435_image__subsetI,axiom,
! [A: set_nat,F: nat > nat > nat,B2: set_nat_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_436_image__subsetI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > set_nat,B2: set_set_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7432509271690132940et_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_437_image__subsetI,axiom,
! [A: set_set_nat,F: set_nat > nat > nat,B2: set_nat_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_nat_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_8569768528772619084at_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_438_image__subsetI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat > nat,B2: set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( member_nat_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_439_image__subsetI,axiom,
! [A: set_nat,F: nat > ( nat > nat ) > nat > nat,B2: set_nat_nat_nat_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member952132173341509300at_nat @ ( F @ X4 ) @ B2 ) )
=> ( ord_le5260717879541182899at_nat @ ( image_6393715451659844596at_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_440_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_441_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_442_image__mono,axiom,
! [A: set_nat_nat,B2: set_nat_nat,F: ( nat > nat ) > nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ ( image_3205354838064109189at_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_443_Int__mono,axiom,
! [A: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_444_Int__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B2 ) @ ( inf_inf_set_nat_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_445_Int__lower1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ A ) ).
% Int_lower1
thf(fact_446_Int__lower1,axiom,
! [A: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B2 ) @ A ) ).
% Int_lower1
thf(fact_447_Int__lower2,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_448_Int__lower2,axiom,
! [A: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_449_Int__absorb1,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( inf_inf_set_nat @ A @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_450_Int__absorb1,axiom,
! [B2: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A )
=> ( ( inf_inf_set_nat_nat @ A @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_451_Int__absorb2,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( inf_inf_set_nat @ A @ B2 )
= A ) ) ).
% Int_absorb2
thf(fact_452_Int__absorb2,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( inf_inf_set_nat_nat @ A @ B2 )
= A ) ) ).
% Int_absorb2
thf(fact_453_Int__greatest,axiom,
! [C2: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_454_Int__greatest,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A )
=> ( ( ord_le9059583361652607317at_nat @ C2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_455_Int__Collect__mono,axiom,
! [A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat,P: ( ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > nat > nat ) > $o] :
( ( ord_le5260717879541182899at_nat @ A @ B2 )
=> ( ! [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 @ B2 @ ( collec3567154360959927026at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_456_Int__Collect__mono,axiom,
! [A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat,P: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o,Q: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o] :
( ( ord_le8099187209609443857at_nat @ A @ B2 )
=> ( ! [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 @ B2 @ ( collec6535634078845029456at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_457_Int__Collect__mono,axiom,
! [A: set_set_nat,B2: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ! [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 @ B2 @ ( collect_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_458_Int__Collect__mono,axiom,
! [A: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [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 @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_459_Int__Collect__mono,axiom,
! [A: set_nat_nat,B2: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ! [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 @ B2 @ ( collect_nat_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_460_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_461_Un__mono,axiom,
! [A: set_set_nat,C2: set_set_nat,B2: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B2 ) @ ( sup_sup_set_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_462_Un__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B2 ) @ ( sup_sup_set_nat_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_463_Un__least,axiom,
! [A: set_nat,C2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_464_Un__least,axiom,
! [A: set_set_nat,C2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_465_Un__least,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_466_Un__upper1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% Un_upper1
thf(fact_467_Un__upper1,axiom,
! [A: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ ( sup_sup_set_set_nat @ A @ B2 ) ) ).
% Un_upper1
thf(fact_468_Un__upper1,axiom,
! [A: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ ( sup_sup_set_nat_nat @ A @ B2 ) ) ).
% Un_upper1
thf(fact_469_Un__upper2,axiom,
! [B2: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% Un_upper2
thf(fact_470_Un__upper2,axiom,
! [B2: set_set_nat,A: set_set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ ( sup_sup_set_set_nat @ A @ B2 ) ) ).
% Un_upper2
thf(fact_471_Un__upper2,axiom,
! [B2: set_nat_nat,A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B2 @ ( sup_sup_set_nat_nat @ A @ B2 ) ) ).
% Un_upper2
thf(fact_472_Un__absorb1,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_473_Un__absorb1,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( sup_sup_set_set_nat @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_474_Un__absorb1,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( sup_sup_set_nat_nat @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_475_Un__absorb2,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_476_Un__absorb2,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A )
=> ( ( sup_sup_set_set_nat @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_477_Un__absorb2,axiom,
! [B2: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A )
=> ( ( sup_sup_set_nat_nat @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_478_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B2 ) )
=> ~ ! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ A )
=> ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ B2 )
=> ( C2
!= ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_479_subset__UnE,axiom,
! [C2: set_set_nat,A: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B2 ) )
=> ~ ! [A4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ A )
=> ! [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ B2 )
=> ( C2
!= ( sup_sup_set_set_nat @ A4 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_480_subset__UnE,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B2 ) )
=> ~ ! [A4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ A )
=> ! [B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ B2 )
=> ( C2
!= ( sup_sup_set_nat_nat @ A4 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_481_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_482_subset__Un__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_483_subset__Un__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_484_diff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% diff_add
thf(fact_485_le__add__diff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_486_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_487_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_488_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_489_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_490_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_491_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_492_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_493_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C )
= ( B
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_494_ivl__disj__un__two_I3_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_495_ivl__disj__int__two_I3_J,axiom,
! [L: $o,M2: $o,U: $o] :
( ( inf_inf_set_o @ ( set_or7139685690850216873Than_o @ L @ M2 ) @ ( set_or7139685690850216873Than_o @ M2 @ U ) )
= bot_bot_set_o ) ).
% ivl_disj_int_two(3)
thf(fact_496_ivl__disj__int__two_I3_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(3)
thf(fact_497_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_498_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_499_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_500_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_501_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_502_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_503_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_504_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_505_ivl__disj__un__one_I2_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( set_ord_lessThan_nat @ U ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_506_ivl__disj__int__one_I2_J,axiom,
! [L: $o,U: $o] :
( ( inf_inf_set_o @ ( set_ord_lessThan_o @ L ) @ ( set_or7139685690850216873Than_o @ L @ U ) )
= bot_bot_set_o ) ).
% ivl_disj_int_one(2)
thf(fact_507_ivl__disj__int__one_I2_J,axiom,
! [L: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or4665077453230672383an_nat @ L @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(2)
thf(fact_508_image__Int__subset,axiom,
! [F: nat > set_nat,A: set_nat,B2: set_nat] : ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( inf_inf_set_nat @ A @ B2 ) ) @ ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_509_image__Int__subset,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A @ B2 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_510_image__Int__subset,axiom,
! [F: nat > nat > nat,A: set_nat,B2: set_nat] : ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ ( inf_inf_set_nat @ A @ B2 ) ) @ ( inf_inf_set_nat_nat @ ( image_nat_nat_nat2 @ F @ A ) @ ( image_nat_nat_nat2 @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_511_Un__Int__assoc__eq,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_512_Un__Int__assoc__eq,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B2 @ C2 ) ) )
= ( ord_le6893508408891458716et_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_513_Un__Int__assoc__eq,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B2 @ C2 ) ) )
= ( ord_le9059583361652607317at_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_514_set__incr__altdef,axiom,
( hales_set_incr
= ( ^ [N3: nat] : ( image_nat_nat @ ( plus_plus_nat @ N3 ) ) ) ) ).
% set_incr_altdef
thf(fact_515_atLeastLessThan__add__Un,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_516_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_517_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_518_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_519_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_520_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_521_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_522_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B5: nat] : ( plus_plus_nat @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_523_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_524_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_525_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_526_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_527_inf__sup__aci_I4_J,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ X2 @ Y3 ) )
= ( inf_inf_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_528_inf__sup__aci_I3_J,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) )
= ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_529_inf__sup__aci_I2_J,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Z )
= ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_530_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_531_inf_Oassoc,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_532_inf__assoc,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Z )
= ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_533_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( inf_inf_set_nat @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_534_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ Y2 @ X3 ) ) ) ).
% inf_commute
thf(fact_535_inf_Oleft__commute,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A2 @ C ) )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_536_inf__left__commute,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) )
= ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_537_inf__sup__aci_I8_J,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) )
= ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_538_inf__sup__aci_I8_J,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y3 ) )
= ( sup_sup_set_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_539_inf__sup__aci_I7_J,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_540_inf__sup__aci_I7_J,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_set_nat @ Y3 @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_541_inf__sup__aci_I6_J,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ Z )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_542_inf__sup__aci_I6_J,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y3 ) @ Z )
= ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y3 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_543_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_544_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_nat
= ( ^ [X3: set_set_nat,Y2: set_set_nat] : ( sup_sup_set_set_nat @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_545_sup_Oassoc,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_546_sup_Oassoc,axiom,
! [A2: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ C )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_547_sup__assoc,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ Z )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% sup_assoc
thf(fact_548_sup__assoc,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y3 ) @ Z )
= ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y3 @ Z ) ) ) ).
% sup_assoc
thf(fact_549_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_550_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_551_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X3 ) ) ) ).
% sup_commute
thf(fact_552_sup__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [X3: set_set_nat,Y2: set_set_nat] : ( sup_sup_set_set_nat @ Y2 @ X3 ) ) ) ).
% sup_commute
thf(fact_553_sup_Oleft__commute,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_554_sup_Oleft__commute,axiom,
! [B: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ B @ ( sup_sup_set_set_nat @ A2 @ C ) )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_555_sup__left__commute,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_556_sup__left__commute,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_set_nat @ Y3 @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_557_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_558_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_559_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_560_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_561_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_562_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_563_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_564_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_565_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_566_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_567_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_568_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_569_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_570_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A2 @ C4 ) ) ) ).
% less_eqE
thf(fact_571_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_572_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
? [C5: nat] :
( B5
= ( plus_plus_nat @ A5 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_573_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_574_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_575_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_576_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_577_inf__sup__ord_I2_J,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_578_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_579_inf__sup__ord_I2_J,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_580_inf__sup__ord_I1_J,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_581_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_582_inf__sup__ord_I1_J,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_583_inf__le1,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_584_inf__le1,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_585_inf__le1,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_586_inf__le2,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_587_inf__le2,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_588_inf__le2,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_589_le__infE,axiom,
! [X2: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_590_le__infE,axiom,
! [X2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_591_le__infE,axiom,
! [X2: set_nat_nat,A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ A2 @ B ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ X2 @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_592_le__infI,axiom,
! [X2: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ B )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_593_le__infI,axiom,
! [X2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_594_le__infI,axiom,
! [X2: set_nat_nat,A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ B )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_595_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_596_inf__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_597_inf__mono,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ( ord_le9059583361652607317at_nat @ B @ D )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ ( inf_inf_set_nat_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_598_le__infI1,axiom,
! [A2: set_nat,X2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_599_le__infI1,axiom,
! [A2: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_600_le__infI1,axiom,
! [A2: set_nat_nat,X2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_601_le__infI2,axiom,
! [B: set_nat,X2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_602_le__infI2,axiom,
! [B: nat,X2: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_603_le__infI2,axiom,
! [B: set_nat_nat,X2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_604_inf_OorderE,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_605_inf_OorderE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( A2
= ( inf_inf_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_606_inf_OorderE,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( A2
= ( inf_inf_set_nat_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_607_inf_OorderI,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_608_inf_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_609_inf_OorderI,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( A2
= ( inf_inf_set_nat_nat @ A2 @ B ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_610_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y3: 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 @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_611_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y3: 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 @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_612_inf__unique,axiom,
! [F: set_nat_nat > set_nat_nat > set_nat_nat,X2: set_nat_nat,Y3: 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 @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_613_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_614_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( inf_inf_nat @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_615_le__iff__inf,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X3: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_616_inf_Oabsorb1,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( inf_inf_set_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_617_inf_Oabsorb1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_618_inf_Oabsorb1,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( inf_inf_set_nat_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_619_inf_Oabsorb2,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_620_inf_Oabsorb2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_621_inf_Oabsorb2,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_622_inf__absorb1,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( inf_inf_set_nat @ X2 @ Y3 )
= X2 ) ) ).
% inf_absorb1
thf(fact_623_inf__absorb1,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( inf_inf_nat @ X2 @ Y3 )
= X2 ) ) ).
% inf_absorb1
thf(fact_624_inf__absorb1,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( inf_inf_set_nat_nat @ X2 @ Y3 )
= X2 ) ) ).
% inf_absorb1
thf(fact_625_inf__absorb2,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( inf_inf_set_nat @ X2 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_626_inf__absorb2,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_627_inf__absorb2,axiom,
! [Y3: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y3 @ X2 )
=> ( ( inf_inf_set_nat_nat @ X2 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_628_inf_OboundedE,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_629_inf_OboundedE,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_630_inf_OboundedE,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ~ ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_631_inf_OboundedI,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_632_inf_OboundedI,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_633_inf_OboundedI,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_634_inf__greatest,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_635_inf__greatest,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_636_inf__greatest,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Z )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ Y3 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_637_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( A5
= ( inf_inf_set_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_638_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( A5
= ( inf_inf_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_639_inf_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( A5
= ( inf_inf_set_nat_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_640_inf_Ocobounded1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_641_inf_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_642_inf_Ocobounded1,axiom,
! [A2: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_643_inf_Ocobounded2,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_644_inf_Ocobounded2,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_645_inf_Ocobounded2,axiom,
! [A2: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_646_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_647_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_648_inf_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_649_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( inf_inf_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_650_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_651_inf_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B5: set_nat_nat,A5: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_652_inf_OcoboundedI1,axiom,
! [A2: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_653_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_654_inf_OcoboundedI1,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_655_inf_OcoboundedI2,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_656_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_657_inf_OcoboundedI2,axiom,
! [B: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_658_inf__sup__ord_I4_J,axiom,
! [Y3: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_659_inf__sup__ord_I4_J,axiom,
! [Y3: set_set_nat,X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y3 @ ( sup_sup_set_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_660_inf__sup__ord_I4_J,axiom,
! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_661_inf__sup__ord_I4_J,axiom,
! [Y3: set_nat_nat,X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y3 @ ( sup_sup_set_nat_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_662_inf__sup__ord_I3_J,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_663_inf__sup__ord_I3_J,axiom,
! [X2: set_set_nat,Y3: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_664_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_665_inf__sup__ord_I3_J,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_666_le__supE,axiom,
! [A2: set_nat,B: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ X2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_set_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_667_le__supE,axiom,
! [A2: set_set_nat,B: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ X2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ X2 )
=> ~ ( ord_le6893508408891458716et_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_668_le__supE,axiom,
! [A2: nat,B: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_669_le__supE,axiom,
! [A2: set_nat_nat,B: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B ) @ X2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ X2 )
=> ~ ( ord_le9059583361652607317at_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_670_le__supI,axiom,
! [A2: set_nat,X2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ( ( ord_less_eq_set_nat @ B @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_671_le__supI,axiom,
! [A2: set_set_nat,X2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_672_le__supI,axiom,
! [A2: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_673_le__supI,axiom,
! [A2: set_nat_nat,X2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_674_sup__ge1,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_675_sup__ge1,axiom,
! [X2: set_set_nat,Y3: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_676_sup__ge1,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_677_sup__ge1,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_678_sup__ge2,axiom,
! [Y3: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_679_sup__ge2,axiom,
! [Y3: set_set_nat,X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y3 @ ( sup_sup_set_set_nat @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_680_sup__ge2,axiom,
! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_681_sup__ge2,axiom,
! [Y3: set_nat_nat,X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y3 @ ( sup_sup_set_nat_nat @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_682_le__supI1,axiom,
! [X2: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_683_le__supI1,axiom,
! [X2: set_set_nat,A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_684_le__supI1,axiom,
! [X2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_685_le__supI1,axiom,
! [X2: set_nat_nat,A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_686_le__supI2,axiom,
! [X2: set_nat,B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ B )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_687_le__supI2,axiom,
! [X2: set_set_nat,B: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ B )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_688_le__supI2,axiom,
! [X2: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_689_le__supI2,axiom,
! [X2: set_nat_nat,B: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ B )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_690_sup_Omono,axiom,
! [C: set_nat,A2: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_691_sup_Omono,axiom,
! [C: set_set_nat,A2: set_set_nat,D: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ D @ B )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ C @ D ) @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_692_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_693_sup_Omono,axiom,
! [C: set_nat_nat,A2: set_nat_nat,D: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ D @ B )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ C @ D ) @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_694_sup__mono,axiom,
! [A2: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_695_sup__mono,axiom,
! [A2: set_set_nat,C: set_set_nat,B: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B @ D )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ ( sup_sup_set_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_696_sup__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_697_sup__mono,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ( ord_le9059583361652607317at_nat @ B @ D )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B ) @ ( sup_sup_set_nat_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_698_sup__least,axiom,
! [Y3: set_nat,X2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_699_sup__least,axiom,
! [Y3: set_set_nat,X2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y3 @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ Z @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_700_sup__least,axiom,
! [Y3: nat,X2: nat,Z: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_701_sup__least,axiom,
! [Y3: set_nat_nat,X2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y3 @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ Z @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_702_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X3 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_703_le__iff__sup,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_704_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y2: nat] :
( ( sup_sup_nat @ X3 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_705_le__iff__sup,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X3: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_706_sup_OorderE,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_707_sup_OorderE,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( A2
= ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_708_sup_OorderE,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_709_sup_OorderE,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( A2
= ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_710_sup_OorderI,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B ) )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_711_sup_OorderI,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( A2
= ( sup_sup_set_set_nat @ A2 @ B ) )
=> ( ord_le6893508408891458716et_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_712_sup_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_713_sup_OorderI,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( A2
= ( sup_sup_set_nat_nat @ A2 @ B ) )
=> ( ord_le9059583361652607317at_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_714_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y3: set_nat] :
( ! [X4: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X4 @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ Y @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X4 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X4 )
=> ( ord_less_eq_set_nat @ ( F @ Y @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_nat @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_715_sup__unique,axiom,
! [F: set_set_nat > set_set_nat > set_set_nat,X2: set_set_nat,Y3: set_set_nat] :
( ! [X4: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ X4 @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X4 )
=> ( ( ord_le6893508408891458716et_nat @ Z3 @ X4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ Y @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_set_nat @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_716_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y3: nat] :
( ! [X4: nat,Y: nat] : ( ord_less_eq_nat @ X4 @ ( F @ X4 @ Y ) )
=> ( ! [X4: nat,Y: nat] : ( ord_less_eq_nat @ Y @ ( F @ X4 @ Y ) )
=> ( ! [X4: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ( ( ord_less_eq_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ ( F @ Y @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_nat @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_717_sup__unique,axiom,
! [F: set_nat_nat > set_nat_nat > set_nat_nat,X2: set_nat_nat,Y3: set_nat_nat] :
( ! [X4: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X4 @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_nat_nat,Y: set_nat_nat,Z3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X4 )
=> ( ( ord_le9059583361652607317at_nat @ Z3 @ X4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ Y @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_nat_nat @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_718_sup_Oabsorb1,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_719_sup_Oabsorb1,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_720_sup_Oabsorb1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_721_sup_Oabsorb1,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( ( sup_sup_set_nat_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_722_sup_Oabsorb2,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_723_sup_Oabsorb2,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( sup_sup_set_set_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_724_sup_Oabsorb2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_725_sup_Oabsorb2,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( sup_sup_set_nat_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_726_sup__absorb1,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( sup_sup_set_nat @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_727_sup__absorb1,axiom,
! [Y3: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y3 @ X2 )
=> ( ( sup_sup_set_set_nat @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_728_sup__absorb1,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_729_sup__absorb1,axiom,
! [Y3: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y3 @ X2 )
=> ( ( sup_sup_set_nat_nat @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_730_sup__absorb2,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( sup_sup_set_nat @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_731_sup__absorb2,axiom,
! [X2: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y3 )
=> ( ( sup_sup_set_set_nat @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_732_sup__absorb2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_733_sup__absorb2,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( sup_sup_set_nat_nat @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_734_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B @ A2 )
=> ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_735_sup_OboundedE,axiom,
! [B: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ~ ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_736_sup_OboundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_737_sup_OboundedE,axiom,
! [B: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_738_sup_OboundedI,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_739_sup_OboundedI,axiom,
! [B: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_740_sup_OboundedI,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_741_sup_OboundedI,axiom,
! [B: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_742_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( A5
= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_743_sup_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B5: set_set_nat,A5: set_set_nat] :
( A5
= ( sup_sup_set_set_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_744_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_745_sup_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B5: set_nat_nat,A5: set_nat_nat] :
( A5
= ( sup_sup_set_nat_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_746_sup_Ocobounded1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_747_sup_Ocobounded1,axiom,
! [A2: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_748_sup_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_749_sup_Ocobounded1,axiom,
! [A2: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_750_sup_Ocobounded2,axiom,
! [B: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_751_sup_Ocobounded2,axiom,
! [B: set_set_nat,A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ B @ ( sup_sup_set_set_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_752_sup_Ocobounded2,axiom,
! [B: nat,A2: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_753_sup_Ocobounded2,axiom,
! [B: set_nat_nat,A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_754_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_755_sup_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B5: set_set_nat,A5: set_set_nat] :
( ( sup_sup_set_set_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_756_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_757_sup_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B5: set_nat_nat,A5: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_758_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_759_sup_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
( ( sup_sup_set_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_760_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_761_sup_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_762_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_763_sup_OcoboundedI1,axiom,
! [C: set_set_nat,A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_764_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_765_sup_OcoboundedI1,axiom,
! [C: set_nat_nat,A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ A2 )
=> ( ord_le9059583361652607317at_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_766_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_767_sup_OcoboundedI2,axiom,
! [C: set_set_nat,B: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_768_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_769_sup_OcoboundedI2,axiom,
! [C: set_nat_nat,B: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_le9059583361652607317at_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_770_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_771_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_772_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_773_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_774_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_775_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_776_distrib__imp1,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ! [X4: set_nat,Y: set_nat,Z3: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( sup_sup_set_nat @ Y @ Z3 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X4 @ Y ) @ ( inf_inf_set_nat @ X4 @ Z3 ) ) )
=> ( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_777_distrib__imp1,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] :
( ! [X4: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( inf_inf_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ Y @ Z3 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X4 @ Y ) @ ( inf_inf_set_set_nat @ X4 @ Z3 ) ) )
=> ( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y3 @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y3 ) @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_778_distrib__imp2,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ! [X4: set_nat,Y: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( inf_inf_set_nat @ Y @ Z3 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X4 @ Y ) @ ( sup_sup_set_nat @ X4 @ Z3 ) ) )
=> ( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_779_distrib__imp2,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] :
( ! [X4: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ ( inf_inf_set_set_nat @ Y @ Z3 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X4 @ Y ) @ ( sup_sup_set_set_nat @ X4 @ Z3 ) ) )
=> ( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y3 ) @ ( inf_inf_set_set_nat @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_780_inf__sup__distrib1,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_781_inf__sup__distrib1,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y3 ) @ ( inf_inf_set_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_782_inf__sup__distrib2,axiom,
! [Y3: set_nat,Z: set_nat,X2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ Z ) @ X2 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ X2 ) @ ( inf_inf_set_nat @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_783_inf__sup__distrib2,axiom,
! [Y3: set_set_nat,Z: set_set_nat,X2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y3 @ Z ) @ X2 )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y3 @ X2 ) @ ( inf_inf_set_set_nat @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_784_sup__inf__distrib1,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_785_sup__inf__distrib1,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y3 @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y3 ) @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_786_sup__inf__distrib2,axiom,
! [Y3: set_nat,Z: set_nat,X2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ Z ) @ X2 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ X2 ) @ ( sup_sup_set_nat @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_787_sup__inf__distrib2,axiom,
! [Y3: set_set_nat,Z: set_set_nat,X2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y3 @ Z ) @ X2 )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y3 @ X2 ) @ ( sup_sup_set_set_nat @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_788_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_789_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_790_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_791_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_792_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_793_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_794_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_795_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_796_trans__le__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_797_trans__le__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_798_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_799_add__decreasing,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_800_add__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_801_add__decreasing2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_802_add__increasing2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_803_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_804_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_805_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_806_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_807_distrib__inf__le,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) @ ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_808_distrib__inf__le,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y3 ) @ ( inf_inf_set_set_nat @ X2 @ Z ) ) @ ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y3 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_809_distrib__inf__le,axiom,
! [X2: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ ( inf_inf_nat @ X2 @ Z ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y3 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_810_distrib__inf__le,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y3 ) @ ( inf_inf_set_nat_nat @ X2 @ Z ) ) @ ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y3 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_811_distrib__sup__le,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_812_distrib__sup__le,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y3 @ Z ) ) @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y3 ) @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_813_distrib__sup__le,axiom,
! [X2: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ ( sup_sup_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_814_distrib__sup__le,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y3 @ Z ) ) @ ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y3 ) @ ( sup_sup_set_nat_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_815_Iio__eq__empty__iff,axiom,
! [N2: $o] :
( ( ( set_ord_lessThan_o @ N2 )
= bot_bot_set_o )
= ( N2 = bot_bot_o ) ) ).
% Iio_eq_empty_iff
thf(fact_816_Iio__eq__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_817_lessThan__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_818_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_819_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_820_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_821_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X2 )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_822_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_823_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_o] :
( ( inf_inf_set_o @ X2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_824_d__def,axiom,
( d
= ( minus_minus_nat @ m @ ( plus_plus_nat @ n @ m2 ) ) ) ).
% d_def
thf(fact_825_diff__shunt__var,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_826_diff__shunt__var,axiom,
! [X2: set_o,Y3: set_o] :
( ( ( minus_minus_set_o @ X2 @ Y3 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X2 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_827_diff__shunt__var,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ( minus_8121590178497047118at_nat @ X2 @ Y3 )
= bot_bot_set_nat_nat )
= ( ord_le9059583361652607317at_nat @ X2 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_828_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_829_M_H__prop,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ n @ m2 ) @ m ).
% M'_prop
thf(fact_830_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_831_Diff__cancel,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ A )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_832_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_833_empty__Diff,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_834_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_835_Diff__empty,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ bot_bot_set_o )
= A ) ).
% Diff_empty
thf(fact_836_Un__Diff__cancel2,axiom,
! [B2: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A ) @ A )
= ( sup_sup_set_nat @ B2 @ A ) ) ).
% Un_Diff_cancel2
thf(fact_837_Un__Diff__cancel2,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A ) @ A )
= ( sup_sup_set_set_nat @ B2 @ A ) ) ).
% Un_Diff_cancel2
thf(fact_838_Un__Diff__cancel,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B2 @ A ) )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_839_Un__Diff__cancel,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( minus_2163939370556025621et_nat @ B2 @ A ) )
= ( sup_sup_set_set_nat @ A @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_840__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_841_Diff__eq__empty__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_842_Diff__eq__empty__iff,axiom,
! [A: set_o,B2: set_o] :
( ( ( minus_minus_set_o @ A @ B2 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_843_Diff__eq__empty__iff,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( ( minus_8121590178497047118at_nat @ A @ B2 )
= bot_bot_set_nat_nat )
= ( ord_le9059583361652607317at_nat @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_844_ivl__diff,axiom,
! [I2: nat,N2: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ M2 ) @ ( set_or4665077453230672383an_nat @ I2 @ N2 ) )
= ( set_or4665077453230672383an_nat @ N2 @ M2 ) ) ) ).
% ivl_diff
thf(fact_845_Diff__disjoint,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B2 @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_846_Diff__disjoint,axiom,
! [A: set_o,B2: set_o] :
( ( inf_inf_set_o @ A @ ( minus_minus_set_o @ B2 @ A ) )
= bot_bot_set_o ) ).
% Diff_disjoint
thf(fact_847_lessThan__minus__lessThan,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( set_ord_lessThan_nat @ M2 ) )
= ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ).
% lessThan_minus_lessThan
thf(fact_848_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_849_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_850_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_851_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_852_double__diff,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
=> ( ( minus_8121590178497047118at_nat @ B2 @ ( minus_8121590178497047118at_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_853_Diff__subset,axiom,
! [A: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_854_Diff__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,D2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ D2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B2 ) @ ( minus_8121590178497047118at_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_855_Diff__Int__distrib2,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ C2 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_856_Diff__Int__distrib,axiom,
! [C2: set_nat,A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A ) @ ( inf_inf_set_nat @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_857_Diff__Diff__Int,axiom,
! [A: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ A @ ( minus_minus_set_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_858_Diff__Int2,axiom,
! [A: set_nat,C2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_859_Int__Diff,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_860_Un__Diff,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C2 ) @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_861_Un__Diff,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ ( sup_sup_set_set_nat @ A @ B2 ) @ C2 )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ C2 ) @ ( minus_2163939370556025621et_nat @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_862_image__diff__subset,axiom,
! [F: nat > set_nat,A: set_nat,B2: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B2 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_863_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% image_diff_subset
thf(fact_864_Int__Diff__disjoint,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ ( minus_minus_set_nat @ A @ B2 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_865_Int__Diff__disjoint,axiom,
! [A: set_o,B2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A @ B2 ) @ ( minus_minus_set_o @ A @ B2 ) )
= bot_bot_set_o ) ).
% Int_Diff_disjoint
thf(fact_866_Diff__triv,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_867_Diff__triv,axiom,
! [A: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A @ B2 )
= bot_bot_set_o )
=> ( ( minus_minus_set_o @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_868_Diff__partition,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B2 @ A ) )
= B2 ) ) ).
% Diff_partition
thf(fact_869_Diff__partition,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( sup_sup_set_set_nat @ A @ ( minus_2163939370556025621et_nat @ B2 @ A ) )
= B2 ) ) ).
% Diff_partition
thf(fact_870_Diff__partition,axiom,
! [A: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B2 )
=> ( ( sup_sup_set_nat_nat @ A @ ( minus_8121590178497047118at_nat @ B2 @ A ) )
= B2 ) ) ).
% Diff_partition
thf(fact_871_Diff__subset__conv,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ C2 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_872_Diff__subset__conv,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ B2 ) @ C2 )
= ( ord_le6893508408891458716et_nat @ A @ ( sup_sup_set_set_nat @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_873_Diff__subset__conv,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B2 ) @ C2 )
= ( ord_le9059583361652607317at_nat @ A @ ( sup_sup_set_nat_nat @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_874_Un__Diff__Int,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ ( inf_inf_set_nat @ A @ B2 ) )
= A ) ).
% Un_Diff_Int
thf(fact_875_Un__Diff__Int,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B2 ) @ ( inf_inf_set_set_nat @ A @ B2 ) )
= A ) ).
% Un_Diff_Int
thf(fact_876_Int__Diff__Un,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ ( minus_minus_set_nat @ A @ B2 ) )
= A ) ).
% Int_Diff_Un
thf(fact_877_Int__Diff__Un,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B2 ) @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
= A ) ).
% Int_Diff_Un
thf(fact_878_Diff__Int,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ ( minus_minus_set_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_879_Diff__Int,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( inf_inf_set_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B2 ) @ ( minus_2163939370556025621et_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_880_Diff__Un,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ ( minus_minus_set_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_881_Diff__Un,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( sup_sup_set_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B2 ) @ ( minus_2163939370556025621et_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_882_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A2: set_nat] :
( ( B2
= ( inf_inf_set_nat @ K @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_883_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B: set_nat] :
( ( A
= ( inf_inf_set_nat @ K @ A2 ) )
=> ( ( inf_inf_set_nat @ A @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_884_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A2: set_nat] :
( ( B2
= ( sup_sup_set_nat @ K @ B ) )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_885_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_set_nat,K: set_set_nat,B: set_set_nat,A2: set_set_nat] :
( ( B2
= ( sup_sup_set_set_nat @ K @ B ) )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= ( sup_sup_set_set_nat @ K @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_886_boolean__algebra__cancel_Osup1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ K @ A2 ) )
=> ( ( sup_sup_set_nat @ A @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_887_boolean__algebra__cancel_Osup1,axiom,
! [A: set_set_nat,K: set_set_nat,A2: set_set_nat,B: set_set_nat] :
( ( A
= ( sup_sup_set_set_nat @ K @ A2 ) )
=> ( ( sup_sup_set_set_nat @ A @ B )
= ( sup_sup_set_set_nat @ K @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_888_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_889_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_890_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_891_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ bot_bot_set_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_892_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_893_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_894_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y3: set_nat,Z: set_nat,X2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ Z ) @ X2 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ X2 ) @ ( sup_sup_set_nat @ Z @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_895_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y3: set_set_nat,Z: set_set_nat,X2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y3 @ Z ) @ X2 )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y3 @ X2 ) @ ( sup_sup_set_set_nat @ Z @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_896_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y3: set_nat,Z: set_nat,X2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ Z ) @ X2 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ X2 ) @ ( inf_inf_set_nat @ Z @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_897_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y3: set_set_nat,Z: set_set_nat,X2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y3 @ Z ) @ X2 )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y3 @ X2 ) @ ( inf_inf_set_set_nat @ Z @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_898_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_899_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y3 @ Z ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y3 ) @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_900_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_901_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_set_nat,Y3: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y3 ) @ ( inf_inf_set_set_nat @ X2 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_902_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_903_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_904_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_905_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N2: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_906_fT__def,axiom,
( fT
= ( ^ [X3: nat] : ( if_nat @ ( member_nat @ X3 @ ( bl @ one_one_nat ) ) @ ( fL @ X3 ) @ ( if_nat @ ( member_nat @ X3 @ ( hales_set_incr @ n2 @ ( bs @ k ) ) ) @ ( fS @ ( minus_minus_nat @ X3 @ n2 ) ) @ undefined_nat ) ) ) ) ).
% fT_def
thf(fact_907_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_908_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_909_dual__order_Orefl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_910_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_911_order__refl,axiom,
! [X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_912_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_913_Diff__iff,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
= ( ( member_set_nat @ C @ A )
& ~ ( member_set_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_914_Diff__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_915_Diff__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B2 ) )
= ( ( member_nat_nat @ C @ A )
& ~ ( member_nat_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_916_Diff__iff,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A @ B2 ) )
= ( ( member952132173341509300at_nat @ C @ A )
& ~ ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_917_Diff__iff,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( minus_9165053394918225162at_nat @ A @ B2 ) )
= ( ( member8881365325514865170at_nat @ C @ A )
& ~ ( member8881365325514865170at_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_918_DiffI,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ~ ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_919_DiffI,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_920_DiffI,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( ~ ( member_nat_nat @ C @ B2 )
=> ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_921_DiffI,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ A )
=> ( ~ ( member952132173341509300at_nat @ C @ B2 )
=> ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_922_DiffI,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ A )
=> ( ~ ( member8881365325514865170at_nat @ C @ B2 )
=> ( member8881365325514865170at_nat @ C @ ( minus_9165053394918225162at_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_923_atMost__eq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ( set_ord_atMost_nat @ X2 )
= ( set_ord_atMost_nat @ Y3 ) )
= ( X2 = Y3 ) ) ).
% atMost_eq_iff
thf(fact_924_atMost__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_925_atMost__iff,axiom,
! [I2: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I2 @ ( set_or9140604705432621368at_nat @ K ) )
= ( ord_less_eq_nat_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_926_atMost__iff,axiom,
! [I2: ( nat > nat ) > nat > nat,K: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I2 @ ( set_or3591701359631937174at_nat @ K ) )
= ( ord_le747776305331315197at_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_927_atMost__iff,axiom,
! [I2: ( nat > nat ) > ( nat > nat ) > nat > nat,K: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ I2 @ ( set_or9155507668907256820at_nat @ K ) )
= ( ord_le5526148332077535835at_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_928_atMost__iff,axiom,
! [I2: set_nat_nat,K: set_nat_nat] :
( ( member_set_nat_nat @ I2 @ ( set_or250740698829186286at_nat @ K ) )
= ( ord_le9059583361652607317at_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_929_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_930_Sup__atMost,axiom,
! [Y3: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ Y3 ) )
= Y3 ) ).
% Sup_atMost
thf(fact_931_Sup__atMost,axiom,
! [Y3: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ Y3 ) )
= Y3 ) ).
% Sup_atMost
thf(fact_932_BfL__props_I1_J,axiom,
disjoi6798895846410478970at_nat @ bl @ ( set_ord_atMost_nat @ one_one_nat ) ).
% BfL_props(1)
thf(fact_933_atMost__subset__iff,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ ( set_or250740698829186286at_nat @ X2 ) @ ( set_or250740698829186286at_nat @ Y3 ) )
= ( ord_le9059583361652607317at_nat @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_934_atMost__subset__iff,axiom,
! [X2: nat > nat,Y3: nat > nat] :
( ( ord_le9059583361652607317at_nat @ ( set_or9140604705432621368at_nat @ X2 ) @ ( set_or9140604705432621368at_nat @ Y3 ) )
= ( ord_less_eq_nat_nat @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_935_atMost__subset__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_936_BfS__props_I1_J,axiom,
disjoi6798895846410478970at_nat @ bs @ ( set_ord_atMost_nat @ k ) ).
% BfS_props(1)
thf(fact_937_F3,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ bt @ ( set_ord_atMost_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ) )
= ( set_ord_lessThan_nat @ ( plus_plus_nat @ n2 @ m2 ) ) ) ).
% F3
thf(fact_938_DiffD2,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
=> ~ ( member_set_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_939_DiffD2,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_940_DiffD2,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B2 ) )
=> ~ ( member_nat_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_941_DiffD2,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A @ B2 ) )
=> ~ ( member952132173341509300at_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_942_DiffD2,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( minus_9165053394918225162at_nat @ A @ B2 ) )
=> ~ ( member8881365325514865170at_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_943_DiffD1,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
=> ( member_set_nat @ C @ A ) ) ).
% DiffD1
thf(fact_944_DiffD1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_945_DiffD1,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B2 ) )
=> ( member_nat_nat @ C @ A ) ) ).
% DiffD1
thf(fact_946_DiffD1,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A @ B2 ) )
=> ( member952132173341509300at_nat @ C @ A ) ) ).
% DiffD1
thf(fact_947_DiffD1,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( minus_9165053394918225162at_nat @ A @ B2 ) )
=> ( member8881365325514865170at_nat @ C @ A ) ) ).
% DiffD1
thf(fact_948_DiffE,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_949_DiffE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_950_DiffE,axiom,
! [C: nat > nat,A: set_nat_nat,B2: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B2 ) )
=> ~ ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_951_DiffE,axiom,
! [C: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ C @ ( minus_4646100876039749548at_nat @ A @ B2 ) )
=> ~ ( ( member952132173341509300at_nat @ C @ A )
=> ( member952132173341509300at_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_952_DiffE,axiom,
! [C: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B2: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ C @ ( minus_9165053394918225162at_nat @ A @ B2 ) )
=> ~ ( ( member8881365325514865170at_nat @ C @ A )
=> ( member8881365325514865170at_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_953_not__empty__eq__Iic__eq__empty,axiom,
! [H: $o] :
( bot_bot_set_o
!= ( set_ord_atMost_o @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_954_not__empty__eq__Iic__eq__empty,axiom,
! [H: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_955_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_956_le__cases3,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_957_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_958_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat_nat,Z2: set_nat_nat] : ( Y4 = Z2 ) )
= ( ^ [X3: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
& ( ord_le9059583361652607317at_nat @ Y2 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_959_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_960_ord__eq__le__trans,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A2 = B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_961_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_962_ord__le__eq__trans,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_963_order__antisym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_964_order__antisym,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ord_le9059583361652607317at_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_965_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_966_order_Otrans,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_967_order__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_968_order__trans,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
=> ( ( ord_le9059583361652607317at_nat @ Y3 @ Z )
=> ( ord_le9059583361652607317at_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_969_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_970_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_971_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat_nat,Z2: set_nat_nat] : ( Y4 = 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_972_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_973_dual__order_Oantisym,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_974_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_975_dual__order_Otrans,axiom,
! [B: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_976_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_977_antisym,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_978_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_979_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat_nat,Z2: set_nat_nat] : ( Y4 = 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_980_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ 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_981_order__subst1,axiom,
! [A2: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ 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_982_order__subst1,axiom,
! [A2: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ 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_983_order__subst1,axiom,
! [A2: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ 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_984_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ 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_985_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ 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_986_order__subst2,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ 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_987_order__subst2,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ ( F @ B ) @ 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_988_order__eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_989_order__eq__refl,axiom,
! [X2: set_nat_nat,Y3: set_nat_nat] :
( ( X2 = Y3 )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_990_linorder__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_991_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ 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_992_ord__eq__le__subst,axiom,
! [A2: set_nat_nat,F: nat > set_nat_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ 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_993_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat_nat > nat,B: set_nat_nat,C: set_nat_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ 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_994_ord__eq__le__subst,axiom,
! [A2: set_nat_nat,F: set_nat_nat > set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le9059583361652607317at_nat @ B @ 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_995_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= 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_996_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat_nat,C: set_nat_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= 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_997_ord__le__eq__subst,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ( F @ B )
= 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_998_ord__le__eq__subst,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: set_nat_nat > set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ( F @ B )
= 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_999_linorder__le__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_1000_order__antisym__conv,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_1001_order__antisym__conv,axiom,
! [Y3: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y3 @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_1002_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_1003_bot_Oextremum,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% bot.extremum
thf(fact_1004_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_1005_bot_Oextremum,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% bot.extremum
thf(fact_1006_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_1007_bot_Oextremum__unique,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_1008_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1009_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_1010_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_1011_bot_Oextremum__uniqueI,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
=> ( A2 = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_1012_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1013_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_1014_join__def,axiom,
( hales_join_nat
= ( ^ [F2: nat > nat,G2: nat > nat,N3: nat,M4: nat,X3: nat] : ( if_nat @ ( member_nat @ X3 @ ( set_ord_lessThan_nat @ N3 ) ) @ ( F2 @ X3 ) @ ( if_nat @ ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ N3 @ ( plus_plus_nat @ N3 @ M4 ) ) ) @ ( G2 @ ( minus_minus_nat @ X3 @ N3 ) ) @ undefined_nat ) ) ) ) ).
% join_def
thf(fact_1015_Union__Un__distrib,axiom,
! [A: set_set_set_nat,B2: set_set_set_nat] :
( ( comple548664676211718543et_nat @ ( sup_su4213647025997063966et_nat @ A @ B2 ) )
= ( sup_sup_set_set_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_1016_Union__Un__distrib,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_1017_Sup__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Sup_empty
thf(fact_1018_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_1019_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_1020_calculation_I2_J,axiom,
disjoi6798895846410478970at_nat @ bt @ ( set_ord_atMost_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ).
% calculation(2)
thf(fact_1021_Union__iff,axiom,
! [A: set_nat,C2: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) )
= ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ C2 )
& ( member_set_nat @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_1022_Union__iff,axiom,
! [A: nat > nat,C2: set_set_nat_nat] :
( ( member_nat_nat @ A @ ( comple5448282615319421384at_nat @ C2 ) )
= ( ? [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ C2 )
& ( member_nat_nat @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_1023_Union__iff,axiom,
! [A: ( nat > nat ) > nat > nat,C2: set_se3022870823424313865at_nat] :
( ( member952132173341509300at_nat @ A @ ( comple2605510978757769510at_nat @ C2 ) )
= ( ? [X3: set_nat_nat_nat_nat] :
( ( member7681264892014656106at_nat @ X3 @ C2 )
& ( member952132173341509300at_nat @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_1024_Union__iff,axiom,
! [A: ( nat > nat ) > ( nat > nat ) > nat > nat,C2: set_se5827506804761348711at_nat] :
( ( member8881365325514865170at_nat @ A @ ( comple3227554028126040196at_nat @ C2 ) )
= ( ? [X3: set_na7233567106578532785at_nat] :
( ( member4685516209270408648at_nat @ X3 @ C2 )
& ( member8881365325514865170at_nat @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_1025_Union__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ C2 )
& ( member_nat @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_1026_UnionI,axiom,
! [X5: set_set_nat,C2: set_set_set_nat,A: set_nat] :
( ( member_set_set_nat @ X5 @ C2 )
=> ( ( member_set_nat @ A @ X5 )
=> ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_1027_UnionI,axiom,
! [X5: set_nat_nat,C2: set_set_nat_nat,A: nat > nat] :
( ( member_set_nat_nat @ X5 @ C2 )
=> ( ( member_nat_nat @ A @ X5 )
=> ( member_nat_nat @ A @ ( comple5448282615319421384at_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_1028_UnionI,axiom,
! [X5: set_nat_nat_nat_nat,C2: set_se3022870823424313865at_nat,A: ( nat > nat ) > nat > nat] :
( ( member7681264892014656106at_nat @ X5 @ C2 )
=> ( ( member952132173341509300at_nat @ A @ X5 )
=> ( member952132173341509300at_nat @ A @ ( comple2605510978757769510at_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_1029_UnionI,axiom,
! [X5: set_na7233567106578532785at_nat,C2: set_se5827506804761348711at_nat,A: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member4685516209270408648at_nat @ X5 @ C2 )
=> ( ( member8881365325514865170at_nat @ A @ X5 )
=> ( member8881365325514865170at_nat @ A @ ( comple3227554028126040196at_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_1030_UnionI,axiom,
! [X5: set_nat,C2: set_set_nat,A: nat] :
( ( member_set_nat @ X5 @ C2 )
=> ( ( member_nat @ A @ X5 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_1031_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ A ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ X3 )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_1032_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ A ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ X3 )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_1033_Sup__bot__conv_I2_J,axiom,
! [A: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A ) )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A )
=> ( X3 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1034_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1035_Sup__bot__conv_I2_J,axiom,
! [A: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( X3 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1036_Sup__bot__conv_I1_J,axiom,
! [A: set_set_o] :
( ( ( comple90263536869209701_set_o @ A )
= bot_bot_set_o )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A )
=> ( X3 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1037_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1038_Sup__bot__conv_I1_J,axiom,
! [A: set_o] :
( ( ( complete_Sup_Sup_o @ A )
= bot_bot_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( X3 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1039_UN__ball__bex__simps_I4_J,axiom,
! [B2: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X3 ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_1040_UN__ball__bex__simps_I2_J,axiom,
! [B2: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X3 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_1041_bex__UN,axiom,
! [B2: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ? [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X3 ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_1042_ball__UN,axiom,
! [B2: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B2 @ X3 ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_1043_disjoint__family__onI,axiom,
! [S: set_set_nat,A: set_nat > set_nat] :
( ! [M3: set_nat,N4: set_nat] :
( ( member_set_nat @ M3 @ S )
=> ( ( member_set_nat @ N4 @ S )
=> ( ( M3 != N4 )
=> ( ( inf_inf_set_nat @ ( A @ M3 ) @ ( A @ N4 ) )
= bot_bot_set_nat ) ) ) )
=> ( disjoi2115144663756723504at_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_1044_disjoint__family__onI,axiom,
! [S: set_nat_nat,A: ( nat > nat ) > set_nat] :
( ! [M3: nat > nat,N4: nat > nat] :
( ( member_nat_nat @ M3 @ S )
=> ( ( member_nat_nat @ N4 @ S )
=> ( ( M3 != N4 )
=> ( ( inf_inf_set_nat @ ( A @ M3 ) @ ( A @ N4 ) )
= bot_bot_set_nat ) ) ) )
=> ( disjoi831272138528337257at_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_1045_disjoint__family__onI,axiom,
! [S: set_nat_nat_nat_nat,A: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ! [M3: ( nat > nat ) > nat > nat,N4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ M3 @ S )
=> ( ( member952132173341509300at_nat @ N4 @ S )
=> ( ( M3 != N4 )
=> ( ( inf_inf_set_nat @ ( A @ M3 ) @ ( A @ N4 ) )
= bot_bot_set_nat ) ) ) )
=> ( disjoi4499352858376688327at_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_1046_disjoint__family__onI,axiom,
! [S: set_na7233567106578532785at_nat,A: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_nat] :
( ! [M3: ( nat > nat ) > ( nat > nat ) > nat > nat,N4: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ M3 @ S )
=> ( ( member8881365325514865170at_nat @ N4 @ S )
=> ( ( M3 != N4 )
=> ( ( inf_inf_set_nat @ ( A @ M3 ) @ ( A @ N4 ) )
= bot_bot_set_nat ) ) ) )
=> ( disjoi2115914870343817253at_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_1047_disjoint__family__onI,axiom,
! [S: set_set_nat,A: set_nat > set_o] :
( ! [M3: set_nat,N4: set_nat] :
( ( member_set_nat @ M3 @ S )
=> ( ( member_set_nat @ N4 @ S )
=> ( ( M3 != N4 )
=> ( ( inf_inf_set_o @ ( A @ M3 ) @ ( A @ N4 ) )
= bot_bot_set_o ) ) ) )
=> ( disjoi7862385731094200888_nat_o @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_1048_disjoint__family__onI,axiom,
! [S: set_nat,A: nat > set_o] :
( ! [M3: nat,N4: nat] :
( ( member_nat @ M3 @ S )
=> ( ( member_nat @ N4 @ S )
=> ( ( M3 != N4 )
=> ( ( inf_inf_set_o @ ( A @ M3 ) @ ( A @ N4 ) )
= bot_bot_set_o ) ) ) )
=> ( disjoi1808054049482533742_nat_o @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_1049_disjoint__family__onI,axiom,
! [S: set_nat_nat,A: ( nat > nat ) > set_o] :
( ! [M3: nat > nat,N4: nat > nat] :
( ( member_nat_nat @ M3 @ S )
=> ( ( member_nat_nat @ N4 @ S )
=> ( ( M3 != N4 )
=> ( ( inf_inf_set_o @ ( A @ M3 ) @ ( A @ N4 ) )
= bot_bot_set_o ) ) ) )
=> ( disjoi7375875898804509119_nat_o @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_1050_disjoint__family__onI,axiom,
! [S: set_nat_nat_nat_nat,A: ( ( nat > nat ) > nat > nat ) > set_o] :
( ! [M3: ( nat > nat ) > nat > nat,N4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ M3 @ S )
=> ( ( member952132173341509300at_nat @ N4 @ S )
=> ( ( M3 != N4 )
=> ( ( inf_inf_set_o @ ( A @ M3 ) @ ( A @ N4 ) )
= bot_bot_set_o ) ) ) )
=> ( disjoi6151367327612141793_nat_o @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_1051_disjoint__family__onI,axiom,
! [S: set_na7233567106578532785at_nat,A: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_o] :
( ! [M3: ( nat > nat ) > ( nat > nat ) > nat > nat,N4: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( member8881365325514865170at_nat @ M3 @ S )
=> ( ( member8881365325514865170at_nat @ N4 @ S )
=> ( ( M3 != N4 )
=> ( ( inf_inf_set_o @ ( A @ M3 ) @ ( A @ N4 ) )
= bot_bot_set_o ) ) ) )
=> ( disjoi3706031484853323779_nat_o @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_1052_disjoint__family__onI,axiom,
! [S: set_nat,A: nat > set_nat] :
( ! [M3: nat,N4: nat] :
( ( member_nat @ M3 @ S )
=> ( ( member_nat @ N4 @ S )
=> ( ( M3 != N4 )
=> ( ( inf_inf_set_nat @ ( A @ M3 ) @ ( A @ N4 ) )
= bot_bot_set_nat ) ) ) )
=> ( disjoi6798895846410478970at_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_1053_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > set_nat,D2: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C2 @ A ) )
= ( Sup @ ( image_nat_set_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1054_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A ) )
= ( Sup @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_1055_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > set_nat,D2: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C2 @ A ) )
= ( Inf @ ( image_nat_set_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1056_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A ) )
= ( Inf @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_1057_UnionE,axiom,
! [A: set_nat,C2: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) )
=> ~ ! [X6: set_set_nat] :
( ( member_set_nat @ A @ X6 )
=> ~ ( member_set_set_nat @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_1058_UnionE,axiom,
! [A: nat > nat,C2: set_set_nat_nat] :
( ( member_nat_nat @ A @ ( comple5448282615319421384at_nat @ C2 ) )
=> ~ ! [X6: set_nat_nat] :
( ( member_nat_nat @ A @ X6 )
=> ~ ( member_set_nat_nat @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_1059_UnionE,axiom,
! [A: ( nat > nat ) > nat > nat,C2: set_se3022870823424313865at_nat] :
( ( member952132173341509300at_nat @ A @ ( comple2605510978757769510at_nat @ C2 ) )
=> ~ ! [X6: set_nat_nat_nat_nat] :
( ( member952132173341509300at_nat @ A @ X6 )
=> ~ ( member7681264892014656106at_nat @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_1060_UnionE,axiom,
! [A: ( nat > nat ) > ( nat > nat ) > nat > nat,C2: set_se5827506804761348711at_nat] :
( ( member8881365325514865170at_nat @ A @ ( comple3227554028126040196at_nat @ C2 ) )
=> ~ ! [X6: set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ A @ X6 )
=> ~ ( member4685516209270408648at_nat @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_1061_UnionE,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X6: set_nat] :
( ( member_nat @ A @ X6 )
=> ~ ( member_set_nat @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_1062_Sup__eqI,axiom,
! [A: set_set_nat_nat,X2: set_nat_nat] :
( ! [Y: set_nat_nat] :
( ( member_set_nat_nat @ Y @ A )
=> ( ord_le9059583361652607317at_nat @ Y @ X2 ) )
=> ( ! [Y: set_nat_nat] :
( ! [Z4: set_nat_nat] :
( ( member_set_nat_nat @ Z4 @ A )
=> ( ord_le9059583361652607317at_nat @ Z4 @ Y ) )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y ) )
=> ( ( comple5448282615319421384at_nat @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1063_Sup__eqI,axiom,
! [A: set_set_nat,X2: set_nat] :
( ! [Y: set_nat] :
( ( member_set_nat @ Y @ A )
=> ( ord_less_eq_set_nat @ Y @ X2 ) )
=> ( ! [Y: set_nat] :
( ! [Z4: set_nat] :
( ( member_set_nat @ Z4 @ A )
=> ( ord_less_eq_set_nat @ Z4 @ Y ) )
=> ( ord_less_eq_set_nat @ X2 @ Y ) )
=> ( ( comple7399068483239264473et_nat @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1064_Sup__eqI,axiom,
! [A: set_o,X2: $o] :
( ! [Y: $o] :
( ( member_o @ Y @ A )
=> ( ord_less_eq_o @ Y @ X2 ) )
=> ( ! [Y: $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ A )
=> ( ord_less_eq_o @ Z4 @ Y ) )
=> ( ord_less_eq_o @ X2 @ Y ) )
=> ( ( complete_Sup_Sup_o @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1065_Sup__mono,axiom,
! [A: set_set_nat_nat,B2: set_set_nat_nat] :
( ! [A6: set_nat_nat] :
( ( member_set_nat_nat @ A6 @ A )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ B2 )
& ( ord_le9059583361652607317at_nat @ A6 @ X ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_1066_Sup__mono,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A6 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_1067_Sup__mono,axiom,
! [A: set_o,B2: set_o] :
( ! [A6: $o] :
( ( member_o @ A6 @ A )
=> ? [X: $o] :
( ( member_o @ X @ B2 )
& ( ord_less_eq_o @ A6 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_1068_Sup__least,axiom,
! [A: set_set_nat_nat,Z: set_nat_nat] :
( ! [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ A )
=> ( ord_le9059583361652607317at_nat @ X4 @ Z ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A ) @ Z ) ) ).
% Sup_least
thf(fact_1069_Sup__least,axiom,
! [A: set_set_nat,Z: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ X4 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ Z ) ) ).
% Sup_least
thf(fact_1070_Sup__least,axiom,
! [A: set_o,Z: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ X4 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z ) ) ).
% Sup_least
thf(fact_1071_Sup__upper,axiom,
! [X2: set_nat_nat,A: set_set_nat_nat] :
( ( member_set_nat_nat @ X2 @ A )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( comple5448282615319421384at_nat @ A ) ) ) ).
% Sup_upper
thf(fact_1072_Sup__upper,axiom,
! [X2: set_nat,A: set_set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_1073_Sup__upper,axiom,
! [X2: $o,A: set_o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_upper
thf(fact_1074_Sup__le__iff,axiom,
! [A: set_set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A ) @ B )
= ( ! [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ A )
=> ( ord_le9059583361652607317at_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1075_Sup__le__iff,axiom,
! [A: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ B )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1076_Sup__le__iff,axiom,
! [A: set_o,B: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ B )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1077_Sup__upper2,axiom,
! [U: set_nat_nat,A: set_set_nat_nat,V: set_nat_nat] :
( ( member_set_nat_nat @ U @ A )
=> ( ( ord_le9059583361652607317at_nat @ V @ U )
=> ( ord_le9059583361652607317at_nat @ V @ ( comple5448282615319421384at_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1078_Sup__upper2,axiom,
! [U: set_nat,A: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1079_Sup__upper2,axiom,
! [U: $o,A: set_o,V: $o] :
( ( member_o @ U @ A )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1080_SUP__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat] :
( ( A = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1081_SUP__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > $o,D2: nat > $o] :
( ( A = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1082_SUP__cong,axiom,
! [A: set_nat,B2: set_nat,C2: nat > set_nat,D2: nat > set_nat] :
( ( A = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1083_SUP__cong,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_nat > nat,D2: set_nat > nat] :
( ( A = B2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1084_SUP__cong,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_nat > $o,D2: set_nat > $o] :
( ( A = B2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1085_SUP__cong,axiom,
! [A: set_set_nat,B2: set_set_nat,C2: set_nat > set_nat,D2: set_nat > set_nat] :
( ( A = B2 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C2 @ A ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1086_SUP__cong,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C2: ( nat > nat ) > nat,D2: ( nat > nat ) > nat] :
( ( A = B2 )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1087_SUP__cong,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C2: ( nat > nat ) > $o,D2: ( nat > nat ) > $o] :
( ( A = B2 )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_nat_o @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1088_SUP__cong,axiom,
! [A: set_nat_nat,B2: set_nat_nat,C2: ( nat > nat ) > set_nat,D2: ( nat > nat ) > set_nat] :
( ( A = B2 )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ C2 @ A ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1089_SUP__cong,axiom,
! [A: set_nat_nat_nat_nat,B2: set_nat_nat_nat_nat,C2: ( ( nat > nat ) > nat > nat ) > nat,D2: ( ( nat > nat ) > nat > nat ) > nat] :
( ( A = B2 )
=> ( ! [X4: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_8194121248528334964at_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_8194121248528334964at_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_1090_Union__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Union_empty
thf(fact_1091_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_1092_Union__empty__conv,axiom,
! [A: set_set_o] :
( ( ( comple90263536869209701_set_o @ A )
= bot_bot_set_o )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A )
=> ( X3 = bot_bot_set_o ) ) ) ) ).
% Union_empty_conv
thf(fact_1093_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_1094_empty__Union__conv,axiom,
! [A: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A ) )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A )
=> ( X3 = bot_bot_set_o ) ) ) ) ).
% empty_Union_conv
thf(fact_1095_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( X3 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_1096_Union__subsetI,axiom,
! [A: set_set_nat_nat,B2: set_set_nat_nat] :
( ! [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ A )
=> ? [Y5: set_nat_nat] :
( ( member_set_nat_nat @ Y5 @ B2 )
& ( ord_le9059583361652607317at_nat @ X4 @ Y5 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1097_Union__subsetI,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ? [Y5: set_nat] :
( ( member_set_nat @ Y5 @ B2 )
& ( ord_less_eq_set_nat @ X4 @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_1098_Union__upper,axiom,
! [B2: set_nat_nat,A: set_set_nat_nat] :
( ( member_set_nat_nat @ B2 @ A )
=> ( ord_le9059583361652607317at_nat @ B2 @ ( comple5448282615319421384at_nat @ A ) ) ) ).
% Union_upper
thf(fact_1099_Union__upper,axiom,
! [B2: set_nat,A: set_set_nat] :
( ( member_set_nat @ B2 @ A )
=> ( ord_less_eq_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Union_upper
thf(fact_1100_Union__least,axiom,
! [A: set_set_nat_nat,C2: set_nat_nat] :
( ! [X6: set_nat_nat] :
( ( member_set_nat_nat @ X6 @ A )
=> ( ord_le9059583361652607317at_nat @ X6 @ C2 ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A ) @ C2 ) ) ).
% Union_least
thf(fact_1101_Union__least,axiom,
! [A: set_set_nat,C2: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A )
=> ( ord_less_eq_set_nat @ X6 @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ C2 ) ) ).
% Union_least
thf(fact_1102_Union__mono,axiom,
! [A: set_set_nat_nat,B2: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ A @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_1103_Union__mono,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_1104_SUP__eq,axiom,
! [A: set_nat,B2: set_nat,F: nat > $o,G: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1105_SUP__eq,axiom,
! [A: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1106_SUP__eq,axiom,
! [A: set_set_nat,B2: set_nat,F: set_nat > $o,G: nat > $o] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1107_SUP__eq,axiom,
! [A: set_nat,B2: set_set_nat,F: nat > $o,G: set_nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X ) ) ) )
=> ( ! [J2: set_nat] :
( ( member_set_nat @ J2 @ B2 )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1108_SUP__eq,axiom,
! [A: set_set_nat,B2: set_nat,F: set_nat > set_nat,G: nat > set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1109_SUP__eq,axiom,
! [A: set_nat,B2: set_set_nat,F: nat > set_nat,G: set_nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X ) ) ) )
=> ( ! [J2: set_nat] :
( ( member_set_nat @ J2 @ B2 )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1110_SUP__eq,axiom,
! [A: set_set_nat,B2: set_set_nat,F: set_nat > $o,G: set_nat > $o] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X ) ) ) )
=> ( ! [J2: set_nat] :
( ( member_set_nat @ J2 @ B2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1111_SUP__eq,axiom,
! [A: set_nat,B2: set_nat_nat,F: nat > $o,G: ( nat > nat ) > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X ) ) ) )
=> ( ! [J2: nat > nat] :
( ( member_nat_nat @ J2 @ B2 )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1112_SUP__eq,axiom,
! [A: set_nat_nat,B2: set_nat,F: ( nat > nat ) > $o,G: nat > $o] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1113_SUP__eq,axiom,
! [A: set_nat,B2: set_nat,F: nat > set_nat_nat,G: nat > set_nat_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ B2 )
& ( ord_le9059583361652607317at_nat @ ( F @ I3 ) @ ( G @ X ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_le9059583361652607317at_nat @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ A ) )
= ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1114_less__eq__Sup,axiom,
! [A: set_set_nat_nat,U: set_nat_nat] :
( ! [V2: set_nat_nat] :
( ( member_set_nat_nat @ V2 @ A )
=> ( ord_le9059583361652607317at_nat @ U @ V2 ) )
=> ( ( A != bot_bo7376149671870096959at_nat )
=> ( ord_le9059583361652607317at_nat @ U @ ( comple5448282615319421384at_nat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1115_less__eq__Sup,axiom,
! [A: set_set_nat,U: set_nat] :
( ! [V2: set_nat] :
( ( member_set_nat @ V2 @ A )
=> ( ord_less_eq_set_nat @ U @ V2 ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1116_less__eq__Sup,axiom,
! [A: set_o,U: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A )
=> ( ord_less_eq_o @ U @ V2 ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1117_Sup__subset__mono,axiom,
! [A: set_set_nat_nat,B2: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ A @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_1118_Sup__subset__mono,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_1119_Sup__subset__mono,axiom,
! [A: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A @ B2 )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_1120_SUP__eq__const,axiom,
! [I4: set_nat,F: nat > $o,X2: $o] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I4 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_1121_SUP__eq__const,axiom,
! [I4: set_o,F: $o > $o,X2: $o] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I4 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_1122_SUP__eq__const,axiom,
! [I4: set_nat,F: nat > set_nat,X2: set_nat] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I4 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_1123_SUP__eq__const,axiom,
! [I4: set_o,F: $o > set_nat,X2: set_nat] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I4 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_1124_SUP__eq__const,axiom,
! [I4: set_set_nat,F: set_nat > $o,X2: $o] :
( ( I4 != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ I4 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_1125_SUP__eq__const,axiom,
! [I4: set_set_nat,F: set_nat > set_nat,X2: set_nat] :
( ( I4 != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ I4 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_1126_SUP__eq__const,axiom,
! [I4: set_nat_nat,F: ( nat > nat ) > $o,X2: $o] :
( ( I4 != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ I4 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_1127_SUP__eq__const,axiom,
! [I4: set_nat_nat,F: ( nat > nat ) > set_nat,X2: set_nat] :
( ( I4 != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ I4 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_1128_SUP__eq__const,axiom,
! [I4: set_nat_nat_nat_nat,F: ( ( nat > nat ) > nat > nat ) > $o,X2: $o] :
( ( I4 != bot_bo3919185967433191911at_nat )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_8690456353314504180_nat_o @ F @ I4 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_1129_SUP__eq__const,axiom,
! [I4: set_nat_nat_nat_nat,F: ( ( nat > nat ) > nat > nat ) > set_nat,X2: set_nat] :
( ( I4 != bot_bo3919185967433191911at_nat )
=> ( ! [I3: ( nat > nat ) > nat > nat] :
( ( member952132173341509300at_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ F @ I4 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_1130_Union__disjoint,axiom,
! [C2: set_set_o,A: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ C2 ) @ A )
= bot_bot_set_o )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ C2 )
=> ( ( inf_inf_set_o @ X3 @ A )
= bot_bot_set_o ) ) ) ) ).
% Union_disjoint
thf(fact_1131_Union__disjoint,axiom,
! [C2: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C2 ) @ A )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C2 )
=> ( ( inf_inf_set_nat @ X3 @ A )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_1132_Sup__union__distrib,axiom,
! [A: set_set_set_nat,B2: set_set_set_nat] :
( ( comple548664676211718543et_nat @ ( sup_su4213647025997063966et_nat @ A @ B2 ) )
= ( sup_sup_set_set_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_1133_Sup__union__distrib,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_1134_Sup__union__distrib,axiom,
! [A: set_o,B2: set_o] :
( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A @ B2 ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_1135_Union__Int__subset,axiom,
! [A: set_set_nat_nat,B2: set_set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( inf_in710756014367367485at_nat @ A @ B2 ) ) @ ( inf_inf_set_nat_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Union_Int_subset
thf(fact_1136_Union__Int__subset,axiom,
! [A: set_set_nat,B2: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A @ B2 ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_Int_subset
thf(fact_1137_SUP__eq__iff,axiom,
! [I4: set_nat,C: $o,F: nat > $o] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I4 ) )
= C )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I4 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1138_SUP__eq__iff,axiom,
! [I4: set_o,C: $o,F: $o > $o] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I4 ) )
= C )
= ( ! [X3: $o] :
( ( member_o @ X3 @ I4 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1139_SUP__eq__iff,axiom,
! [I4: set_nat,C: set_nat,F: nat > set_nat] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I4 ) )
= C )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I4 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1140_SUP__eq__iff,axiom,
! [I4: set_o,C: set_nat,F: $o > set_nat] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I4 ) )
= C )
= ( ! [X3: $o] :
( ( member_o @ X3 @ I4 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1141_SUP__eq__iff,axiom,
! [I4: set_set_nat,C: $o,F: set_nat > $o] :
( ( I4 != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ I4 ) )
= C )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ I4 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1142_SUP__eq__iff,axiom,
! [I4: set_set_nat,C: set_nat,F: set_nat > set_nat] :
( ( I4 != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ I4 ) )
= C )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ I4 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1143_SUP__eq__iff,axiom,
! [I4: set_nat_nat,C: $o,F: ( nat > nat ) > $o] :
( ( I4 != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I4 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ I4 ) )
= C )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I4 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1144_SUP__eq__iff,axiom,
! [I4: set_nat,C: set_nat_nat,F: nat > set_nat_nat] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_le9059583361652607317at_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ I4 ) )
= C )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I4 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1145_SUP__eq__iff,axiom,
! [I4: set_o,C: set_nat_nat,F: $o > set_nat_nat] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ord_le9059583361652607317at_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple5448282615319421384at_nat @ ( image_o_set_nat_nat @ F @ I4 ) )
= C )
= ( ! [X3: $o] :
( ( member_o @ X3 @ I4 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1146_SUP__eq__iff,axiom,
! [I4: set_nat_nat,C: set_nat,F: ( nat > nat ) > set_nat] :
( ( I4 != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I4 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ I4 ) )
= C )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ I4 )
=> ( ( F @ X3 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1147_Sup__inter__less__eq,axiom,
! [A: set_set_nat_nat,B2: set_set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( inf_in710756014367367485at_nat @ A @ B2 ) ) @ ( inf_inf_set_nat_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B2 ) ) ) ).
% Sup_inter_less_eq
thf(fact_1148_Sup__inter__less__eq,axiom,
! [A: set_set_nat,B2: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A @ B2 ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_inter_less_eq
thf(fact_1149_Sup__inter__less__eq,axiom,
! [A: set_o,B2: set_o] : ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( inf_inf_set_o @ A @ B2 ) ) @ ( inf_inf_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_inter_less_eq
thf(fact_1150_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1151_cSup__atMost,axiom,
! [X2: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ X2 ) )
= X2 ) ).
% cSup_atMost
thf(fact_1152_cSup__atMost,axiom,
! [X2: nat] :
( ( complete_Sup_Sup_nat @ ( set_ord_atMost_nat @ X2 ) )
= X2 ) ).
% cSup_atMost
thf(fact_1153_cSup__atMost,axiom,
! [X2: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ X2 ) )
= X2 ) ).
% cSup_atMost
thf(fact_1154_Union__image__empty,axiom,
! [A: set_set_nat,F: nat > set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ bot_bot_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_1155_Union__image__empty,axiom,
! [A: set_set_nat,F: $o > set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ F @ bot_bot_set_o ) ) )
= A ) ).
% Union_image_empty
thf(fact_1156_Union__image__empty,axiom,
! [A: set_nat,F: nat > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_1157_Union__image__empty,axiom,
! [A: set_nat,F: $o > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ bot_bot_set_o ) ) )
= A ) ).
% Union_image_empty
thf(fact_1158_cSup__eq__maximum,axiom,
! [Z: set_nat_nat,X5: set_set_nat_nat] :
( ( member_set_nat_nat @ Z @ X5 )
=> ( ! [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ X5 )
=> ( ord_le9059583361652607317at_nat @ X4 @ Z ) )
=> ( ( comple5448282615319421384at_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1159_cSup__eq__maximum,axiom,
! [Z: set_nat,X5: set_set_nat] :
( ( member_set_nat @ Z @ X5 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ Z ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1160_cSup__eq__maximum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat @ Z @ X5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1161_cSup__eq__maximum,axiom,
! [Z: $o,X5: set_o] :
( ( member_o @ Z @ X5 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ Z ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1162_cSup__least,axiom,
! [X5: set_set_nat_nat,Z: set_nat_nat] :
( ( X5 != bot_bo7376149671870096959at_nat )
=> ( ! [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ X5 )
=> ( ord_le9059583361652607317at_nat @ X4 @ Z ) )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1163_cSup__least,axiom,
! [X5: set_set_nat,Z: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1164_cSup__least,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1165_cSup__least,axiom,
! [X5: set_o,Z: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1166_cSup__eq__non__empty,axiom,
! [X5: set_set_nat_nat,A2: set_nat_nat] :
( ( X5 != bot_bo7376149671870096959at_nat )
=> ( ! [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ X5 )
=> ( ord_le9059583361652607317at_nat @ X4 @ A2 ) )
=> ( ! [Y: set_nat_nat] :
( ! [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ X5 )
=> ( ord_le9059583361652607317at_nat @ X @ Y ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ Y ) )
=> ( ( comple5448282615319421384at_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1167_cSup__eq__non__empty,axiom,
! [X5: set_set_nat,A2: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ A2 ) )
=> ( ! [Y: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ X5 )
=> ( ord_less_eq_set_nat @ X @ Y ) )
=> ( ord_less_eq_set_nat @ A2 @ Y ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1168_cSup__eq__non__empty,axiom,
! [X5: set_nat,A2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ A2 ) )
=> ( ! [Y: nat] :
( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ord_less_eq_nat @ X @ Y ) )
=> ( ord_less_eq_nat @ A2 @ Y ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1169_cSup__eq__non__empty,axiom,
! [X5: set_o,A2: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ A2 ) )
=> ( ! [Y: $o] :
( ! [X: $o] :
( ( member_o @ X @ X5 )
=> ( ord_less_eq_o @ X @ Y ) )
=> ( ord_less_eq_o @ A2 @ Y ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1170_cSUP__least,axiom,
! [A: set_nat,F: nat > nat,M: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1171_cSUP__least,axiom,
! [A: set_o,F: $o > nat,M: nat] :
( ( A != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1172_cSUP__least,axiom,
! [A: set_nat,F: nat > $o,M: $o] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1173_cSUP__least,axiom,
! [A: set_o,F: $o > $o,M: $o] :
( ( A != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1174_cSUP__least,axiom,
! [A: set_nat,F: nat > set_nat,M: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1175_cSUP__least,axiom,
! [A: set_o,F: $o > set_nat,M: set_nat] :
( ( A != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1176_cSUP__least,axiom,
! [A: set_set_nat,F: set_nat > nat,M: nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1177_cSUP__least,axiom,
! [A: set_set_nat,F: set_nat > $o,M: $o] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1178_cSUP__least,axiom,
! [A: set_set_nat,F: set_nat > set_nat,M: set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1179_cSUP__least,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat,M: nat] :
( ( A != bot_bot_set_nat_nat )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1180_disjoint__family__onD,axiom,
! [A: set_nat > set_nat,I4: set_set_nat,I2: set_nat,J: set_nat] :
( ( disjoi2115144663756723504at_nat @ A @ I4 )
=> ( ( member_set_nat @ I2 @ I4 )
=> ( ( member_set_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_1181_disjoint__family__onD,axiom,
! [A: ( nat > nat ) > set_nat,I4: set_nat_nat,I2: nat > nat,J: nat > nat] :
( ( disjoi831272138528337257at_nat @ A @ I4 )
=> ( ( member_nat_nat @ I2 @ I4 )
=> ( ( member_nat_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_1182_disjoint__family__onD,axiom,
! [A: ( ( nat > nat ) > nat > nat ) > set_nat,I4: set_nat_nat_nat_nat,I2: ( nat > nat ) > nat > nat,J: ( nat > nat ) > nat > nat] :
( ( disjoi4499352858376688327at_nat @ A @ I4 )
=> ( ( member952132173341509300at_nat @ I2 @ I4 )
=> ( ( member952132173341509300at_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_1183_disjoint__family__onD,axiom,
! [A: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_nat,I4: set_na7233567106578532785at_nat,I2: ( nat > nat ) > ( nat > nat ) > nat > nat,J: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( disjoi2115914870343817253at_nat @ A @ I4 )
=> ( ( member8881365325514865170at_nat @ I2 @ I4 )
=> ( ( member8881365325514865170at_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_1184_disjoint__family__onD,axiom,
! [A: set_nat > set_o,I4: set_set_nat,I2: set_nat,J: set_nat] :
( ( disjoi7862385731094200888_nat_o @ A @ I4 )
=> ( ( member_set_nat @ I2 @ I4 )
=> ( ( member_set_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_o @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_o ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_1185_disjoint__family__onD,axiom,
! [A: nat > set_o,I4: set_nat,I2: nat,J: nat] :
( ( disjoi1808054049482533742_nat_o @ A @ I4 )
=> ( ( member_nat @ I2 @ I4 )
=> ( ( member_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_o @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_o ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_1186_disjoint__family__onD,axiom,
! [A: ( nat > nat ) > set_o,I4: set_nat_nat,I2: nat > nat,J: nat > nat] :
( ( disjoi7375875898804509119_nat_o @ A @ I4 )
=> ( ( member_nat_nat @ I2 @ I4 )
=> ( ( member_nat_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_o @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_o ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_1187_disjoint__family__onD,axiom,
! [A: ( ( nat > nat ) > nat > nat ) > set_o,I4: set_nat_nat_nat_nat,I2: ( nat > nat ) > nat > nat,J: ( nat > nat ) > nat > nat] :
( ( disjoi6151367327612141793_nat_o @ A @ I4 )
=> ( ( member952132173341509300at_nat @ I2 @ I4 )
=> ( ( member952132173341509300at_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_o @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_o ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_1188_disjoint__family__onD,axiom,
! [A: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_o,I4: set_na7233567106578532785at_nat,I2: ( nat > nat ) > ( nat > nat ) > nat > nat,J: ( nat > nat ) > ( nat > nat ) > nat > nat] :
( ( disjoi3706031484853323779_nat_o @ A @ I4 )
=> ( ( member8881365325514865170at_nat @ I2 @ I4 )
=> ( ( member8881365325514865170at_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_o @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_o ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_1189_disjoint__family__onD,axiom,
! [A: nat > set_nat,I4: set_nat,I2: nat,J: nat] :
( ( disjoi6798895846410478970at_nat @ A @ I4 )
=> ( ( member_nat @ I2 @ I4 )
=> ( ( member_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_1190_disjoint__family__on__def,axiom,
( disjoi6798895846410478970at_nat
= ( ^ [A3: nat > set_nat,S2: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ S2 )
=> ( ( X3 != Y2 )
=> ( ( inf_inf_set_nat @ ( A3 @ X3 ) @ ( A3 @ Y2 ) )
= bot_bot_set_nat ) ) ) ) ) ) ).
% disjoint_family_on_def
thf(fact_1191_disjoint__family__on__bisimulation,axiom,
! [F: nat > set_o,S: set_nat,G: nat > set_o] :
( ( disjoi1808054049482533742_nat_o @ F @ S )
=> ( ! [N4: nat,M3: nat] :
( ( member_nat @ N4 @ S )
=> ( ( member_nat @ M3 @ S )
=> ( ( N4 != M3 )
=> ( ( ( inf_inf_set_o @ ( F @ N4 ) @ ( F @ M3 ) )
= bot_bot_set_o )
=> ( ( inf_inf_set_o @ ( G @ N4 ) @ ( G @ M3 ) )
= bot_bot_set_o ) ) ) ) )
=> ( disjoi1808054049482533742_nat_o @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_1192_disjoint__family__on__bisimulation,axiom,
! [F: nat > set_o,S: set_nat,G: nat > set_nat] :
( ( disjoi1808054049482533742_nat_o @ F @ S )
=> ( ! [N4: nat,M3: nat] :
( ( member_nat @ N4 @ S )
=> ( ( member_nat @ M3 @ S )
=> ( ( N4 != M3 )
=> ( ( ( inf_inf_set_o @ ( F @ N4 ) @ ( F @ M3 ) )
= bot_bot_set_o )
=> ( ( inf_inf_set_nat @ ( G @ N4 ) @ ( G @ M3 ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi6798895846410478970at_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_1193_disjoint__family__on__bisimulation,axiom,
! [F: nat > set_nat,S: set_nat,G: nat > set_o] :
( ( disjoi6798895846410478970at_nat @ F @ S )
=> ( ! [N4: nat,M3: nat] :
( ( member_nat @ N4 @ S )
=> ( ( member_nat @ M3 @ S )
=> ( ( N4 != M3 )
=> ( ( ( inf_inf_set_nat @ ( F @ N4 ) @ ( F @ M3 ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_o @ ( G @ N4 ) @ ( G @ M3 ) )
= bot_bot_set_o ) ) ) ) )
=> ( disjoi1808054049482533742_nat_o @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_1194_disjoint__family__on__bisimulation,axiom,
! [F: nat > set_nat,S: set_nat,G: nat > set_nat] :
( ( disjoi6798895846410478970at_nat @ F @ S )
=> ( ! [N4: nat,M3: nat] :
( ( member_nat @ N4 @ S )
=> ( ( member_nat @ M3 @ S )
=> ( ( N4 != M3 )
=> ( ( ( inf_inf_set_nat @ ( F @ N4 ) @ ( F @ M3 ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ ( G @ N4 ) @ ( G @ M3 ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi6798895846410478970at_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_1195_disjoint__family__on__bisimulation,axiom,
! [F: set_nat > set_nat,S: set_set_nat,G: set_nat > set_nat] :
( ( disjoi2115144663756723504at_nat @ F @ S )
=> ( ! [N4: set_nat,M3: set_nat] :
( ( member_set_nat @ N4 @ S )
=> ( ( member_set_nat @ M3 @ S )
=> ( ( N4 != M3 )
=> ( ( ( inf_inf_set_nat @ ( F @ N4 ) @ ( F @ M3 ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ ( G @ N4 ) @ ( G @ M3 ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi2115144663756723504at_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_1196_disjoint__family__on__bisimulation,axiom,
! [F: set_nat > set_nat,S: set_set_nat,G: set_nat > set_o] :
( ( disjoi2115144663756723504at_nat @ F @ S )
=> ( ! [N4: set_nat,M3: set_nat] :
( ( member_set_nat @ N4 @ S )
=> ( ( member_set_nat @ M3 @ S )
=> ( ( N4 != M3 )
=> ( ( ( inf_inf_set_nat @ ( F @ N4 ) @ ( F @ M3 ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_o @ ( G @ N4 ) @ ( G @ M3 ) )
= bot_bot_set_o ) ) ) ) )
=> ( disjoi7862385731094200888_nat_o @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_1197_disjoint__family__on__bisimulation,axiom,
! [F: set_nat > set_o,S: set_set_nat,G: set_nat > set_nat] :
( ( disjoi7862385731094200888_nat_o @ F @ S )
=> ( ! [N4: set_nat,M3: set_nat] :
( ( member_set_nat @ N4 @ S )
=> ( ( member_set_nat @ M3 @ S )
=> ( ( N4 != M3 )
=> ( ( ( inf_inf_set_o @ ( F @ N4 ) @ ( F @ M3 ) )
= bot_bot_set_o )
=> ( ( inf_inf_set_nat @ ( G @ N4 ) @ ( G @ M3 ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi2115144663756723504at_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_1198_disjoint__family__on__bisimulation,axiom,
! [F: set_nat > set_o,S: set_set_nat,G: set_nat > set_o] :
( ( disjoi7862385731094200888_nat_o @ F @ S )
=> ( ! [N4: set_nat,M3: set_nat] :
( ( member_set_nat @ N4 @ S )
=> ( ( member_set_nat @ M3 @ S )
=> ( ( N4 != M3 )
=> ( ( ( inf_inf_set_o @ ( F @ N4 ) @ ( F @ M3 ) )
= bot_bot_set_o )
=> ( ( inf_inf_set_o @ ( G @ N4 ) @ ( G @ M3 ) )
= bot_bot_set_o ) ) ) ) )
=> ( disjoi7862385731094200888_nat_o @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_1199_disjoint__family__on__bisimulation,axiom,
! [F: ( nat > nat ) > set_nat,S: set_nat_nat,G: ( nat > nat ) > set_nat] :
( ( disjoi831272138528337257at_nat @ F @ S )
=> ( ! [N4: nat > nat,M3: nat > nat] :
( ( member_nat_nat @ N4 @ S )
=> ( ( member_nat_nat @ M3 @ S )
=> ( ( N4 != M3 )
=> ( ( ( inf_inf_set_nat @ ( F @ N4 ) @ ( F @ M3 ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ ( G @ N4 ) @ ( G @ M3 ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi831272138528337257at_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_1200_disjoint__family__on__bisimulation,axiom,
! [F: ( nat > nat ) > set_nat,S: set_nat_nat,G: ( nat > nat ) > set_o] :
( ( disjoi831272138528337257at_nat @ F @ S )
=> ( ! [N4: nat > nat,M3: nat > nat] :
( ( member_nat_nat @ N4 @ S )
=> ( ( member_nat_nat @ M3 @ S )
=> ( ( N4 != M3 )
=> ( ( ( inf_inf_set_nat @ ( F @ N4 ) @ ( F @ M3 ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_o @ ( G @ N4 ) @ ( G @ M3 ) )
= bot_bot_set_o ) ) ) ) )
=> ( disjoi7375875898804509119_nat_o @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_1201_inf__Sup,axiom,
! [A2: set_nat,B2: set_set_nat] :
( ( inf_inf_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A2 ) @ B2 ) ) ) ).
% inf_Sup
thf(fact_1202_inf__Sup,axiom,
! [A2: $o,B2: set_o] :
( ( inf_inf_o @ A2 @ ( complete_Sup_Sup_o @ B2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ ( inf_inf_o @ A2 ) @ B2 ) ) ) ).
% inf_Sup
thf(fact_1203_disjoint__family__on__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( disjoi6798895846410478970at_nat @ F @ B2 )
=> ( disjoi6798895846410478970at_nat @ F @ A ) ) ) ).
% disjoint_family_on_mono
thf(fact_1204_Sup__inf__eq__bot__iff,axiom,
! [B2: set_set_o,A2: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ B2 ) @ A2 )
= bot_bot_set_o )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ B2 )
=> ( ( inf_inf_set_o @ X3 @ A2 )
= bot_bot_set_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1205_Sup__inf__eq__bot__iff,axiom,
! [B2: set_set_nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B2 ) @ A2 )
= bot_bot_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B2 )
=> ( ( inf_inf_set_nat @ X3 @ A2 )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1206_Sup__inf__eq__bot__iff,axiom,
! [B2: set_o,A2: $o] :
( ( ( inf_inf_o @ ( complete_Sup_Sup_o @ B2 ) @ A2 )
= bot_bot_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ B2 )
=> ( ( inf_inf_o @ X3 @ A2 )
= bot_bot_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1207_fact2,axiom,
( ( inf_inf_set_nat @ ( bl @ zero_zero_nat )
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( hales_set_incr @ n2 @ ( bs @ I ) )
@ ( set_ord_lessThan_nat @ k ) ) ) )
= bot_bot_set_nat ) ).
% fact2
thf(fact_1208_Collect__empty__eq__bot,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( P = bot_bot_set_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1209_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1210_Collect__empty__eq__bot,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( P = bot_bot_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_1211_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1212_bot__empty__eq,axiom,
( bot_bot_nat_nat_o
= ( ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ bot_bot_set_nat_nat ) ) ) ).
% bot_empty_eq
thf(fact_1213_bot__empty__eq,axiom,
( bot_bo1568108970253895006_nat_o
= ( ^ [X3: ( nat > nat ) > nat > nat] : ( member952132173341509300at_nat @ X3 @ bot_bo3919185967433191911at_nat ) ) ) ).
% bot_empty_eq
thf(fact_1214_bot__empty__eq,axiom,
( bot_bo5587768346753192576_nat_o
= ( ^ [X3: ( nat > nat ) > ( nat > nat ) > nat > nat] : ( member8881365325514865170at_nat @ X3 @ bot_bo2676777031303994949at_nat ) ) ) ).
% bot_empty_eq
thf(fact_1215_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1216_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X3: $o] : ( member_o @ X3 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_1217_image__ident,axiom,
! [Y6: set_nat] :
( ( image_nat_nat
@ ^ [X3: nat] : X3
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_1218_SUP__identity__eq,axiom,
! [A: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X3: set_nat] : X3
@ A ) )
= ( comple7399068483239264473et_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_1219_SUP__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_1220_SUP__identity__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X3: $o] : X3
@ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_identity_eq
thf(fact_1221_UN__I,axiom,
! [A2: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B: ( nat > nat ) > ( nat > nat ) > nat > nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_na7233567106578532785at_nat] :
( ( member8881365325514865170at_nat @ A2 @ A )
=> ( ( member8881365325514865170at_nat @ B @ ( B2 @ A2 ) )
=> ( member8881365325514865170at_nat @ B @ ( comple3227554028126040196at_nat @ ( image_757773569841663411at_nat @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1222_UN__I,axiom,
! [A2: set_nat,A: set_set_nat,B: nat,B2: set_nat > set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( member_nat @ B @ ( B2 @ A2 ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1223_UN__I,axiom,
! [A2: nat,A: set_nat,B: nat,B2: nat > set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member_nat @ B @ ( B2 @ A2 ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1224_UN__I,axiom,
! [A2: nat > nat,A: set_nat_nat,B: nat,B2: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ A2 @ A )
=> ( ( member_nat @ B @ ( B2 @ A2 ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1225_UN__I,axiom,
! [A2: ( nat > nat ) > nat > nat,A: set_nat_nat_nat_nat,B: nat,B2: ( ( nat > nat ) > nat > nat ) > set_nat] :
( ( member952132173341509300at_nat @ A2 @ A )
=> ( ( member_nat @ B @ ( B2 @ A2 ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_1946857609996606506et_nat @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1226_UN__I,axiom,
! [A2: ( nat > nat ) > ( nat > nat ) > nat > nat,A: set_na7233567106578532785at_nat,B: nat,B2: ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > set_nat] :
( ( member8881365325514865170at_nat @ A2 @ A )
=> ( ( member_nat @ B @ ( B2 @ A2 ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_2666519055618792072et_nat @ B2 @ A ) ) ) ) ) ).
% UN_I
thf(fact_1227_set__incr__def,axiom,
( hales_set_incr
= ( ^ [N3: nat] :
( image_nat_nat
@ ^ [A5: nat] : ( plus_plus_nat @ A5 @ N3 ) ) ) ) ).
% set_incr_def
thf(fact_1228__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] :
? [I: nat] :
( ( Uu
= ( hales_set_incr @ n2 @ ( bs @ I ) ) )
& ( member_nat @ I @ ( 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_1229__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] :
? [I: nat] :
( ( Uu
= ( hales_set_incr @ n2 @ ( bs @ I ) ) )
& ( member_nat @ I @ ( set_ord_lessThan_nat @ k ) ) ) ) ) ).
% \<open>{} \<notin> {set_incr n (BS i) |i. i \<in> {..<k}}\<close>
thf(fact_1230_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_1231_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1232_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
= ( P @ B6 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ zero_zero_nat )
=> ( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
=> ( P @ A6 @ ( plus_plus_nat @ A6 @ B6 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1233_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_1234_UN__atMost__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atMost_UNIV
thf(fact_1235_BfS__props_I4_J,axiom,
( member_nat_nat @ fS
@ ( piE_nat_nat @ ( bs @ k )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% BfS_props(4)
thf(fact_1236_BfL__props_I4_J,axiom,
( member_nat_nat @ fL
@ ( piE_nat_nat @ ( bl @ one_one_nat )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% BfL_props(4)
thf(fact_1237_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ t3 ).
% assms(1)
thf(fact_1238_assms_I4_J,axiom,
! [K3: nat,R: nat] :
( ( ord_less_eq_nat @ K3 @ k )
=> ( hales_lhj @ R @ t3 @ K3 ) ) ).
% assms(4)
thf(fact_1239_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_1240_F4,axiom,
( member_nat_nat @ fT
@ ( piE_nat_nat @ ( bt @ ( plus_plus_nat @ k @ one_one_nat ) )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% F4
thf(fact_1241_that,axiom,
member_nat_nat @ y @ ( hales_cube @ ( plus_plus_nat @ k @ one_one_nat ) @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ).
% that
thf(fact_1242_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1243_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1244_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_1245_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1246_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1247_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1248_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_1249_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 ) )
@ ^ [I: nat > nat] : ( hales_cube @ n2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% BfL_props(5)
thf(fact_1250_BfS__props_I5_J,axiom,
( member952132173341509300at_nat @ s
@ ( piE_nat_nat_nat_nat @ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I: nat > nat] : ( hales_cube @ m2 @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% BfS_props(5)
thf(fact_1251_BfL__props_I6_J,axiom,
! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( bl @ one_one_nat ) )
=> ( ( l @ X @ Xa )
= ( fL @ Xa ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ one_one_nat )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( bl @ J3 ) )
=> ( ( l @ X @ Xa )
= ( X @ J3 ) ) ) ) ) ) ).
% BfL_props(6)
thf(fact_1252_assms_I5_J,axiom,
ord_less_nat @ zero_zero_nat @ r ).
% assms(5)
thf(fact_1253_BfS__props_I6_J,axiom,
! [X: nat > nat] :
( ( member_nat_nat @ X @ ( hales_cube @ k @ ( plus_plus_nat @ t3 @ one_one_nat ) ) )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( bs @ k ) )
=> ( ( s @ X @ Xa )
= ( fS @ Xa ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ k )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( bs @ J3 ) )
=> ( ( s @ X @ Xa )
= ( X @ J3 ) ) ) ) ) ) ).
% BfS_props(6)
thf(fact_1254_T__def,axiom,
( t2
= ( restri4446420529079022766at_nat
@ ^ [X3: nat > nat] :
( t @ ( restrict_nat_nat @ X3 @ ( set_ord_lessThan_nat @ one_one_nat ) )
@ ( restrict_nat_nat
@ ^ [Y2: nat] : ( X3 @ ( plus_plus_nat @ Y2 @ 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_1255__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_1256_T_H__prop,axiom,
( member8881365325514865170at_nat @ t
@ ( piE_na5223350113562215832at_nat @ ( hales_cube @ one_one_nat @ ( plus_plus_nat @ t3 @ one_one_nat ) )
@ ^ [I: 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_1257_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 ) )
@ ^ [I: nat > nat] : ( hales_cube @ ( plus_plus_nat @ n2 @ m2 ) @ ( plus_plus_nat @ t3 @ one_one_nat ) ) ) ) ).
% T_prop
thf(fact_1258__092_060chi_062L__def,axiom,
( chi_L
= ( restri6011711336257459485at_nat
@ ^ [X3: nat > nat] :
( restrict_nat_nat_nat
@ ^ [Y2: nat > nat] : ( chi @ ( hales_join_nat @ X3 @ Y2 @ 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_1259_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1260_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I2 @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1261_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N2 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1262_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_1263_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1264_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1265_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1266_less__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1267_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1268_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1269_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1270_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1271_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1272_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1273_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
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,
! [X2: set_nat,Y3: set_nat] :
( ( if_set_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( if_set_nat @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_nat @ i @ bstat ).
%------------------------------------------------------------------------------