TPTP Problem File: SLH0851^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% 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 : SCC_Bloemen_Sequential/0000_SCC_Bloemen_Sequential/prob_01796_060861__6089606_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1524 ( 613 unt; 241 typ; 0 def)
% Number of atoms : 3783 (1307 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 12595 ( 450 ~; 60 |; 290 &;10037 @)
% ( 0 <=>;1758 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 662 ( 662 >; 0 *; 0 +; 0 <<)
% Number of symbols : 220 ( 217 usr; 20 con; 0-9 aty)
% Number of variables : 3664 ( 152 ^;3394 !; 118 ?;3664 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:53:40.262
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J_J,type,
set_Pr7499474215547700295od_v_v: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
produc1504107476793160551od_v_v: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J_J,type,
set_se5707775751431548583od_v_v: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
set_Pr2149350503807050951od_v_v: $tType ).
thf(ty_n_t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Ounit_J,type,
sCC_Bl7326425374436813197t_unit: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
produc5741669702376414499t_unit: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
produc206430290419586791od_v_v: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_Itf__v_J_J_J,type,
set_Pr6206931691796273479list_v: $tType ).
thf(ty_n_t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J,type,
sCC_Bl1394983891496994913t_unit: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_Itf__v_J_J,type,
produc1391462591744249447list_v: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
list_l4795378083388841843od_v_v: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
set_li2323639185124838733od_v_v: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
set_se8455005133513928103od_v_v: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
list_P7986770385144383213od_v_v: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
set_Product_prod_v_v: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
product_prod_v_v: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__v_J_J,type,
list_list_v: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__v_J_J,type,
set_list_v: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_Itf__v_J_J,type,
list_set_v: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
set_set_v: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__List__Olist_Itf__v_J,type,
list_v: $tType ).
thf(ty_n_t__Set__Oset_Itf__v_J,type,
set_v: $tType ).
thf(ty_n_tf__v,type,
v: $tType ).
% Explicit typings (217)
thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001tf__v,type,
bNF_Ca4386975739854426340inal_v: set_v > set_Product_prod_v_v > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001tf__v_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
bNF_Ca66581456156439064od_v_v: set_Product_prod_v_v > ( v > set_Product_prod_v_v ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001tf__v_001t__Set__Oset_Itf__v_J,type,
bNF_Ca2468909317164667716_set_v: set_Product_prod_v_v > ( v > set_v ) > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
bNF_We8684021896157364691od_v_v: set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001tf__v,type,
bNF_We1162827675446710015_rel_v: set_Product_prod_v_v > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
bNF_We6235008509051751325od_v_v: set_Pr2149350503807050951od_v_v > set_Product_prod_v_v > product_prod_v_v > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001tf__v,type,
bNF_We6697304935525757641inim_v: set_Product_prod_v_v > set_v > v > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
bNF_We3854103423653685557od_v_v: set_Pr2149350503807050951od_v_v > product_prod_v_v > product_prod_v_v > product_prod_v_v ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001tf__v,type,
bNF_We3763454674811381857max2_v: set_Product_prod_v_v > v > v > v ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
bNF_We5492458111348578227od_v_v: set_Pr2149350503807050951od_v_v > set_Product_prod_v_v > product_prod_v_v ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001tf__v,type,
bNF_We5615626441682584799inim_v: set_Product_prod_v_v > set_v > v ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
comple514088740646613812od_v_v: set_se5707775751431548583od_v_v > set_Pr2149350503807050951od_v_v ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
comple5788137035815166516od_v_v: set_se8455005133513928103od_v_v > set_Product_prod_v_v ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__v_J,type,
comple2307003700295860064_set_v: set_set_v > set_v ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
finite5952053201251911184od_v_v: set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
finite3348123685078250256od_v_v: set_Product_prod_v_v > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
finite6084192165098772208od_v_v: set_se8455005133513928103od_v_v > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__v_J,type,
finite_finite_set_v: set_set_v > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__v,type,
finite_finite_v: set_v > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
minus_5255927943254941998od_v_v: set_Pr2149350503807050951od_v_v > set_Pr2149350503807050951od_v_v > set_Pr2149350503807050951od_v_v ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
minus_4183494784930505774od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
minus_7679383599658060814od_v_v: set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
minus_7228012346218142266_set_v: set_set_v > set_set_v > set_set_v ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__v_J,type,
minus_minus_set_v: set_v > set_v > set_v ).
thf(sy_c_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
if_set4279007504652509325od_v_v: $o > set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_If_001t__Set__Oset_Itf__v_J,type,
if_set_v: $o > set_v > set_v > set_v ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_Itf__v_J_J,type,
inf_inf_set_list_v: set_list_v > set_list_v > set_list_v ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
inf_in6271465464967711157od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
inf_inf_set_set_v: set_set_v > set_set_v > set_set_v ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__v_J,type,
inf_inf_set_v: set_v > set_v > set_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_Itf__v_J_J,type,
sup_sup_set_list_v: set_list_v > set_list_v > set_list_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
sup_su1742609618068805275od_v_v: set_Pr2149350503807050951od_v_v > set_Pr2149350503807050951od_v_v > set_Pr2149350503807050951od_v_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
sup_su414716646722978715od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
sup_su335656005089752955od_v_v: set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
sup_sup_set_set_v: set_set_v > set_set_v > set_set_v ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__v_J,type,
sup_sup_set_v: set_v > set_v > set_v ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
lattic4767070952889939172od_v_v: set_se8455005133513928103od_v_v > set_Product_prod_v_v ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_Itf__v_J,type,
lattic8209813555532694032_set_v: set_set_v > set_v ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
lattic5151207300795964030od_v_v: set_se8455005133513928103od_v_v > set_Product_prod_v_v ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_Itf__v_J,type,
lattic2918178447194608042_set_v: set_set_v > set_v ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
append2138873909117096322od_v_v: list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Oappend_001tf__v,type,
append_v: list_v > list_v > list_v ).
thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
concat2875663619778446888od_v_v: list_l4795378083388841843od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Oconcat_001tf__v,type,
concat_v: list_list_v > list_v ).
thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
coset_766761627116920666od_v_v: list_P7986770385144383213od_v_v > set_Product_prod_v_v ).
thf(sy_c_List_Ocoset_001tf__v,type,
coset_v: list_v > set_v ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
distin913317783593574886od_v_v: list_l4795378083388841843od_v_v > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_Itf__v_J,type,
distinct_list_v: list_list_v > $o ).
thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
distin6159370996967099744od_v_v: list_P7986770385144383213od_v_v > $o ).
thf(sy_c_List_Odistinct_001tf__v,type,
distinct_v: list_v > $o ).
thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
insert4539780211034306307od_v_v: product_prod_v_v > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Oinsert_001tf__v,type,
insert_v: v > list_v > list_v ).
thf(sy_c_List_Olexord_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
lexord8601710409828808922od_v_v: set_Pr2149350503807050951od_v_v > set_Pr7499474215547700295od_v_v ).
thf(sy_c_List_Olexord_001tf__v,type,
lexord_v: set_Product_prod_v_v > set_Pr6206931691796273479list_v ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
cons_P4120604216776828829od_v_v: product_prod_v_v > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Olist_OCons_001tf__v,type,
cons_v: v > list_v > list_v ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
nil_Product_prod_v_v: list_P7986770385144383213od_v_v ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_Itf__v_J,type,
nil_set_v: list_set_v ).
thf(sy_c_List_Olist_ONil_001tf__v,type,
nil_v: list_v ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
hd_Product_prod_v_v: list_P7986770385144383213od_v_v > product_prod_v_v ).
thf(sy_c_List_Olist_Ohd_001tf__v,type,
hd_v: list_v > v ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
set_li2340707408155270402od_v_v: list_l4795378083388841843od_v_v > set_li2323639185124838733od_v_v ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__v_J,type,
set_list_v2: list_list_v > set_list_v ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
set_Product_prod_v_v2: list_P7986770385144383213od_v_v > set_Product_prod_v_v ).
thf(sy_c_List_Olist_Oset_001tf__v,type,
set_v2: list_v > set_v ).
thf(sy_c_List_Olist_Otl_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
tl_Product_prod_v_v: list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Olist_Otl_001tf__v,type,
tl_v: list_v > list_v ).
thf(sy_c_List_Olistrel1_001tf__v,type,
listrel1_v: set_Product_prod_v_v > set_Pr6206931691796273479list_v ).
thf(sy_c_List_Olists_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
lists_5865669170805476827od_v_v: set_Product_prod_v_v > set_li2323639185124838733od_v_v ).
thf(sy_c_List_Olists_001tf__v,type,
lists_v: set_v > set_list_v ).
thf(sy_c_List_Olistset_001tf__v,type,
listset_v: list_set_v > set_list_v ).
thf(sy_c_List_Onull_001tf__v,type,
null_v: list_v > $o ).
thf(sy_c_List_Oremove1_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
remove333779696311199107od_v_v: product_prod_v_v > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_Oremove1_001tf__v,type,
remove1_v: v > list_v > list_v ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
remove5095778601549809401od_v_v: list_P7986770385144383213od_v_v > list_l4795378083388841843od_v_v > list_l4795378083388841843od_v_v ).
thf(sy_c_List_OremoveAll_001t__List__Olist_Itf__v_J,type,
removeAll_list_v: list_v > list_list_v > list_list_v ).
thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
remove481895986417801203od_v_v: product_prod_v_v > list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v ).
thf(sy_c_List_OremoveAll_001tf__v,type,
removeAll_v: v > list_v > list_v ).
thf(sy_c_List_Oshuffles_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
shuffl71542398924059522od_v_v: list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v > set_li2323639185124838733od_v_v ).
thf(sy_c_List_Oshuffles_001tf__v,type,
shuffles_v: list_v > list_v > set_list_v ).
thf(sy_c_Order__Relation_OaboveS_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
order_1156346741491923410od_v_v: set_Pr2149350503807050951od_v_v > product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Order__Relation_OaboveS_001tf__v,type,
order_aboveS_v: set_Product_prod_v_v > v > set_v ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
order_6462556390437124636od_v_v: set_Product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Order__Relation_Olinear__order__on_001tf__v,type,
order_8768733634509060168r_on_v: set_v > set_Product_prod_v_v > $o ).
thf(sy_c_Order__Relation_Opartial__order__on_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
order_4212533993404950492od_v_v: set_Product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Order__Relation_Opartial__order__on_001tf__v,type,
order_5272072345360262664r_on_v: set_v > set_Product_prod_v_v > $o ).
thf(sy_c_Order__Relation_OunderS_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
order_5211820470575790509od_v_v: set_Pr2149350503807050951od_v_v > product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Order__Relation_OunderS_001tf__v,type,
order_underS_v: set_Product_prod_v_v > v > set_v ).
thf(sy_c_Order__Relation_Ounder_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
order_6892855479609198156od_v_v: set_Pr2149350503807050951od_v_v > product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Order__Relation_Ounder_001tf__v,type,
order_under_v: set_Product_prod_v_v > v > set_v ).
thf(sy_c_Order__Relation_Owell__order__on_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
order_7541072052284126853od_v_v: set_Product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Order__Relation_Owell__order__on_001tf__v,type,
order_6972113574731384241r_on_v: set_v > set_Product_prod_v_v > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_Itf__v_Mtf__v_J_M_Eo_J,type,
bot_bo8461541820394803818_v_v_o: product_prod_v_v > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__v_J_M_Eo_J,type,
bot_bot_set_v_o: set_v > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__v_M_Eo_J,type,
bot_bot_v_o: v > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
bot_bo54012148666785209od_v_v: set_li2323639185124838733od_v_v ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__v_J_J,type,
bot_bot_set_list_v: set_list_v ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
bot_bo3282589961317712691od_v_v: set_Pr2149350503807050951od_v_v ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
bot_bo723834152578015283od_v_v: set_Product_prod_v_v ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
bot_bo3497076220358800403od_v_v: set_se8455005133513928103od_v_v ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
bot_bot_set_set_v: set_set_v ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__v_J,type,
bot_bot_set_v: set_v ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
ord_le5393391283775026413od_v_v: set_li2323639185124838733od_v_v > set_li2323639185124838733od_v_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__v_J_J,type,
ord_le1129530298279361049list_v: set_list_v > set_list_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_Itf__v_J_J_J,type,
ord_le791731619978752231list_v: set_Pr6206931691796273479list_v > set_Pr6206931691796273479list_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
ord_le6241436655786843239od_v_v: set_Pr2149350503807050951od_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
ord_le7336532860387713383od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
ord_le4714265922333009223od_v_v: set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
ord_le5216385588623774835_set_v: set_set_v > set_set_v > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__v_J,type,
ord_less_eq_set_v: set_v > set_v > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
ord_mi6996445931809003310od_v_v: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_Itf__v_J,type,
ord_min_set_v: set_v > set_v > set_v ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
produc674067373767953879od_v_v: list_P7986770385144383213od_v_v > list_P7986770385144383213od_v_v > produc1504107476793160551od_v_v ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__v_J_001t__List__Olist_Itf__v_J,type,
produc6795410681906604247list_v: list_v > list_v > produc1391462591744249447list_v ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
produc4031800376763917143od_v_v: product_prod_v_v > product_prod_v_v > produc206430290419586791od_v_v ).
thf(sy_c_Product__Type_OPair_001tf__v_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J,type,
produc3862955338007567901t_unit: v > sCC_Bl1394983891496994913t_unit > produc5741669702376414499t_unit ).
thf(sy_c_Product__Type_OPair_001tf__v_001tf__v,type,
product_Pair_v_v: v > v > product_prod_v_v ).
thf(sy_c_Relation_ODomain_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
domain6359000466948879308od_v_v: set_Pr2149350503807050951od_v_v > set_Product_prod_v_v ).
thf(sy_c_Relation_ODomain_001tf__v_001tf__v,type,
domain_v_v: set_Product_prod_v_v > set_v ).
thf(sy_c_Relation_OField_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
field_7153129647634986036od_v_v: set_Pr2149350503807050951od_v_v > set_Product_prod_v_v ).
thf(sy_c_Relation_OField_001tf__v,type,
field_v: set_Product_prod_v_v > set_v ).
thf(sy_c_Relation_OId_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
id_Product_prod_v_v: set_Pr2149350503807050951od_v_v ).
thf(sy_c_Relation_OId_001tf__v,type,
id_v: set_Product_prod_v_v ).
thf(sy_c_Relation_ORange_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
range_7878975032137371189od_v_v: set_Pr2149350503807050951od_v_v > set_Product_prod_v_v ).
thf(sy_c_Relation_ORange_001tf__v_001tf__v,type,
range_v_v: set_Product_prod_v_v > set_v ).
thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
refl_o4548774019903118566od_v_v: set_Product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Relation_Orefl__on_001tf__v,type,
refl_on_v: set_v > set_Product_prod_v_v > $o ).
thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
total_9075964390993782123od_v_v: set_Product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Relation_Ototal__on_001tf__v,type,
total_on_v: set_v > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_O_092_060S_062_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl8440648026628373538t_unit: sCC_Bl7326425374436813197t_unit > product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_O_092_060S_062_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl1280885523602775798t_unit: sCC_Bl1394983891496994913t_unit > v > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ocstack_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl9201514103433284750t_unit: sCC_Bl1394983891496994913t_unit > list_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Oenv__ext_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl8064756265740546429t_unit: v > ( v > set_v ) > set_v > set_v > ( v > set_v ) > set_set_v > list_v > list_v > product_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Oexplored_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl5094201334446601350t_unit: sCC_Bl7326425374436813197t_unit > set_Product_prod_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Oexplored_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl157864678168468314t_unit: sCC_Bl1394983891496994913t_unit > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Omore_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl3567736435408124606t_unit: sCC_Bl1394983891496994913t_unit > product_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Oroot_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl1090238580953940555t_unit: sCC_Bl1394983891496994913t_unit > v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Osccs_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl2536197123907397897t_unit: sCC_Bl1394983891496994913t_unit > set_set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ostack_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl2021302119412358655t_unit: sCC_Bl7326425374436813197t_unit > list_P7986770385144383213od_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ostack_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl8828226123343373779t_unit: sCC_Bl1394983891496994913t_unit > list_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovisited_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl5498988629518860705t_unit: sCC_Bl7326425374436813197t_unit > set_Product_prod_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovisited_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl4645233313691564917t_unit: sCC_Bl1394983891496994913t_unit > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovsuccs_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl3878977043676959280t_unit: sCC_Bl7326425374436813197t_unit > product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovsuccs_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl3795065053823578884t_unit: sCC_Bl1394983891496994913t_unit > v > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovsuccs__update_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl48393358579903213t_unit: ( ( v > set_v ) > v > set_v ) > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl8307124943676871238od_v_v: set_Product_prod_v_v > ( product_prod_v_v > set_Product_prod_v_v ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001tf__v,type,
sCC_Bloemen_graph_v: set_v > ( v > set_v ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Odfs_001tf__v,type,
sCC_Bloemen_dfs_v: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl6242042402218619277od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001tf__v,type,
sCC_Bloemen_is_scc_v: ( v > set_v ) > set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl2301996248249672505od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001tf__v,type,
sCC_Bl5398416737448265317bscc_v: ( v > set_v ) > set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opost__dfs_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl8953792750115413617t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opost__dfss_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl6082031138996704384t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfs_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl36166008131615352t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfss_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl3607325323686918683t_unit: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfss_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl1748261141445803503t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4981926079593201289od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001tf__v,type,
sCC_Bl649662514949026229able_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__avoiding_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl5370300055464682748od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__avoiding_001tf__v,type,
sCC_Bl4291963740693775144ding_v: ( v > set_v ) > v > v > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4714988717384592488od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_001tf__v,type,
sCC_Bl770211535891879572_end_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Osub__env_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl7963838319573962697t_unit: sCC_Bl7326425374436813197t_unit > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Osub__env_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl5768913643336123637t_unit: sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounite_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4702006153222411093od_v_v: product_prod_v_v > product_prod_v_v > sCC_Bl7326425374436813197t_unit > sCC_Bl7326425374436813197t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounite_001tf__v,type,
sCC_Bloemen_unite_v: v > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Owf__env_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl7798947040364291444t_unit: ( product_prod_v_v > set_Product_prod_v_v ) > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Owf__env_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl9196236973127232072t_unit: ( v > set_v ) > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oinit__env_001tf__v,type,
sCC_Bl7693227186847904995_env_v: v > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl2026170059108282219od_v_v: product_prod_v_v > product_prod_v_v > list_P7986770385144383213od_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001tf__v,type,
sCC_Bl4022239298816431255edes_v: v > v > list_v > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
collec140062887454715474od_v_v: ( product_prod_v_v > $o ) > set_Product_prod_v_v ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__v_J,type,
collect_set_v: ( set_v > $o ) > set_set_v ).
thf(sy_c_Set_OCollect_001tf__v,type,
collect_v: ( v > $o ) > set_v ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__v_J_001t__List__Olist_Itf__v_J,type,
image_list_v_list_v: ( list_v > list_v ) > set_list_v > set_list_v ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
image_781944334261467077od_v_v: ( product_prod_v_v > product_prod_v_v ) > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Set__Oset_Itf__v_J,type,
image_2529437795422174673_set_v: ( product_prod_v_v > set_v ) > set_Product_prod_v_v > set_set_v ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001tf__v,type,
image_6152814753742948081_v_v_v: ( product_prod_v_v > v ) > set_Product_prod_v_v > set_v ).
thf(sy_c_Set_Oimage_001tf__v_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
image_9222788639401671577od_v_v: ( v > product_prod_v_v ) > set_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oimage_001tf__v_001t__Set__Oset_Itf__v_J,type,
image_v_set_v: ( v > set_v ) > set_v > set_set_v ).
thf(sy_c_Set_Oimage_001tf__v_001tf__v,type,
image_v_v: ( v > v ) > set_v > set_v ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
insert4087971119735676093od_v_v: list_P7986770385144383213od_v_v > set_li2323639185124838733od_v_v > set_li2323639185124838733od_v_v ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__v_J,type,
insert_list_v: list_v > set_list_v > set_list_v ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
insert5641704497130386615od_v_v: produc206430290419586791od_v_v > set_Pr2149350503807050951od_v_v > set_Pr2149350503807050951od_v_v ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
insert1338601472111419319od_v_v: product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
insert7504383016908236695od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__v_J,type,
insert_set_v: set_v > set_set_v > set_set_v ).
thf(sy_c_Set_Oinsert_001tf__v,type,
insert_v2: v > set_v > set_v ).
thf(sy_c_Set_Ois__empty_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
is_emp8964507351669718201od_v_v: set_Product_prod_v_v > $o ).
thf(sy_c_Set_Ois__empty_001tf__v,type,
is_empty_v: set_v > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
is_sin9198872032823709915od_v_v: set_Product_prod_v_v > $o ).
thf(sy_c_Set_Ois__singleton_001tf__v,type,
is_singleton_v: set_v > $o ).
thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
remove5001965847480235980od_v_v: product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oremove_001tf__v,type,
remove_v: v > set_v > set_v ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
the_el5392834299063928540od_v_v: set_Product_prod_v_v > product_prod_v_v ).
thf(sy_c_Set_Othe__elem_001tf__v,type,
the_elem_v: set_v > v ).
thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
wf_Product_prod_v_v: set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Wellfounded_Owf_001tf__v,type,
wf_v: set_Product_prod_v_v > $o ).
thf(sy_c_Zorn_OChains_001tf__v,type,
chains_v: set_Product_prod_v_v > set_set_v ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member4190458934886417558od_v_v: list_P7986770385144383213od_v_v > set_li2323639185124838733od_v_v > $o ).
thf(sy_c_member_001t__List__Olist_Itf__v_J,type,
member_list_v: list_v > set_list_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
member6382463057129219728od_v_v: produc1504107476793160551od_v_v > set_Pr7499474215547700295od_v_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_Itf__v_J_J,type,
member418487059593946000list_v: produc1391462591744249447list_v > set_Pr6206931691796273479list_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member3038538357316246288od_v_v: produc206430290419586791od_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
member7453568604450474000od_v_v: product_prod_v_v > set_Product_prod_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
member2865299526245254384od_v_v: set_Pr2149350503807050951od_v_v > set_se5707775751431548583od_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member8406446414694345712od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__v_J,type,
member_set_v: set_v > set_set_v > $o ).
thf(sy_c_member_001tf__v,type,
member_v: v > set_v > $o ).
thf(sy_v_e,type,
e: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H,type,
e2: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H_H,type,
e3: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_successors,type,
successors: v > set_v ).
thf(sy_v_v,type,
v2: v ).
thf(sy_v_vertices,type,
vertices: set_v ).
thf(sy_v_w,type,
w: v ).
% Relevant facts (1277)
thf(fact_0_sub__env__trans,axiom,
! [E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ).
% sub_env_trans
thf(fact_1_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_2__092_060open_062wf__env_Ae_092_060close_062,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e ).
% \<open>wf_env e\<close>
thf(fact_3_S__reflexive,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ).
% S_reflexive
thf(fact_4_w_I1_J,axiom,
member_v @ w @ ( successors @ v2 ) ).
% w(1)
thf(fact_5_reachable__end_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y )
=> ~ ( member_v @ A2 @ ( successors @ Y ) ) ) ) ) ).
% reachable_end.cases
thf(fact_6_re__refl,axiom,
! [X: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X @ X ) ).
% re_refl
thf(fact_7_re__succ,axiom,
! [X: v,Y2: v,Z: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y2 )
=> ( ( member_v @ Z @ ( successors @ Y2 ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% re_succ
thf(fact_8_reachable__end_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y3 )
& ( member_v @ Z2 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_9_succ__re,axiom,
! [Y2: v,X: v,Z: v] :
( ( member_v @ Y2 @ ( successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y2 @ Z )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% succ_re
thf(fact_10_init__env__pre__dfs,axiom,
! [V: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ).
% init_env_pre_dfs
thf(fact_11_ra__refl,axiom,
! [X: v,E4: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X @ X @ E4 ) ).
% ra_refl
thf(fact_12_ra__trans,axiom,
! [X: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y2 @ Z @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E4 ) ) ) ).
% ra_trans
thf(fact_13_pre,axiom,
sCC_Bl1748261141445803503t_unit @ successors @ v2 @ e ).
% pre
thf(fact_14_w_I2_J,axiom,
~ ( member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ e ) ) ).
% w(2)
thf(fact_15_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_16_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_17_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_18_e_H__def,axiom,
( e2
= ( sCC_Bloemen_dfs_v @ successors @ w @ e ) ) ).
% e'_def
thf(fact_19_post,axiom,
sCC_Bl8953792750115413617t_unit @ successors @ w @ e @ e2 ).
% post
thf(fact_20__092_060open_062v_A_092_060in_062_Avisited_Ae_092_060close_062,axiom,
member_v @ v2 @ ( sCC_Bl4645233313691564917t_unit @ e ) ).
% \<open>v \<in> visited e\<close>
thf(fact_21_ra__mono,axiom,
! [X: v,Y2: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E5 ) ) ) ).
% ra_mono
thf(fact_22_ra__cases,axiom,
! [X: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( ( X = Y2 )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y2 @ E4 ) ) ) ) ).
% ra_cases
thf(fact_23_edge__ra,axiom,
! [Y2: v,X: v,E4: set_Product_prod_v_v] :
( ( member_v @ Y2 @ ( successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 ) ) ) ).
% edge_ra
thf(fact_24_reachable__avoiding_Osimps,axiom,
! [A1: v,A2: v,A3: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A3 )
= ( ? [X2: v,E6: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A3 = E6 ) )
| ? [X2: v,Y3: v,E6: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A3 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y3 @ E6 )
& ( member_v @ Z2 @ ( successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_25_ra__succ,axiom,
! [X: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( ( member_v @ Z @ ( successors @ Y2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E4 ) ) ) ) ).
% ra_succ
thf(fact_26_reachable__avoiding_Ocases,axiom,
! [A1: v,A2: v,A3: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A3 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y @ A3 )
=> ( ( member_v @ A2 @ ( successors @ Y ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ A2 ) @ A3 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_27_reachable__re,axiom,
! [X: v,Y2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y2 ) ) ).
% reachable_re
thf(fact_28_re__reachable,axiom,
! [X: v,Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).
% re_reachable
thf(fact_29_reachable_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( member_v @ Y @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y @ A2 ) ) ) ) ).
% reachable.cases
thf(fact_30_reachable__refl,axiom,
! [X: v] : ( sCC_Bl649662514949026229able_v @ successors @ X @ X ) ).
% reachable_refl
thf(fact_31_reachable__succ,axiom,
! [Y2: v,X: v,Z: v] :
( ( member_v @ Y2 @ ( successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_succ
thf(fact_32_reachable_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( member_v @ Y3 @ ( successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z2 ) ) ) ) ).
% reachable.simps
thf(fact_33_reachable__edge,axiom,
! [Y2: v,X: v] :
( ( member_v @ Y2 @ ( successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).
% reachable_edge
thf(fact_34_reachable__end__induct,axiom,
! [X: v,Y2: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y: v,Z3: v] :
( ( P @ X3 @ Y )
=> ( ( member_v @ Z3 @ ( successors @ Y ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% reachable_end_induct
thf(fact_35_reachable__trans,axiom,
! [X: v,Y2: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_trans
thf(fact_36_succ__reachable,axiom,
! [X: v,Y2: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ( member_v @ Z @ ( successors @ Y2 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% succ_reachable
thf(fact_37_ra__reachable,axiom,
! [X: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).
% ra_reachable
thf(fact_38__092_060open_062w_A_092_060in_062_Avisited_Ae_H_092_060close_062,axiom,
member_v @ w @ ( sCC_Bl4645233313691564917t_unit @ e2 ) ).
% \<open>w \<in> visited e'\<close>
thf(fact_39_pre__dfss__pre__dfs,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( sCC_Bl36166008131615352t_unit @ successors @ W @ E ) ) ) ) ).
% pre_dfss_pre_dfs
thf(fact_40_mem__Collect__eq,axiom,
! [A: v,P: v > $o] :
( ( member_v @ A @ ( collect_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_41_mem__Collect__eq,axiom,
! [A: product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
! [A: set_v,P: set_v > $o] :
( ( member_set_v @ A @ ( collect_set_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A4: set_v] :
( ( collect_v
@ ^ [X2: v] : ( member_v @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A4: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A4: set_set_v] :
( ( collect_set_v
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X3: set_v] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_v @ P )
= ( collect_set_v @ Q ) ) ) ).
% Collect_cong
thf(fact_47_sccE,axiom,
! [S: set_v,X: v,Y2: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X )
=> ( member_v @ Y2 @ S ) ) ) ) ) ).
% sccE
thf(fact_48_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_49_graph_Opost__dfs_Ocong,axiom,
sCC_Bl8953792750115413617t_unit = sCC_Bl8953792750115413617t_unit ).
% graph.post_dfs.cong
thf(fact_50_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_51_graph_Odfs_Ocong,axiom,
sCC_Bloemen_dfs_v = sCC_Bloemen_dfs_v ).
% graph.dfs.cong
thf(fact_52_is__subscc__def,axiom,
! [S: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y3 ) ) ) ) ) ).
% is_subscc_def
thf(fact_53_ra__empty,axiom,
! [X: v,Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).
% ra_empty
thf(fact_54_subset__antisym,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A4 )
=> ( A4 = B ) ) ) ).
% subset_antisym
thf(fact_55_subset__antisym,axiom,
! [A4: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ( ord_less_eq_set_v @ B @ A4 )
=> ( A4 = B ) ) ) ).
% subset_antisym
thf(fact_56_subsetI,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ( member7453568604450474000od_v_v @ X3 @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ B ) ) ).
% subsetI
thf(fact_57_subsetI,axiom,
! [A4: set_v,B: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A4 )
=> ( member_v @ X3 @ B ) )
=> ( ord_less_eq_set_v @ A4 @ B ) ) ).
% subsetI
thf(fact_58_old_Oprod_Oinject,axiom,
! [A: v,B2: v,A5: v,B3: v] :
( ( ( product_Pair_v_v @ A @ B2 )
= ( product_Pair_v_v @ A5 @ B3 ) )
= ( ( A = A5 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_59_prod_Oinject,axiom,
! [X1: v,X22: v,Y1: v,Y22: v] :
( ( ( product_Pair_v_v @ X1 @ X22 )
= ( product_Pair_v_v @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_60_dual__order_Orefl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% dual_order.refl
thf(fact_61_dual__order_Orefl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% dual_order.refl
thf(fact_62_order__refl,axiom,
! [X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ X ) ).
% order_refl
thf(fact_63_order__refl,axiom,
! [X: set_v] : ( ord_less_eq_set_v @ X @ X ) ).
% order_refl
thf(fact_64_subrelI,axiom,
! [R: set_Product_prod_v_v,S2: set_Product_prod_v_v] :
( ! [X3: v,Y: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ S2 ) )
=> ( ord_le7336532860387713383od_v_v @ R @ S2 ) ) ).
% subrelI
thf(fact_65_reachable__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ W )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_66_relChain__def,axiom,
( bNF_Ca66581456156439064od_v_v
= ( ^ [R2: set_Product_prod_v_v,As: v > set_Product_prod_v_v] :
! [I: v,J: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R2 )
=> ( ord_le7336532860387713383od_v_v @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_67_relChain__def,axiom,
( bNF_Ca2468909317164667716_set_v
= ( ^ [R2: set_Product_prod_v_v,As: v > set_v] :
! [I: v,J: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R2 )
=> ( ord_less_eq_set_v @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_68_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_69_empty__iff,axiom,
! [C: v] :
~ ( member_v @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_70_all__not__in__conv,axiom,
! [A4: set_Product_prod_v_v] :
( ( ! [X2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X2 @ A4 ) )
= ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_71_all__not__in__conv,axiom,
! [A4: set_v] :
( ( ! [X2: v] :
~ ( member_v @ X2 @ A4 ) )
= ( A4 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_72_Collect__empty__eq,axiom,
! [P: set_v > $o] :
( ( ( collect_set_v @ P )
= bot_bot_set_set_v )
= ( ! [X2: set_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_73_Collect__empty__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_74_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_75_empty__Collect__eq,axiom,
! [P: set_v > $o] :
( ( bot_bot_set_set_v
= ( collect_set_v @ P ) )
= ( ! [X2: set_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_76_empty__Collect__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P ) )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_77_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_78_DiffI,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ A4 )
=> ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ ( minus_minus_set_v @ A4 @ B ) ) ) ) ).
% DiffI
thf(fact_79_DiffI,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A4 )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A4 @ B ) ) ) ) ).
% DiffI
thf(fact_80_Diff__iff,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A4 @ B ) )
= ( ( member_v @ C @ A4 )
& ~ ( member_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_81_Diff__iff,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A4 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A4 )
& ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_82_subset__empty,axiom,
! [A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_83_subset__empty,axiom,
! [A4: set_v] :
( ( ord_less_eq_set_v @ A4 @ bot_bot_set_v )
= ( A4 = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_84_empty__subsetI,axiom,
! [A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A4 ) ).
% empty_subsetI
thf(fact_85_empty__subsetI,axiom,
! [A4: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A4 ) ).
% empty_subsetI
thf(fact_86_Diff__empty,axiom,
! [A4: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= A4 ) ).
% Diff_empty
thf(fact_87_Diff__empty,axiom,
! [A4: set_v] :
( ( minus_minus_set_v @ A4 @ bot_bot_set_v )
= A4 ) ).
% Diff_empty
thf(fact_88_empty__Diff,axiom,
! [A4: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A4 )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_89_empty__Diff,axiom,
! [A4: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A4 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_90_Diff__cancel,axiom,
! [A4: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A4 @ A4 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_91_Diff__cancel,axiom,
! [A4: set_v] :
( ( minus_minus_set_v @ A4 @ A4 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_92_Diff__eq__empty__iff,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A4 @ B )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A4 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_93_Diff__eq__empty__iff,axiom,
! [A4: set_v,B: set_v] :
( ( ( minus_minus_set_v @ A4 @ B )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A4 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_94_sclosed,axiom,
! [X4: v] :
( ( member_v @ X4 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X4 ) @ vertices ) ) ).
% sclosed
thf(fact_95_DiffE,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A4 @ B ) )
=> ~ ( ( member_v @ C @ A4 )
=> ( member_v @ C @ B ) ) ) ).
% DiffE
thf(fact_96_DiffE,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A4 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A4 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% DiffE
thf(fact_97_DiffD1,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A4 @ B ) )
=> ( member_v @ C @ A4 ) ) ).
% DiffD1
thf(fact_98_DiffD1,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A4 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A4 ) ) ).
% DiffD1
thf(fact_99_DiffD2,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A4 @ B ) )
=> ~ ( member_v @ C @ B ) ) ).
% DiffD2
thf(fact_100_DiffD2,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A4 @ B ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ).
% DiffD2
thf(fact_101_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_102_emptyE,axiom,
! [A: v] :
~ ( member_v @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_103_equals0D,axiom,
! [A4: set_Product_prod_v_v,A: product_prod_v_v] :
( ( A4 = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A @ A4 ) ) ).
% equals0D
thf(fact_104_equals0D,axiom,
! [A4: set_v,A: v] :
( ( A4 = bot_bot_set_v )
=> ~ ( member_v @ A @ A4 ) ) ).
% equals0D
thf(fact_105_equals0I,axiom,
! [A4: set_Product_prod_v_v] :
( ! [Y: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y @ A4 )
=> ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_106_equals0I,axiom,
! [A4: set_v] :
( ! [Y: v] :
~ ( member_v @ Y @ A4 )
=> ( A4 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_107_ex__in__conv,axiom,
! [A4: set_Product_prod_v_v] :
( ( ? [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A4 ) )
= ( A4 != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_108_ex__in__conv,axiom,
! [A4: set_v] :
( ( ? [X2: v] : ( member_v @ X2 @ A4 ) )
= ( A4 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_109_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_110_bot_Oextremum,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% bot.extremum
thf(fact_111_bot_Oextremum,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% bot.extremum
thf(fact_112_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_unique
thf(fact_113_bot_Oextremum__unique,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
= ( A = bot_bot_set_v ) ) ).
% bot.extremum_unique
thf(fact_114_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
=> ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_uniqueI
thf(fact_115_bot_Oextremum__uniqueI,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
=> ( A = bot_bot_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_116_Diff__mono,axiom,
! [A4: set_Product_prod_v_v,C2: set_Product_prod_v_v,D: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ B ) @ ( minus_4183494784930505774od_v_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_117_Diff__mono,axiom,
! [A4: set_v,C2: set_v,D: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A4 @ C2 )
=> ( ( ord_less_eq_set_v @ D @ B )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A4 @ B ) @ ( minus_minus_set_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_118_Diff__subset,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ B ) @ A4 ) ).
% Diff_subset
thf(fact_119_Diff__subset,axiom,
! [A4: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A4 @ B ) @ A4 ) ).
% Diff_subset
thf(fact_120_double__diff,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C2 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_121_double__diff,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C2 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_122_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_123_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
& ( ord_le7336532860387713383od_v_v @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_124_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
& ( ord_less_eq_set_v @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_125_ord__eq__le__trans,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A = B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_126_ord__eq__le__trans,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( A = B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_127_ord__le__eq__trans,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_128_ord__le__eq__trans,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_129_order__antisym,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_130_order__antisym,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_less_eq_set_v @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_131_order_Otrans,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% order.trans
thf(fact_132_order_Otrans,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% order.trans
thf(fact_133_order__trans,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ Y2 @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_134_order__trans,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_less_eq_set_v @ Y2 @ Z )
=> ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_135_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A6 )
& ( ord_le7336532860387713383od_v_v @ A6 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_136_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ B4 @ A6 )
& ( ord_less_eq_set_v @ A6 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_137_dual__order_Oantisym,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_138_dual__order_Oantisym,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( ord_less_eq_set_v @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_139_dual__order_Otrans,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_140_dual__order_Otrans,axiom,
! [B2: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( ord_less_eq_set_v @ C @ B2 )
=> ( ord_less_eq_set_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_141_antisym,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_142_antisym,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_143_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A6 @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_144_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A6 @ B4 )
& ( ord_less_eq_set_v @ B4 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_145_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_146_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_147_order__subst1,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_148_order__subst1,axiom,
! [A: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_149_order__subst2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_150_order__subst2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_151_order__subst2,axiom,
! [A: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_152_order__subst2,axiom,
! [A: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ ( F @ B2 ) @ C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_153_order__eq__refl,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( X = Y2 )
=> ( ord_le7336532860387713383od_v_v @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_154_order__eq__refl,axiom,
! [X: set_v,Y2: set_v] :
( ( X = Y2 )
=> ( ord_less_eq_set_v @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_155_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_156_ord__eq__le__subst,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_157_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B2: set_v,C: set_v] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_158_ord__eq__le__subst,axiom,
! [A: set_v,F: set_v > set_v,B2: set_v,C: set_v] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_159_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_160_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_161_ord__le__eq__subst,axiom,
! [A: set_v,B2: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_162_ord__le__eq__subst,axiom,
! [A: set_v,B2: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_163_order__antisym__conv,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_164_order__antisym__conv,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( ord_less_eq_set_v @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_165_old_Oprod_Oexhaust,axiom,
! [Y2: product_prod_v_v] :
~ ! [A7: v,B5: v] :
( Y2
!= ( product_Pair_v_v @ A7 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_166_surj__pair,axiom,
! [P2: product_prod_v_v] :
? [X3: v,Y: v] :
( P2
= ( product_Pair_v_v @ X3 @ Y ) ) ).
% surj_pair
thf(fact_167_prod__cases,axiom,
! [P: product_prod_v_v > $o,P2: product_prod_v_v] :
( ! [A7: v,B5: v] : ( P @ ( product_Pair_v_v @ A7 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_168_Pair__inject,axiom,
! [A: v,B2: v,A5: v,B3: v] :
( ( ( product_Pair_v_v @ A @ B2 )
= ( product_Pair_v_v @ A5 @ B3 ) )
=> ~ ( ( A = A5 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_169_in__mono,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( member7453568604450474000od_v_v @ X @ B ) ) ) ).
% in_mono
thf(fact_170_in__mono,axiom,
! [A4: set_v,B: set_v,X: v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ( member_v @ X @ A4 )
=> ( member_v @ X @ B ) ) ) ).
% in_mono
thf(fact_171_subsetD,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ( member7453568604450474000od_v_v @ C @ A4 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% subsetD
thf(fact_172_subsetD,axiom,
! [A4: set_v,B: set_v,C: v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ( member_v @ C @ A4 )
=> ( member_v @ C @ B ) ) ) ).
% subsetD
thf(fact_173_equalityE,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A4 = B )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ A4 ) ) ) ).
% equalityE
thf(fact_174_equalityE,axiom,
! [A4: set_v,B: set_v] :
( ( A4 = B )
=> ~ ( ( ord_less_eq_set_v @ A4 @ B )
=> ~ ( ord_less_eq_set_v @ B @ A4 ) ) ) ).
% equalityE
thf(fact_175_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A8 )
=> ( member7453568604450474000od_v_v @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_176_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A8: set_v,B6: set_v] :
! [X2: v] :
( ( member_v @ X2 @ A8 )
=> ( member_v @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_177_equalityD1,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A4 = B )
=> ( ord_le7336532860387713383od_v_v @ A4 @ B ) ) ).
% equalityD1
thf(fact_178_equalityD1,axiom,
! [A4: set_v,B: set_v] :
( ( A4 = B )
=> ( ord_less_eq_set_v @ A4 @ B ) ) ).
% equalityD1
thf(fact_179_equalityD2,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A4 = B )
=> ( ord_le7336532860387713383od_v_v @ B @ A4 ) ) ).
% equalityD2
thf(fact_180_equalityD2,axiom,
! [A4: set_v,B: set_v] :
( ( A4 = B )
=> ( ord_less_eq_set_v @ B @ A4 ) ) ).
% equalityD2
thf(fact_181_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A8 )
=> ( member7453568604450474000od_v_v @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_182_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A8: set_v,B6: set_v] :
! [T: v] :
( ( member_v @ T @ A8 )
=> ( member_v @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_183_subset__refl,axiom,
! [A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A4 @ A4 ) ).
% subset_refl
thf(fact_184_subset__refl,axiom,
! [A4: set_v] : ( ord_less_eq_set_v @ A4 @ A4 ) ).
% subset_refl
thf(fact_185_Collect__mono,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X3: set_v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_mono
thf(fact_186_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X3: product_prod_v_v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_187_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X3: v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_188_subset__trans,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_189_subset__trans,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_190_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A8 @ B6 )
& ( ord_le7336532860387713383od_v_v @ B6 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_191_set__eq__subset,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A8: set_v,B6: set_v] :
( ( ord_less_eq_set_v @ A8 @ B6 )
& ( ord_less_eq_set_v @ B6 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_192_Collect__mono__iff,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) )
= ( ! [X2: set_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_193_Collect__mono__iff,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) )
= ( ! [X2: product_prod_v_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_194_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X2: v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_195_is__scc__def,axiom,
! [S: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
& ! [S3: set_v] :
( ( ( ord_less_eq_set_v @ S @ S3 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 ) )
=> ( S3 = S ) ) ) ) ).
% is_scc_def
thf(fact_196_subscc__add,axiom,
! [S: set_v,X: v,Y2: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v2 @ Y2 @ S ) ) ) ) ) ) ).
% subscc_add
thf(fact_197_diff__shunt__var,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ X @ Y2 )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_198_diff__shunt__var,axiom,
! [X: set_v,Y2: set_v] :
( ( ( minus_minus_set_v @ X @ Y2 )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_199_scc__partition,axiom,
! [S: set_v,S4: set_v,X: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S4 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S4 ) )
=> ( S = S4 ) ) ) ) ).
% scc_partition
thf(fact_200_less__by__empty,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A4 = bot_bo723834152578015283od_v_v )
=> ( ord_le7336532860387713383od_v_v @ A4 @ B ) ) ).
% less_by_empty
thf(fact_201_subset__emptyI,axiom,
! [A4: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ).
% subset_emptyI
thf(fact_202_subset__emptyI,axiom,
! [A4: set_v] :
( ! [X3: v] :
~ ( member_v @ X3 @ A4 )
=> ( ord_less_eq_set_v @ A4 @ bot_bot_set_v ) ) ).
% subset_emptyI
thf(fact_203_insertCI,axiom,
! [A: v,B: set_v,B2: v] :
( ( ~ ( member_v @ A @ B )
=> ( A = B2 ) )
=> ( member_v @ A @ ( insert_v2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_204_insertCI,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B )
=> ( A = B2 ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_205_insert__iff,axiom,
! [A: v,B2: v,A4: set_v] :
( ( member_v @ A @ ( insert_v2 @ B2 @ A4 ) )
= ( ( A = B2 )
| ( member_v @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_206_insert__iff,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ A4 ) )
= ( ( A = B2 )
| ( member7453568604450474000od_v_v @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_207_insert__absorb2,axiom,
! [X: v,A4: set_v] :
( ( insert_v2 @ X @ ( insert_v2 @ X @ A4 ) )
= ( insert_v2 @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_208_insert__absorb2,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ X @ A4 ) )
= ( insert1338601472111419319od_v_v @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_209_inf__right__idem,axiom,
! [X: set_v,Y2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Y2 )
= ( inf_inf_set_v @ X @ Y2 ) ) ).
% inf_right_idem
thf(fact_210_inf_Oright__idem,axiom,
! [A: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B2 ) @ B2 )
= ( inf_inf_set_v @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_211_inf__left__idem,axiom,
! [X: set_v,Y2: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y2 ) )
= ( inf_inf_set_v @ X @ Y2 ) ) ).
% inf_left_idem
thf(fact_212_inf_Oleft__idem,axiom,
! [A: set_v,B2: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B2 ) )
= ( inf_inf_set_v @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_213_inf__idem,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ X )
= X ) ).
% inf_idem
thf(fact_214_inf_Oidem,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% inf.idem
thf(fact_215_IntI,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A4 )
=> ( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) ) ) ) ).
% IntI
thf(fact_216_IntI,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ A4 )
=> ( ( member_v @ C @ B )
=> ( member_v @ C @ ( inf_inf_set_v @ A4 @ B ) ) ) ) ).
% IntI
thf(fact_217_Int__iff,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A4 )
& ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_218_Int__iff,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A4 @ B ) )
= ( ( member_v @ C @ A4 )
& ( member_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_219_inf_Obounded__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
& ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_220_inf_Obounded__iff,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) )
= ( ( ord_less_eq_set_v @ A @ B2 )
& ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_221_le__inf__iff,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
& ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_222_le__inf__iff,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( ( ord_less_eq_set_v @ X @ Y2 )
& ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_223_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_224_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_225_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_226_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_227_inf__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_right
thf(fact_228_inf__bot__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% inf_bot_right
thf(fact_229_inf__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_left
thf(fact_230_inf__bot__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% inf_bot_left
thf(fact_231_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_232_singletonI,axiom,
! [A: v] : ( member_v @ A @ ( insert_v2 @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_233_insert__subset,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ A4 ) @ B )
= ( ( member7453568604450474000od_v_v @ X @ B )
& ( ord_le7336532860387713383od_v_v @ A4 @ B ) ) ) ).
% insert_subset
thf(fact_234_insert__subset,axiom,
! [X: v,A4: set_v,B: set_v] :
( ( ord_less_eq_set_v @ ( insert_v2 @ X @ A4 ) @ B )
= ( ( member_v @ X @ B )
& ( ord_less_eq_set_v @ A4 @ B ) ) ) ).
% insert_subset
thf(fact_235_Int__subset__iff,axiom,
! [C2: set_Product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) )
= ( ( ord_le7336532860387713383od_v_v @ C2 @ A4 )
& ( ord_le7336532860387713383od_v_v @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_236_Int__subset__iff,axiom,
! [C2: set_v,A4: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A4 @ B ) )
= ( ( ord_less_eq_set_v @ C2 @ A4 )
& ( ord_less_eq_set_v @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_237_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_238_Int__insert__left__if0,axiom,
! [A: v,C2: set_v,B: set_v] :
( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B ) @ C2 )
= ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_239_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_240_Int__insert__left__if1,axiom,
! [A: v,C2: set_v,B: set_v] :
( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B ) @ C2 )
= ( insert_v2 @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_241_insert__inter__insert,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A4 ) @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) ) ) ).
% insert_inter_insert
thf(fact_242_insert__inter__insert,axiom,
! [A: v,A4: set_v,B: set_v] :
( ( inf_inf_set_v @ ( insert_v2 @ A @ A4 ) @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A4 @ B ) ) ) ).
% insert_inter_insert
thf(fact_243_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A4 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_244_Int__insert__right__if0,axiom,
! [A: v,A4: set_v,B: set_v] :
( ~ ( member_v @ A @ A4 )
=> ( ( inf_inf_set_v @ A4 @ ( insert_v2 @ A @ B ) )
= ( inf_inf_set_v @ A4 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_245_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_246_Int__insert__right__if1,axiom,
! [A: v,A4: set_v,B: set_v] :
( ( member_v @ A @ A4 )
=> ( ( inf_inf_set_v @ A4 @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A4 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_247_insert__Diff1,axiom,
! [X: v,B: set_v,A4: set_v] :
( ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A4 ) @ B )
= ( minus_minus_set_v @ A4 @ B ) ) ) ).
% insert_Diff1
thf(fact_248_insert__Diff1,axiom,
! [X: product_prod_v_v,B: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A4 ) @ B )
= ( minus_4183494784930505774od_v_v @ A4 @ B ) ) ) ).
% insert_Diff1
thf(fact_249_Diff__insert0,axiom,
! [X: v,A4: set_v,B: set_v] :
( ~ ( member_v @ X @ A4 )
=> ( ( minus_minus_set_v @ A4 @ ( insert_v2 @ X @ B ) )
= ( minus_minus_set_v @ A4 @ B ) ) ) ).
% Diff_insert0
thf(fact_250_Diff__insert0,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( minus_4183494784930505774od_v_v @ A4 @ B ) ) ) ).
% Diff_insert0
thf(fact_251_singleton__insert__inj__eq,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A4 ) )
= ( ( A = B2 )
& ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_252_singleton__insert__inj__eq,axiom,
! [B2: v,A: v,A4: set_v] :
( ( ( insert_v2 @ B2 @ bot_bot_set_v )
= ( insert_v2 @ A @ A4 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_v @ A4 @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_253_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A4 )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B2 )
& ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_254_singleton__insert__inj__eq_H,axiom,
! [A: v,A4: set_v,B2: v] :
( ( ( insert_v2 @ A @ A4 )
= ( insert_v2 @ B2 @ bot_bot_set_v ) )
= ( ( A = B2 )
& ( ord_less_eq_set_v @ A4 @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_255_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A4 ) @ B )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ A4 @ B )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_256_insert__disjoint_I1_J,axiom,
! [A: v,A4: set_v,B: set_v] :
( ( ( inf_inf_set_v @ ( insert_v2 @ A @ A4 ) @ B )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B )
& ( ( inf_inf_set_v @ A4 @ B )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_257_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A4 ) @ B ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A4 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_258_insert__disjoint_I2_J,axiom,
! [A: v,A4: set_v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v2 @ A @ A4 ) @ B ) )
= ( ~ ( member_v @ A @ B )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A4 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_259_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ A4 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ B @ A4 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_260_disjoint__insert_I1_J,axiom,
! [B: set_v,A: v,A4: set_v] :
( ( ( inf_inf_set_v @ B @ ( insert_v2 @ A @ A4 ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B )
& ( ( inf_inf_set_v @ B @ A4 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_261_disjoint__insert_I2_J,axiom,
! [A4: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B2 @ A4 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A4 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_262_disjoint__insert_I2_J,axiom,
! [A4: set_v,B2: v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A4 @ ( insert_v2 @ B2 @ B ) ) )
= ( ~ ( member_v @ B2 @ A4 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A4 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_263_insert__Diff__single,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_264_insert__Diff__single,axiom,
! [A: v,A4: set_v] :
( ( insert_v2 @ A @ ( minus_minus_set_v @ A4 @ ( insert_v2 @ A @ bot_bot_set_v ) ) )
= ( insert_v2 @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_265_Diff__disjoint,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ B @ A4 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_266_Diff__disjoint,axiom,
! [A4: set_v,B: set_v] :
( ( inf_inf_set_v @ A4 @ ( minus_minus_set_v @ B @ A4 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_267_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_268_IntE,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A4 )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% IntE
thf(fact_269_IntE,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A4 @ B ) )
=> ~ ( ( member_v @ C @ A4 )
=> ~ ( member_v @ C @ B ) ) ) ).
% IntE
thf(fact_270_IntD1,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A4 ) ) ).
% IntD1
thf(fact_271_IntD1,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A4 @ B ) )
=> ( member_v @ C @ A4 ) ) ).
% IntD1
thf(fact_272_IntD2,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ).
% IntD2
thf(fact_273_IntD2,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A4 @ B ) )
=> ( member_v @ C @ B ) ) ).
% IntD2
thf(fact_274_insertE,axiom,
! [A: v,B2: v,A4: set_v] :
( ( member_v @ A @ ( insert_v2 @ B2 @ A4 ) )
=> ( ( A != B2 )
=> ( member_v @ A @ A4 ) ) ) ).
% insertE
thf(fact_275_insertE,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ A4 ) )
=> ( ( A != B2 )
=> ( member7453568604450474000od_v_v @ A @ A4 ) ) ) ).
% insertE
thf(fact_276_insertI1,axiom,
! [A: v,B: set_v] : ( member_v @ A @ ( insert_v2 @ A @ B ) ) ).
% insertI1
thf(fact_277_insertI1,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).
% insertI1
thf(fact_278_insertI2,axiom,
! [A: v,B: set_v,B2: v] :
( ( member_v @ A @ B )
=> ( member_v @ A @ ( insert_v2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_279_insertI2,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_280_Int__assoc,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A4 @ B ) @ C2 )
= ( inf_inf_set_v @ A4 @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_281_Int__absorb,axiom,
! [A4: set_v] :
( ( inf_inf_set_v @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_282_Set_Oset__insert,axiom,
! [X: v,A4: set_v] :
( ( member_v @ X @ A4 )
=> ~ ! [B7: set_v] :
( ( A4
= ( insert_v2 @ X @ B7 ) )
=> ( member_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_283_Set_Oset__insert,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A4 )
=> ~ ! [B7: set_Product_prod_v_v] :
( ( A4
= ( insert1338601472111419319od_v_v @ X @ B7 ) )
=> ( member7453568604450474000od_v_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_284_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A8: set_v,B6: set_v] : ( inf_inf_set_v @ B6 @ A8 ) ) ) ).
% Int_commute
thf(fact_285_insert__ident,axiom,
! [X: v,A4: set_v,B: set_v] :
( ~ ( member_v @ X @ A4 )
=> ( ~ ( member_v @ X @ B )
=> ( ( ( insert_v2 @ X @ A4 )
= ( insert_v2 @ X @ B ) )
= ( A4 = B ) ) ) ) ).
% insert_ident
thf(fact_286_insert__ident,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( ( insert1338601472111419319od_v_v @ X @ A4 )
= ( insert1338601472111419319od_v_v @ X @ B ) )
= ( A4 = B ) ) ) ) ).
% insert_ident
thf(fact_287_insert__absorb,axiom,
! [A: v,A4: set_v] :
( ( member_v @ A @ A4 )
=> ( ( insert_v2 @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_288_insert__absorb,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ( insert1338601472111419319od_v_v @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_289_insert__eq__iff,axiom,
! [A: v,A4: set_v,B2: v,B: set_v] :
( ~ ( member_v @ A @ A4 )
=> ( ~ ( member_v @ B2 @ B )
=> ( ( ( insert_v2 @ A @ A4 )
= ( insert_v2 @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A4 = B ) )
& ( ( A != B2 )
=> ? [C3: set_v] :
( ( A4
= ( insert_v2 @ B2 @ C3 ) )
& ~ ( member_v @ B2 @ C3 )
& ( B
= ( insert_v2 @ A @ C3 ) )
& ~ ( member_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_290_insert__eq__iff,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ~ ( member7453568604450474000od_v_v @ B2 @ B )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A4 )
= ( insert1338601472111419319od_v_v @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A4 = B ) )
& ( ( A != B2 )
=> ? [C3: set_Product_prod_v_v] :
( ( A4
= ( insert1338601472111419319od_v_v @ B2 @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B2 @ C3 )
& ( B
= ( insert1338601472111419319od_v_v @ A @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_291_insert__commute,axiom,
! [X: v,Y2: v,A4: set_v] :
( ( insert_v2 @ X @ ( insert_v2 @ Y2 @ A4 ) )
= ( insert_v2 @ Y2 @ ( insert_v2 @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_292_insert__commute,axiom,
! [X: product_prod_v_v,Y2: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ Y2 @ A4 ) )
= ( insert1338601472111419319od_v_v @ Y2 @ ( insert1338601472111419319od_v_v @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_293_Int__insert__left,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_294_Int__insert__left,axiom,
! [A: v,C2: set_v,B: set_v] :
( ( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B ) @ C2 )
= ( insert_v2 @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) )
& ( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B ) @ C2 )
= ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_295_Int__left__absorb,axiom,
! [A4: set_v,B: set_v] :
( ( inf_inf_set_v @ A4 @ ( inf_inf_set_v @ A4 @ B ) )
= ( inf_inf_set_v @ A4 @ B ) ) ).
% Int_left_absorb
thf(fact_296_Int__insert__right,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A4 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_297_Int__insert__right,axiom,
! [A: v,A4: set_v,B: set_v] :
( ( ( member_v @ A @ A4 )
=> ( ( inf_inf_set_v @ A4 @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A4 @ B ) ) ) )
& ( ~ ( member_v @ A @ A4 )
=> ( ( inf_inf_set_v @ A4 @ ( insert_v2 @ A @ B ) )
= ( inf_inf_set_v @ A4 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_298_Int__left__commute,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ A4 @ ( inf_inf_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A4 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_299_mk__disjoint__insert,axiom,
! [A: v,A4: set_v] :
( ( member_v @ A @ A4 )
=> ? [B7: set_v] :
( ( A4
= ( insert_v2 @ A @ B7 ) )
& ~ ( member_v @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_300_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A4 )
=> ? [B7: set_Product_prod_v_v] :
( ( A4
= ( insert1338601472111419319od_v_v @ A @ B7 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_301_inf__left__commute,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ Y2 @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_302_inf_Oleft__commute,axiom,
! [B2: set_v,A: set_v,C: set_v] :
( ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A @ C ) )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_303_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_v,K: set_v,B2: set_v,A: set_v] :
( ( B
= ( inf_inf_set_v @ K @ B2 ) )
=> ( ( inf_inf_set_v @ A @ B )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_304_boolean__algebra__cancel_Oinf1,axiom,
! [A4: set_v,K: set_v,A: set_v,B2: set_v] :
( ( A4
= ( inf_inf_set_v @ K @ A ) )
=> ( ( inf_inf_set_v @ A4 @ B2 )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_305_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_306_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A6: set_v,B4: set_v] : ( inf_inf_set_v @ B4 @ A6 ) ) ) ).
% inf.commute
thf(fact_307_inf__assoc,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_308_inf_Oassoc,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_309_inf__sup__aci_I1_J,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_310_inf__sup__aci_I2_J,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_311_inf__sup__aci_I3_J,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ Y2 @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_312_inf__sup__aci_I4_J,axiom,
! [X: set_v,Y2: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y2 ) )
= ( inf_inf_set_v @ X @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_313_inf_OcoboundedI2,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_314_inf_OcoboundedI2,axiom,
! [B2: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_315_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_316_inf_OcoboundedI1,axiom,
! [A: set_v,C: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_317_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A6 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_318_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A6: set_v] :
( ( inf_inf_set_v @ A6 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_319_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A6 @ B4 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_320_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B4: set_v] :
( ( inf_inf_set_v @ A6 @ B4 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_321_inf_Ocobounded2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_322_inf_Ocobounded2,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_323_inf_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_324_inf_Ocobounded1,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_325_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( A6
= ( inf_in6271465464967711157od_v_v @ A6 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_326_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B4: set_v] :
( A6
= ( inf_inf_set_v @ A6 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_327_inf__greatest,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_328_inf__greatest,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_less_eq_set_v @ X @ Z )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_329_inf_OboundedI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_330_inf_OboundedI,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_331_inf_OboundedE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_332_inf_OboundedE,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_v @ A @ B2 )
=> ~ ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_333_inf__absorb2,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_334_inf__absorb2,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( inf_inf_set_v @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_335_inf__absorb1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_336_inf__absorb1,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( inf_inf_set_v @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_337_inf_Oabsorb2,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_338_inf_Oabsorb2,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( inf_inf_set_v @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_339_inf_Oabsorb1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_340_inf_Oabsorb1,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( inf_inf_set_v @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_341_le__iff__inf,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_342_le__iff__inf,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( inf_inf_set_v @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_343_inf__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y ) @ X3 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y ) @ Y )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Z3 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_344_inf__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y2: set_v] :
( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y ) @ X3 )
=> ( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y ) @ Y )
=> ( ! [X3: set_v,Y: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ( ord_less_eq_set_v @ X3 @ Z3 )
=> ( ord_less_eq_set_v @ X3 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_inf_set_v @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_345_inf_OorderI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A
= ( inf_in6271465464967711157od_v_v @ A @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B2 ) ) ).
% inf.orderI
thf(fact_346_inf_OorderI,axiom,
! [A: set_v,B2: set_v] :
( ( A
= ( inf_inf_set_v @ A @ B2 ) )
=> ( ord_less_eq_set_v @ A @ B2 ) ) ).
% inf.orderI
thf(fact_347_inf_OorderE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( A
= ( inf_in6271465464967711157od_v_v @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_348_inf_OorderE,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( A
= ( inf_inf_set_v @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_349_le__infI2,axiom,
! [B2: set_Product_prod_v_v,X: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_350_le__infI2,axiom,
! [B2: set_v,X: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_351_le__infI1,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_352_le__infI1,axiom,
! [A: set_v,X: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_353_inf__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_354_inf__mono,axiom,
! [A: set_v,C: set_v,B2: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ ( inf_inf_set_v @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_355_le__infI,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_356_le__infI,axiom,
! [X: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ( ord_less_eq_set_v @ X @ B2 )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_357_le__infE,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ X @ B2 ) ) ) ).
% le_infE
thf(fact_358_le__infE,axiom,
! [X: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_v @ X @ A )
=> ~ ( ord_less_eq_set_v @ X @ B2 ) ) ) ).
% le_infE
thf(fact_359_inf__le2,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_360_inf__le2,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_361_inf__le1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_362_inf__le1,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_363_inf__sup__ord_I1_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_364_inf__sup__ord_I1_J,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_365_inf__sup__ord_I2_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_366_inf__sup__ord_I2_J,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_367_insert__subsetI,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,X5: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ X5 @ A4 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_368_insert__subsetI,axiom,
! [X: v,A4: set_v,X5: set_v] :
( ( member_v @ X @ A4 )
=> ( ( ord_less_eq_set_v @ X5 @ A4 )
=> ( ord_less_eq_set_v @ ( insert_v2 @ X @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_369_bot__empty__eq,axiom,
( bot_bo8461541820394803818_v_v_o
= ( ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ).
% bot_empty_eq
thf(fact_370_bot__empty__eq,axiom,
( bot_bot_v_o
= ( ^ [X2: v] : ( member_v @ X2 @ bot_bot_set_v ) ) ) ).
% bot_empty_eq
thf(fact_371_bot__set__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v @ bot_bot_set_v_o ) ) ).
% bot_set_def
thf(fact_372_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_373_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_374_singleton__inject,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_375_singleton__inject,axiom,
! [A: v,B2: v] :
( ( ( insert_v2 @ A @ bot_bot_set_v )
= ( insert_v2 @ B2 @ bot_bot_set_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_376_insert__not__empty,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ A4 )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_377_insert__not__empty,axiom,
! [A: v,A4: set_v] :
( ( insert_v2 @ A @ A4 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_378_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,C: product_prod_v_v,D2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D2 @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_379_doubleton__eq__iff,axiom,
! [A: v,B2: v,C: v,D2: v] :
( ( ( insert_v2 @ A @ ( insert_v2 @ B2 @ bot_bot_set_v ) )
= ( insert_v2 @ C @ ( insert_v2 @ D2 @ bot_bot_set_v ) ) )
= ( ( ( A = C )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_380_singleton__iff,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_381_singleton__iff,axiom,
! [B2: v,A: v] :
( ( member_v @ B2 @ ( insert_v2 @ A @ bot_bot_set_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_382_singletonD,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_383_singletonD,axiom,
! [B2: v,A: v] :
( ( member_v @ B2 @ ( insert_v2 @ A @ bot_bot_set_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_384_disjoint__iff__not__equal,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A4 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_385_disjoint__iff__not__equal,axiom,
! [A4: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A4 @ B )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A4 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_386_Int__empty__right,axiom,
! [A4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_387_Int__empty__right,axiom,
! [A4: set_v] :
( ( inf_inf_set_v @ A4 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_388_Int__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_389_Int__empty__left,axiom,
! [B: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_390_disjoint__iff,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A4 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
=> ~ ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_391_disjoint__iff,axiom,
! [A4: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A4 @ B )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A4 )
=> ~ ( member_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_392_Int__emptyI,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B ) )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ B )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_393_Int__emptyI,axiom,
! [A4: set_v,B: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A4 )
=> ~ ( member_v @ X3 @ B ) )
=> ( ( inf_inf_set_v @ A4 @ B )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_394_subset__insertI2,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_395_subset__insertI2,axiom,
! [A4: set_v,B: set_v,B2: v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ord_less_eq_set_v @ A4 @ ( insert_v2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_396_subset__insertI,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).
% subset_insertI
thf(fact_397_subset__insertI,axiom,
! [B: set_v,A: v] : ( ord_less_eq_set_v @ B @ ( insert_v2 @ A @ B ) ) ).
% subset_insertI
thf(fact_398_subset__insert,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ord_le7336532860387713383od_v_v @ A4 @ B ) ) ) ).
% subset_insert
thf(fact_399_subset__insert,axiom,
! [X: v,A4: set_v,B: set_v] :
( ~ ( member_v @ X @ A4 )
=> ( ( ord_less_eq_set_v @ A4 @ ( insert_v2 @ X @ B ) )
= ( ord_less_eq_set_v @ A4 @ B ) ) ) ).
% subset_insert
thf(fact_400_insert__mono,axiom,
! [C2: set_Product_prod_v_v,D: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C2 ) @ ( insert1338601472111419319od_v_v @ A @ D ) ) ) ).
% insert_mono
thf(fact_401_insert__mono,axiom,
! [C2: set_v,D: set_v,A: v] :
( ( ord_less_eq_set_v @ C2 @ D )
=> ( ord_less_eq_set_v @ ( insert_v2 @ A @ C2 ) @ ( insert_v2 @ A @ D ) ) ) ).
% insert_mono
thf(fact_402_Int__Collect__mono,axiom,
! [A4: set_set_v,B: set_set_v,P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A4 @ B )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A4 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( inf_inf_set_set_v @ A4 @ ( collect_set_v @ P ) ) @ ( inf_inf_set_set_v @ B @ ( collect_set_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_403_Int__Collect__mono,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_404_Int__Collect__mono,axiom,
! [A4: set_v,B: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ! [X3: v] :
( ( member_v @ X3 @ A4 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_405_Int__greatest,axiom,
! [C2: set_Product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) ) ) ) ).
% Int_greatest
thf(fact_406_Int__greatest,axiom,
! [C2: set_v,A4: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ A4 )
=> ( ( ord_less_eq_set_v @ C2 @ B )
=> ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A4 @ B ) ) ) ) ).
% Int_greatest
thf(fact_407_Int__absorb2,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ B )
= A4 ) ) ).
% Int_absorb2
thf(fact_408_Int__absorb2,axiom,
! [A4: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ( inf_inf_set_v @ A4 @ B )
= A4 ) ) ).
% Int_absorb2
thf(fact_409_Int__absorb1,axiom,
! [B: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_410_Int__absorb1,axiom,
! [B: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B @ A4 )
=> ( ( inf_inf_set_v @ A4 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_411_Int__lower2,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) @ B ) ).
% Int_lower2
thf(fact_412_Int__lower2,axiom,
! [A4: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B ) @ B ) ).
% Int_lower2
thf(fact_413_Int__lower1,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) @ A4 ) ).
% Int_lower1
thf(fact_414_Int__lower1,axiom,
! [A4: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B ) @ A4 ) ).
% Int_lower1
thf(fact_415_Int__mono,axiom,
! [A4: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_416_Int__mono,axiom,
! [A4: set_v,C2: set_v,B: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A4 @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B ) @ ( inf_inf_set_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_417_insert__Diff__if,axiom,
! [X: v,B: set_v,A4: set_v] :
( ( ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A4 ) @ B )
= ( minus_minus_set_v @ A4 @ B ) ) )
& ( ~ ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A4 ) @ B )
= ( insert_v2 @ X @ ( minus_minus_set_v @ A4 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_418_insert__Diff__if,axiom,
! [X: product_prod_v_v,B: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A4 ) @ B )
= ( minus_4183494784930505774od_v_v @ A4 @ B ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A4 ) @ B )
= ( insert1338601472111419319od_v_v @ X @ ( minus_4183494784930505774od_v_v @ A4 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_419_Diff__Int__distrib2,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A4 @ B ) @ C2 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A4 @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_420_Diff__Int__distrib,axiom,
! [C2: set_v,A4: set_v,B: set_v] :
( ( inf_inf_set_v @ C2 @ ( minus_minus_set_v @ A4 @ B ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C2 @ A4 ) @ ( inf_inf_set_v @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_421_Diff__Diff__Int,axiom,
! [A4: set_v,B: set_v] :
( ( minus_minus_set_v @ A4 @ ( minus_minus_set_v @ A4 @ B ) )
= ( inf_inf_set_v @ A4 @ B ) ) ).
% Diff_Diff_Int
thf(fact_422_Diff__Int2,axiom,
! [A4: set_v,C2: set_v,B: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A4 @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A4 @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_423_Int__Diff,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A4 @ B ) @ C2 )
= ( inf_inf_set_v @ A4 @ ( minus_minus_set_v @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_424_subset__singletonD,axiom,
! [A4: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
=> ( ( A4 = bot_bo723834152578015283od_v_v )
| ( A4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_425_subset__singletonD,axiom,
! [A4: set_v,X: v] :
( ( ord_less_eq_set_v @ A4 @ ( insert_v2 @ X @ bot_bot_set_v ) )
=> ( ( A4 = bot_bot_set_v )
| ( A4
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_426_subset__singleton__iff,axiom,
! [X5: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X5 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( ( X5 = bot_bo723834152578015283od_v_v )
| ( X5
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_427_subset__singleton__iff,axiom,
! [X5: set_v,A: v] :
( ( ord_less_eq_set_v @ X5 @ ( insert_v2 @ A @ bot_bot_set_v ) )
= ( ( X5 = bot_bot_set_v )
| ( X5
= ( insert_v2 @ A @ bot_bot_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_428_Diff__insert__absorb,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A4 ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_429_Diff__insert__absorb,axiom,
! [X: v,A4: set_v] :
( ~ ( member_v @ X @ A4 )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A4 ) @ ( insert_v2 @ X @ bot_bot_set_v ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_430_Diff__insert2,axiom,
! [A4: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_431_Diff__insert2,axiom,
! [A4: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A4 @ ( insert_v2 @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A4 @ ( insert_v2 @ A @ bot_bot_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_432_insert__Diff,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_433_insert__Diff,axiom,
! [A: v,A4: set_v] :
( ( member_v @ A @ A4 )
=> ( ( insert_v2 @ A @ ( minus_minus_set_v @ A4 @ ( insert_v2 @ A @ bot_bot_set_v ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_434_Diff__insert,axiom,
! [A4: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ B ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_435_Diff__insert,axiom,
! [A4: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A4 @ ( insert_v2 @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A4 @ B ) @ ( insert_v2 @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_436_Int__Diff__disjoint,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) @ ( minus_4183494784930505774od_v_v @ A4 @ B ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_437_Int__Diff__disjoint,axiom,
! [A4: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A4 @ B ) @ ( minus_minus_set_v @ A4 @ B ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_438_Diff__triv,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A4 @ B )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A4 @ B )
= A4 ) ) ).
% Diff_triv
thf(fact_439_Diff__triv,axiom,
! [A4: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A4 @ B )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A4 @ B )
= A4 ) ) ).
% Diff_triv
thf(fact_440_subset__Diff__insert,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ B @ ( insert1338601472111419319od_v_v @ X @ C2 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ B @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_441_subset__Diff__insert,axiom,
! [A4: set_v,B: set_v,X: v,C2: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( minus_minus_set_v @ B @ ( insert_v2 @ X @ C2 ) ) )
= ( ( ord_less_eq_set_v @ A4 @ ( minus_minus_set_v @ B @ C2 ) )
& ~ ( member_v @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_442_subset__insert__iff,axiom,
! [A4: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_443_subset__insert__iff,axiom,
! [A4: set_v,X: v,B: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( insert_v2 @ X @ B ) )
= ( ( ( member_v @ X @ A4 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A4 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B ) )
& ( ~ ( member_v @ X @ A4 )
=> ( ord_less_eq_set_v @ A4 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_444_Diff__single__insert,axiom,
! [A4: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_445_Diff__single__insert,axiom,
! [A4: set_v,X: v,B: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A4 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B )
=> ( ord_less_eq_set_v @ A4 @ ( insert_v2 @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_446_ssubst__Pair__rhs,axiom,
! [R: v,S2: v,R3: set_Product_prod_v_v,S5: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R @ S2 ) @ R3 )
=> ( ( S5 = S2 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R @ S5 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_447_the__elem__eq,axiom,
! [X: product_prod_v_v] :
( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= X ) ).
% the_elem_eq
thf(fact_448_the__elem__eq,axiom,
! [X: v] :
( ( the_elem_v @ ( insert_v2 @ X @ bot_bot_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_449_is__singletonI,axiom,
! [X: product_prod_v_v] : ( is_sin9198872032823709915od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ).
% is_singletonI
thf(fact_450_is__singletonI,axiom,
! [X: v] : ( is_singleton_v @ ( insert_v2 @ X @ bot_bot_set_v ) ) ).
% is_singletonI
thf(fact_451_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_452_unite__wf__env,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ successors @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ).
% unite_wf_env
thf(fact_453_remove__def,axiom,
( remove5001965847480235980od_v_v
= ( ^ [X2: product_prod_v_v,A8: set_Product_prod_v_v] : ( minus_4183494784930505774od_v_v @ A8 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% remove_def
thf(fact_454_remove__def,axiom,
( remove_v
= ( ^ [X2: v,A8: set_v] : ( minus_minus_set_v @ A8 @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ).
% remove_def
thf(fact_455_graph_Oreachable__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ W )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( Successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_456_Set_Ois__empty__def,axiom,
( is_emp8964507351669718201od_v_v
= ( ^ [A8: set_Product_prod_v_v] : ( A8 = bot_bo723834152578015283od_v_v ) ) ) ).
% Set.is_empty_def
thf(fact_457_Set_Ois__empty__def,axiom,
( is_empty_v
= ( ^ [A8: set_v] : ( A8 = bot_bot_set_v ) ) ) ).
% Set.is_empty_def
thf(fact_458_is__singleton__def,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A8: set_Product_prod_v_v] :
? [X2: product_prod_v_v] :
( A8
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_def
thf(fact_459_is__singleton__def,axiom,
( is_singleton_v
= ( ^ [A8: set_v] :
? [X2: v] :
( A8
= ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ).
% is_singleton_def
thf(fact_460_is__singletonE,axiom,
! [A4: set_Product_prod_v_v] :
( ( is_sin9198872032823709915od_v_v @ A4 )
=> ~ ! [X3: product_prod_v_v] :
( A4
!= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ).
% is_singletonE
thf(fact_461_is__singletonE,axiom,
! [A4: set_v] :
( ( is_singleton_v @ A4 )
=> ~ ! [X3: v] :
( A4
!= ( insert_v2 @ X3 @ bot_bot_set_v ) ) ) ).
% is_singletonE
thf(fact_462_member__remove,axiom,
! [X: v,Y2: v,A4: set_v] :
( ( member_v @ X @ ( remove_v @ Y2 @ A4 ) )
= ( ( member_v @ X @ A4 )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_463_member__remove,axiom,
! [X: product_prod_v_v,Y2: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( remove5001965847480235980od_v_v @ Y2 @ A4 ) )
= ( ( member7453568604450474000od_v_v @ X @ A4 )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_464_unite__sub__env,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ).
% unite_sub_env
thf(fact_465_graph_Odfss_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: produc5741669702376414499t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ~ ! [V2: v,E7: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V2 @ E7 ) ) ) ).
% graph.dfss.cases
thf(fact_466_graph_Ointro,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_467_graph_Ointro,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( finite_finite_v @ Vertices )
=> ( ! [X3: v] :
( ( member_v @ X3 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bloemen_graph_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_468_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bl8307124943676871238od_v_v
= ( ^ [Vertices2: set_Product_prod_v_v,Successors2: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices2 )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ Vertices2 )
=> ( ord_le7336532860387713383od_v_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_469_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bloemen_graph_v
= ( ^ [Vertices2: set_v,Successors2: v > set_v] :
( ( finite_finite_v @ Vertices2 )
& ! [X2: v] :
( ( member_v @ X2 @ Vertices2 )
=> ( ord_less_eq_set_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_470_graph_Ovfin,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( finite_finite_v @ Vertices ) ) ).
% graph.vfin
thf(fact_471_graph_Ounite__sub__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7963838319573962697t_unit @ E @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_472_graph_Ounite__sub__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_473_graph_Osclosed,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_474_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X4: v] :
( ( member_v @ X4 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_475_graph_Oreachable__edge,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_476_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_477_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_478_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ( member_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_479_graph_Oreachable_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_480_graph_Oreachable_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( member_v @ Y @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_481_graph_Oreachable_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_482_graph_Oreachable_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( member_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_483_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_trans
thf(fact_484_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v,P: product_prod_v_v > product_prod_v_v > $o] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
=> ( ! [X3: product_prod_v_v] : ( P @ X3 @ X3 )
=> ( ! [X3: product_prod_v_v,Y: product_prod_v_v,Z3: product_prod_v_v] :
( ( P @ X3 @ Y )
=> ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_485_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y: v,Z3: v] :
( ( P @ X3 @ Y )
=> ( ( member_v @ Z3 @ ( Successors @ Y ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_486_graph_Oreachable__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ X ) ) ).
% graph.reachable_refl
thf(fact_487_graph_Oreachable__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_488_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_489_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y2 @ Z @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E4 ) ) ) ) ).
% graph.ra_trans
thf(fact_490_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ X @ E4 ) ) ).
% graph.ra_refl
thf(fact_491_graph_Ounite__wf__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7798947040364291444t_unit @ Successors @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_492_graph_Ounite__wf__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ Successors @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_493_graph_Osucc__re,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_494_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_495_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y )
=> ~ ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_496_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y )
=> ~ ( member_v @ A2 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_497_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Y3 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_498_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y3 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_499_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ X ) ) ).
% graph.re_refl
thf(fact_500_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_501_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y2 )
=> ( ( member_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_502_graph_Osub__env__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ) ).
% graph.sub_env_trans
thf(fact_503_graph_Oedge__ra,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y2 ) @ E4 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y2 @ E4 ) ) ) ) ).
% graph.edge_ra
thf(fact_504_graph_Oedge__ra,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 ) ) ) ) ).
% graph.edge_ra
thf(fact_505_graph_Ora__cases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y2 @ E4 )
=> ( ( X = Y2 )
| ? [Z3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ X ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Z3 ) @ E4 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ Z3 @ Y2 @ E4 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_506_graph_Ora__cases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( ( X = Y2 )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( Successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ Z3 @ Y2 @ E4 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_507_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A3: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A3 )
=> ( ( A2 != A1 )
=> ~ ! [Y: product_prod_v_v] :
( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ Y @ A3 )
=> ( ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y ) )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ A2 ) @ A3 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_508_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A3: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A3 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ Y @ A3 )
=> ( ( member_v @ A2 @ ( Successors @ Y ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ A2 ) @ A3 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_509_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A3: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A3 )
= ( ? [X2: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A3 = E6 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A3 = E6 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y3 @ E6 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_510_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A3: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A3 )
= ( ? [X2: v,E6: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A3 = E6 ) )
| ? [X2: v,Y3: v,E6: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A3 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y3 @ E6 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_511_graph_Ora__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y2 @ E4 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y2 @ Z ) @ E4 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Z @ E4 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_512_graph_Ora__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( ( member_v @ Z @ ( Successors @ Y2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E4 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_513_graph_Ora__mono,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E5 ) ) ) ) ).
% graph.ra_mono
thf(fact_514_graph_Ora__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).
% graph.ra_reachable
thf(fact_515_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ).
% graph.S_reflexive
thf(fact_516_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).
% graph.re_reachable
thf(fact_517_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y2 ) ) ) ).
% graph.reachable_re
thf(fact_518_graph_Oscc__partition,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,S4: set_Product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S4 )
=> ( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ S @ S4 ) )
=> ( S = S4 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_519_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,S4: set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S4 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S4 ) )
=> ( S = S4 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_520_is__singleton__the__elem,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A8: set_Product_prod_v_v] :
( A8
= ( insert1338601472111419319od_v_v @ ( the_el5392834299063928540od_v_v @ A8 ) @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_521_is__singleton__the__elem,axiom,
( is_singleton_v
= ( ^ [A8: set_v] :
( A8
= ( insert_v2 @ ( the_elem_v @ A8 ) @ bot_bot_set_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_522_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y3 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_523_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X )
=> ( member7453568604450474000od_v_v @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_524_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X )
=> ( member_v @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_525_is__singletonI_H,axiom,
! [A4: set_Product_prod_v_v] :
( ( A4 != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ( ( member7453568604450474000od_v_v @ Y @ A4 )
=> ( X3 = Y ) ) )
=> ( is_sin9198872032823709915od_v_v @ A4 ) ) ) ).
% is_singletonI'
thf(fact_526_is__singletonI_H,axiom,
! [A4: set_v] :
( ( A4 != bot_bot_set_v )
=> ( ! [X3: v,Y: v] :
( ( member_v @ X3 @ A4 )
=> ( ( member_v @ Y @ A4 )
=> ( X3 = Y ) ) )
=> ( is_singleton_v @ A4 ) ) ) ).
% is_singletonI'
thf(fact_527_graph_Ora__empty,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).
% graph.ra_empty
thf(fact_528_graph_Osubscc__add,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y2 @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_529_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v2 @ Y2 @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_530_graph_Opre__dfss__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ W @ E ) ) ) ) ) ).
% graph.pre_dfss_pre_dfs
thf(fact_531_graph_Oinit__env__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ) ).
% graph.init_env_pre_dfs
thf(fact_532_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
= ( ( S != bot_bo723834152578015283od_v_v )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
& ! [S3: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S @ S3 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S3 ) )
=> ( S3 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_533_graph_Ois__scc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
& ! [S3: set_v] :
( ( ( ord_less_eq_set_v @ S @ S3 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 ) )
=> ( S3 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_534_finite__Diff__insert,axiom,
! [A4: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B ) ) )
= ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_535_finite__Diff__insert,axiom,
! [A4: set_v,A: v,B: set_v] :
( ( finite_finite_v @ ( minus_minus_set_v @ A4 @ ( insert_v2 @ A @ B ) ) )
= ( finite_finite_v @ ( minus_minus_set_v @ A4 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_536_avoiding__explored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,X: v,Y2: v,E4: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( ~ ( member_v @ Y2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% avoiding_explored
thf(fact_537_finite__Diff,axiom,
! [A4: set_v,B: set_v] :
( ( finite_finite_v @ A4 )
=> ( finite_finite_v @ ( minus_minus_set_v @ A4 @ B ) ) ) ).
% finite_Diff
thf(fact_538_finite__Diff2,axiom,
! [B: set_v,A4: set_v] :
( ( finite_finite_v @ B )
=> ( ( finite_finite_v @ ( minus_minus_set_v @ A4 @ B ) )
= ( finite_finite_v @ A4 ) ) ) ).
% finite_Diff2
thf(fact_539_finite__Int,axiom,
! [F2: set_v,G: set_v] :
( ( ( finite_finite_v @ F2 )
| ( finite_finite_v @ G ) )
=> ( finite_finite_v @ ( inf_inf_set_v @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_540_finite__remove__induct,axiom,
! [B: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A9: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A9 )
=> ( ( A9 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A9 @ B )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A9 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_541_finite__remove__induct,axiom,
! [B: set_v,P: set_v > $o] :
( ( finite_finite_v @ B )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A9: set_v] :
( ( finite_finite_v @ A9 )
=> ( ( A9 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A9 @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A9 )
=> ( P @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_542_sup_Oidem,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_543_sup_Oidem,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_544_sup__idem,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_545_sup__idem,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_546_sup_Oleft__idem,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B2 ) )
= ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_547_sup_Oleft__idem,axiom,
! [A: set_v,B2: set_v] :
( ( sup_sup_set_v @ A @ ( sup_sup_set_v @ A @ B2 ) )
= ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_548_sup__left__idem,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
= ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_549_sup__left__idem,axiom,
! [X: set_v,Y2: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) )
= ( sup_sup_set_v @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_550_sup_Oright__idem,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ B2 )
= ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_551_sup_Oright__idem,axiom,
! [A: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B2 ) @ B2 )
= ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_552_UnCI,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ A4 ) )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B ) ) ) ).
% UnCI
thf(fact_553_UnCI,axiom,
! [C: v,B: set_v,A4: set_v] :
( ( ~ ( member_v @ C @ B )
=> ( member_v @ C @ A4 ) )
=> ( member_v @ C @ ( sup_sup_set_v @ A4 @ B ) ) ) ).
% UnCI
thf(fact_554_Un__iff,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A4 )
| ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_555_Un__iff,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A4 @ B ) )
= ( ( member_v @ C @ A4 )
| ( member_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_556_sup_Obounded__iff,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A )
= ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
& ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_557_sup_Obounded__iff,axiom,
! [B2: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_v @ B2 @ A )
& ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_558_le__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ Z )
= ( ( ord_le7336532860387713383od_v_v @ X @ Z )
& ( ord_le7336532860387713383od_v_v @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_559_le__sup__iff,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ Z )
= ( ( ord_less_eq_set_v @ X @ Z )
& ( ord_less_eq_set_v @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_560_sup__bot_Oright__neutral,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ bot_bo723834152578015283od_v_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_561_sup__bot_Oright__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_562_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A @ B2 ) )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_563_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_v,B2: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A @ B2 ) )
= ( ( A = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_564_sup__bot_Oleft__neutral,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_565_sup__bot_Oleft__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_566_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_567_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_v,B2: set_v] :
( ( ( sup_sup_set_v @ A @ B2 )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_568_sup__eq__bot__iff,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
= bot_bo723834152578015283od_v_v )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_569_sup__eq__bot__iff,axiom,
! [X: set_v,Y2: set_v] :
( ( ( sup_sup_set_v @ X @ Y2 )
= bot_bot_set_v )
= ( ( X = bot_bot_set_v )
& ( Y2 = bot_bot_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_570_bot__eq__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y2 = bot_bo723834152578015283od_v_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_571_bot__eq__sup__iff,axiom,
! [X: set_v,Y2: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ X @ Y2 ) )
= ( ( X = bot_bot_set_v )
& ( Y2 = bot_bot_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_572_sup__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% sup_bot_right
thf(fact_573_sup__bot__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% sup_bot_right
thf(fact_574_sup__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ X )
= X ) ).
% sup_bot_left
thf(fact_575_sup__bot__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_576_finite__insert,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A @ A4 ) )
= ( finite3348123685078250256od_v_v @ A4 ) ) ).
% finite_insert
thf(fact_577_finite__insert,axiom,
! [A: v,A4: set_v] :
( ( finite_finite_v @ ( insert_v2 @ A @ A4 ) )
= ( finite_finite_v @ A4 ) ) ).
% finite_insert
thf(fact_578_sup__inf__absorb,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_579_sup__inf__absorb,axiom,
! [X: set_v,Y2: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_580_inf__sup__absorb,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_581_inf__sup__absorb,axiom,
! [X: set_v,Y2: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_582_Un__empty,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A4 @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A4 = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_583_Un__empty,axiom,
! [A4: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A4 @ B )
= bot_bot_set_v )
= ( ( A4 = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_584_finite__Un,axiom,
! [F2: set_Product_prod_v_v,G: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ F2 @ G ) )
= ( ( finite3348123685078250256od_v_v @ F2 )
& ( finite3348123685078250256od_v_v @ G ) ) ) ).
% finite_Un
thf(fact_585_finite__Un,axiom,
! [F2: set_v,G: set_v] :
( ( finite_finite_v @ ( sup_sup_set_v @ F2 @ G ) )
= ( ( finite_finite_v @ F2 )
& ( finite_finite_v @ G ) ) ) ).
% finite_Un
thf(fact_586_Un__subset__iff,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B ) @ C2 )
= ( ( ord_le7336532860387713383od_v_v @ A4 @ C2 )
& ( ord_le7336532860387713383od_v_v @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_587_Un__subset__iff,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A4 @ B ) @ C2 )
= ( ( ord_less_eq_set_v @ A4 @ C2 )
& ( ord_less_eq_set_v @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_588_Un__insert__left,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_589_Un__insert__left,axiom,
! [A: v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( insert_v2 @ A @ B ) @ C2 )
= ( insert_v2 @ A @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_590_Un__insert__right,axiom,
! [A4: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A4 @ B ) ) ) ).
% Un_insert_right
thf(fact_591_Un__insert__right,axiom,
! [A4: set_v,A: v,B: set_v] :
( ( sup_sup_set_v @ A4 @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ A @ ( sup_sup_set_v @ A4 @ B ) ) ) ).
% Un_insert_right
thf(fact_592_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T2 @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_593_Int__Un__eq_I4_J,axiom,
! [T2: set_v,S: set_v] :
( ( sup_sup_set_v @ T2 @ ( inf_inf_set_v @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_594_Int__Un__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_595_Int__Un__eq_I3_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ S @ ( inf_inf_set_v @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_596_Int__Un__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_597_Int__Un__eq_I2_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_598_Int__Un__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_599_Int__Un__eq_I1_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_600_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T2 @ ( sup_su414716646722978715od_v_v @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_601_Un__Int__eq_I4_J,axiom,
! [T2: set_v,S: set_v] :
( ( inf_inf_set_v @ T2 @ ( sup_sup_set_v @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_602_Un__Int__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S @ ( sup_su414716646722978715od_v_v @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_603_Un__Int__eq_I3_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ S @ ( sup_sup_set_v @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_604_Un__Int__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_605_Un__Int__eq_I2_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_606_Un__Int__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_607_Un__Int__eq_I1_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_608_Un__Diff__cancel2,axiom,
! [B: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B @ A4 ) @ A4 )
= ( sup_su414716646722978715od_v_v @ B @ A4 ) ) ).
% Un_Diff_cancel2
thf(fact_609_Un__Diff__cancel2,axiom,
! [B: set_v,A4: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ B @ A4 ) @ A4 )
= ( sup_sup_set_v @ B @ A4 ) ) ).
% Un_Diff_cancel2
thf(fact_610_Un__Diff__cancel,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ B @ A4 ) )
= ( sup_su414716646722978715od_v_v @ A4 @ B ) ) ).
% Un_Diff_cancel
thf(fact_611_Un__Diff__cancel,axiom,
! [A4: set_v,B: set_v] :
( ( sup_sup_set_v @ A4 @ ( minus_minus_set_v @ B @ A4 ) )
= ( sup_sup_set_v @ A4 @ B ) ) ).
% Un_Diff_cancel
thf(fact_612_ra__add__edge,axiom,
! [X: v,Y2: v,E4: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ V @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_613_finite__UnI,axiom,
! [F2: set_Product_prod_v_v,G: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( finite3348123685078250256od_v_v @ G )
=> ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_614_finite__UnI,axiom,
! [F2: set_v,G: set_v] :
( ( finite_finite_v @ F2 )
=> ( ( finite_finite_v @ G )
=> ( finite_finite_v @ ( sup_sup_set_v @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_615_Un__infinite,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_616_Un__infinite,axiom,
! [S: set_v,T2: set_v] :
( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( sup_sup_set_v @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_617_infinite__Un,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) ) )
= ( ~ ( finite3348123685078250256od_v_v @ S )
| ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).
% infinite_Un
thf(fact_618_infinite__Un,axiom,
! [S: set_v,T2: set_v] :
( ( ~ ( finite_finite_v @ ( sup_sup_set_v @ S @ T2 ) ) )
= ( ~ ( finite_finite_v @ S )
| ~ ( finite_finite_v @ T2 ) ) ) ).
% infinite_Un
thf(fact_619_UnE,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B ) )
=> ( ~ ( member7453568604450474000od_v_v @ C @ A4 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% UnE
thf(fact_620_UnE,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A4 @ B ) )
=> ( ~ ( member_v @ C @ A4 )
=> ( member_v @ C @ B ) ) ) ).
% UnE
thf(fact_621_UnI1,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A4 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B ) ) ) ).
% UnI1
thf(fact_622_UnI1,axiom,
! [C: v,A4: set_v,B: set_v] :
( ( member_v @ C @ A4 )
=> ( member_v @ C @ ( sup_sup_set_v @ A4 @ B ) ) ) ).
% UnI1
thf(fact_623_UnI2,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B ) ) ) ).
% UnI2
thf(fact_624_UnI2,axiom,
! [C: v,B: set_v,A4: set_v] :
( ( member_v @ C @ B )
=> ( member_v @ C @ ( sup_sup_set_v @ A4 @ B ) ) ) ).
% UnI2
thf(fact_625_bex__Un,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A4 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
& ( P @ X2 ) )
| ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_626_bex__Un,axiom,
! [A4: set_v,B: set_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( sup_sup_set_v @ A4 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: v] :
( ( member_v @ X2 @ A4 )
& ( P @ X2 ) )
| ? [X2: v] :
( ( member_v @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_627_ball__Un,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A4 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
=> ( P @ X2 ) )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_628_ball__Un,axiom,
! [A4: set_v,B: set_v,P: v > $o] :
( ( ! [X2: v] :
( ( member_v @ X2 @ ( sup_sup_set_v @ A4 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ A4 )
=> ( P @ X2 ) )
& ! [X2: v] :
( ( member_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_629_Un__assoc,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B ) @ C2 )
= ( sup_su414716646722978715od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_630_Un__assoc,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A4 @ B ) @ C2 )
= ( sup_sup_set_v @ A4 @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_631_Un__absorb,axiom,
! [A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_632_Un__absorb,axiom,
! [A4: set_v] :
( ( sup_sup_set_v @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_633_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B6 @ A8 ) ) ) ).
% Un_commute
thf(fact_634_Un__commute,axiom,
( sup_sup_set_v
= ( ^ [A8: set_v,B6: set_v] : ( sup_sup_set_v @ B6 @ A8 ) ) ) ).
% Un_commute
thf(fact_635_Un__left__absorb,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ A4 @ B ) )
= ( sup_su414716646722978715od_v_v @ A4 @ B ) ) ).
% Un_left_absorb
thf(fact_636_Un__left__absorb,axiom,
! [A4: set_v,B: set_v] :
( ( sup_sup_set_v @ A4 @ ( sup_sup_set_v @ A4 @ B ) )
= ( sup_sup_set_v @ A4 @ B ) ) ).
% Un_left_absorb
thf(fact_637_Un__left__commute,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A4 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_638_Un__left__commute,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ A4 @ ( sup_sup_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ B @ ( sup_sup_set_v @ A4 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_639_inf__sup__aci_I8_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
= ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_640_inf__sup__aci_I8_J,axiom,
! [X: set_v,Y2: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) )
= ( sup_sup_set_v @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_641_inf__sup__aci_I7_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_642_inf__sup__aci_I7_J,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ Y2 @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_643_inf__sup__aci_I6_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ Z )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_644_inf__sup__aci_I6_J,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ Z )
= ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_645_inf__sup__aci_I5_J,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_646_inf__sup__aci_I5_J,axiom,
( sup_sup_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_647_sup_Oassoc,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ C )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_648_sup_Oassoc,axiom,
! [A: set_v,B2: set_v,C: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B2 ) @ C )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_649_sup__assoc,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ Z )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_650_sup__assoc,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ Z )
= ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_651_sup_Ocommute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B4 @ A6 ) ) ) ).
% sup.commute
thf(fact_652_sup_Ocommute,axiom,
( sup_sup_set_v
= ( ^ [A6: set_v,B4: set_v] : ( sup_sup_set_v @ B4 @ A6 ) ) ) ).
% sup.commute
thf(fact_653_sup__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_654_sup__commute,axiom,
( sup_sup_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( sup_sup_set_v @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_655_boolean__algebra__cancel_Osup1,axiom,
! [A4: set_Product_prod_v_v,K: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A4
= ( sup_su414716646722978715od_v_v @ K @ A ) )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ B2 )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_656_boolean__algebra__cancel_Osup1,axiom,
! [A4: set_v,K: set_v,A: set_v,B2: set_v] :
( ( A4
= ( sup_sup_set_v @ K @ A ) )
=> ( ( sup_sup_set_v @ A4 @ B2 )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_657_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Product_prod_v_v,K: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( B
= ( sup_su414716646722978715od_v_v @ K @ B2 ) )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_658_boolean__algebra__cancel_Osup2,axiom,
! [B: set_v,K: set_v,B2: set_v,A: set_v] :
( ( B
= ( sup_sup_set_v @ K @ B2 ) )
=> ( ( sup_sup_set_v @ A @ B )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_659_sup_Oleft__commute,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A @ C ) )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_660_sup_Oleft__commute,axiom,
! [B2: set_v,A: set_v,C: set_v] :
( ( sup_sup_set_v @ B2 @ ( sup_sup_set_v @ A @ C ) )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_661_sup__left__commute,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_662_sup__left__commute,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ Y2 @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_663_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_664_sup_OcoboundedI2,axiom,
! [C: set_v,B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ C @ B2 )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_665_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_666_sup_OcoboundedI1,axiom,
! [C: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_667_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A6 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_668_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B4: set_v] :
( ( sup_sup_set_v @ A6 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_669_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A6 @ B4 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_670_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A6: set_v] :
( ( sup_sup_set_v @ A6 @ B4 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_671_sup_Ocobounded2,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_672_sup_Ocobounded2,axiom,
! [B2: set_v,A: set_v] : ( ord_less_eq_set_v @ B2 @ ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_673_sup_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_674_sup_Ocobounded1,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_675_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B4: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( A6
= ( sup_su414716646722978715od_v_v @ A6 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_676_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B4: set_v,A6: set_v] :
( A6
= ( sup_sup_set_v @ A6 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_677_sup_OboundedI,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_678_sup_OboundedI,axiom,
! [B2: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_679_sup_OboundedE,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_680_sup_OboundedE,axiom,
! [B2: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_v @ B2 @ A )
=> ~ ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_681_sup__absorb2,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_682_sup__absorb2,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( sup_sup_set_v @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_683_sup__absorb1,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_684_sup__absorb1,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( sup_sup_set_v @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_685_sup_Oabsorb2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_686_sup_Oabsorb2,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( sup_sup_set_v @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_687_sup_Oabsorb1,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_688_sup_Oabsorb1,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( sup_sup_set_v @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_689_sup__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X3 )
=> ( ( ord_le7336532860387713383od_v_v @ Z3 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ Y @ Z3 ) @ X3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_690_sup__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y2: set_v] :
( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ X3 @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ Y @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_v,Y: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ Y @ X3 )
=> ( ( ord_less_eq_set_v @ Z3 @ X3 )
=> ( ord_less_eq_set_v @ ( F @ Y @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_v @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_691_sup_OorderI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ A @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ B2 @ A ) ) ).
% sup.orderI
thf(fact_692_sup_OorderI,axiom,
! [A: set_v,B2: set_v] :
( ( A
= ( sup_sup_set_v @ A @ B2 ) )
=> ( ord_less_eq_set_v @ B2 @ A ) ) ).
% sup.orderI
thf(fact_693_sup_OorderE,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( A
= ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_694_sup_OorderE,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( A
= ( sup_sup_set_v @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_695_le__iff__sup,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_696_le__iff__sup,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( sup_sup_set_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_697_sup__least,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( ord_le7336532860387713383od_v_v @ Z @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_698_sup__least,axiom,
! [Y2: set_v,X: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( ord_less_eq_set_v @ Z @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_699_sup__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ ( sup_su414716646722978715od_v_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_700_sup__mono,axiom,
! [A: set_v,C: set_v,B2: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ ( sup_sup_set_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_701_sup_Omono,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,D2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D2 ) @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_702_sup_Omono,axiom,
! [C: set_v,A: set_v,D2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ( ord_less_eq_set_v @ D2 @ B2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D2 ) @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_703_le__supI2,axiom,
! [X: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_704_le__supI2,axiom,
! [X: set_v,B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ X @ B2 )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_705_le__supI1,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_706_le__supI1,axiom,
! [X: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_707_sup__ge2,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_708_sup__ge2,axiom,
! [Y2: set_v,X: set_v] : ( ord_less_eq_set_v @ Y2 @ ( sup_sup_set_v @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_709_sup__ge1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_710_sup__ge1,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_711_le__supI,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_712_le__supI,axiom,
! [A: set_v,X: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ( ord_less_eq_set_v @ B2 @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_713_le__supE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ X )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ~ ( ord_le7336532860387713383od_v_v @ B2 @ X ) ) ) ).
% le_supE
thf(fact_714_le__supE,axiom,
! [A: set_v,B2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_v @ A @ X )
=> ~ ( ord_less_eq_set_v @ B2 @ X ) ) ) ).
% le_supE
thf(fact_715_inf__sup__ord_I3_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_716_inf__sup__ord_I3_J,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_717_inf__sup__ord_I4_J,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_718_inf__sup__ord_I4_J,axiom,
! [Y2: set_v,X: set_v] : ( ord_less_eq_set_v @ Y2 @ ( sup_sup_set_v @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_719_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_720_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_721_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_722_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_723_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_724_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_725_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_726_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_727_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_728_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_729_sup__inf__distrib2,axiom,
! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_730_sup__inf__distrib2,axiom,
! [Y2: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_731_sup__inf__distrib1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_732_sup__inf__distrib1,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_733_inf__sup__distrib2,axiom,
! [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_734_inf__sup__distrib2,axiom,
! [Y2: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_735_inf__sup__distrib1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_736_inf__sup__distrib1,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_737_distrib__imp2,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y @ Z3 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y ) @ ( sup_su414716646722978715od_v_v @ X3 @ Z3 ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_738_distrib__imp2,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ! [X3: set_v,Y: set_v,Z3: set_v] :
( ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ Y @ Z3 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X3 @ Y ) @ ( sup_sup_set_v @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_739_distrib__imp1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y @ Z3 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y ) @ ( inf_in6271465464967711157od_v_v @ X3 @ Z3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_740_distrib__imp1,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ! [X3: set_v,Y: set_v,Z3: set_v] :
( ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ Y @ Z3 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X3 @ Y ) @ ( inf_inf_set_v @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_741_Un__empty__right,axiom,
! [A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= A4 ) ).
% Un_empty_right
thf(fact_742_Un__empty__right,axiom,
! [A4: set_v] :
( ( sup_sup_set_v @ A4 @ bot_bot_set_v )
= A4 ) ).
% Un_empty_right
thf(fact_743_Un__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B )
= B ) ).
% Un_empty_left
thf(fact_744_Un__empty__left,axiom,
! [B: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B )
= B ) ).
% Un_empty_left
thf(fact_745_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A8: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A8 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_746_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A8: set_v,B6: set_v] :
( ( sup_sup_set_v @ A8 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_747_subset__UnE,axiom,
! [C2: set_Product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A4 @ B ) )
=> ~ ! [A10: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A10 @ A4 )
=> ! [B8: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B8 @ B )
=> ( C2
!= ( sup_su414716646722978715od_v_v @ A10 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_748_subset__UnE,axiom,
! [C2: set_v,A4: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A4 @ B ) )
=> ~ ! [A10: set_v] :
( ( ord_less_eq_set_v @ A10 @ A4 )
=> ! [B8: set_v] :
( ( ord_less_eq_set_v @ B8 @ B )
=> ( C2
!= ( sup_sup_set_v @ A10 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_749_Un__absorb2,axiom,
! [B: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A4 )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ B )
= A4 ) ) ).
% Un_absorb2
thf(fact_750_Un__absorb2,axiom,
! [B: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B @ A4 )
=> ( ( sup_sup_set_v @ A4 @ B )
= A4 ) ) ).
% Un_absorb2
thf(fact_751_Un__absorb1,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_752_Un__absorb1,axiom,
! [A4: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ( sup_sup_set_v @ A4 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_753_Un__upper2,axiom,
! [B: set_Product_prod_v_v,A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A4 @ B ) ) ).
% Un_upper2
thf(fact_754_Un__upper2,axiom,
! [B: set_v,A4: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A4 @ B ) ) ).
% Un_upper2
thf(fact_755_Un__upper1,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ A4 @ B ) ) ).
% Un_upper1
thf(fact_756_Un__upper1,axiom,
! [A4: set_v,B: set_v] : ( ord_less_eq_set_v @ A4 @ ( sup_sup_set_v @ A4 @ B ) ) ).
% Un_upper1
thf(fact_757_Un__least,axiom,
! [A4: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_758_Un__least,axiom,
! [A4: set_v,C2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A4 @ C2 )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A4 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_759_Un__mono,axiom,
! [A4: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B ) @ ( sup_su414716646722978715od_v_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_760_Un__mono,axiom,
! [A4: set_v,C2: set_v,B: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A4 @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A4 @ B ) @ ( sup_sup_set_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_761_Un__Int__crazy,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A4 ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B ) @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) @ ( sup_su414716646722978715od_v_v @ C2 @ A4 ) ) ) ).
% Un_Int_crazy
thf(fact_762_Un__Int__crazy,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A4 @ B ) @ ( inf_inf_set_v @ B @ C2 ) ) @ ( inf_inf_set_v @ C2 @ A4 ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A4 @ B ) @ ( sup_sup_set_v @ B @ C2 ) ) @ ( sup_sup_set_v @ C2 @ A4 ) ) ) ).
% Un_Int_crazy
thf(fact_763_Int__Un__distrib,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) @ ( inf_in6271465464967711157od_v_v @ A4 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_764_Int__Un__distrib,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ A4 @ ( sup_sup_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A4 @ B ) @ ( inf_inf_set_v @ A4 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_765_Un__Int__distrib,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B ) @ ( sup_su414716646722978715od_v_v @ A4 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_766_Un__Int__distrib,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ A4 @ ( inf_inf_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A4 @ B ) @ ( sup_sup_set_v @ A4 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_767_Int__Un__distrib2,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C2 ) @ A4 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ A4 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A4 ) ) ) ).
% Int_Un_distrib2
thf(fact_768_Int__Un__distrib2,axiom,
! [B: set_v,C2: set_v,A4: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B @ C2 ) @ A4 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B @ A4 ) @ ( inf_inf_set_v @ C2 @ A4 ) ) ) ).
% Int_Un_distrib2
thf(fact_769_Un__Int__distrib2,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) @ A4 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ A4 ) @ ( sup_su414716646722978715od_v_v @ C2 @ A4 ) ) ) ).
% Un_Int_distrib2
thf(fact_770_Un__Int__distrib2,axiom,
! [B: set_v,C2: set_v,A4: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B @ C2 ) @ A4 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B @ A4 ) @ ( sup_sup_set_v @ C2 @ A4 ) ) ) ).
% Un_Int_distrib2
thf(fact_771_Un__Diff,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B ) @ C2 )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ C2 ) @ ( minus_4183494784930505774od_v_v @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_772_Un__Diff,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( sup_sup_set_v @ A4 @ B ) @ C2 )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A4 @ C2 ) @ ( minus_minus_set_v @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_773_distrib__sup__le,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_774_distrib__sup__le,axiom,
! [X: set_v,Y2: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_775_distrib__inf__le,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_776_distrib__inf__le,axiom,
! [X: set_v,Y2: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) @ ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_777_singleton__Un__iff,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A4 @ B ) )
= ( ( ( A4 = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_778_singleton__Un__iff,axiom,
! [X: v,A4: set_v,B: set_v] :
( ( ( insert_v2 @ X @ bot_bot_set_v )
= ( sup_sup_set_v @ A4 @ B ) )
= ( ( ( A4 = bot_bot_set_v )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) )
| ( ( A4
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A4
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_779_Un__singleton__iff,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A4 @ B )
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A4 = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_780_Un__singleton__iff,axiom,
! [A4: set_v,B: set_v,X: v] :
( ( ( sup_sup_set_v @ A4 @ B )
= ( insert_v2 @ X @ bot_bot_set_v ) )
= ( ( ( A4 = bot_bot_set_v )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) )
| ( ( A4
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A4
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_781_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A6: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A6 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_782_insert__is__Un,axiom,
( insert_v2
= ( ^ [A6: v] : ( sup_sup_set_v @ ( insert_v2 @ A6 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_783_Un__Int__assoc__eq,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) )
= ( ord_le7336532860387713383od_v_v @ C2 @ A4 ) ) ).
% Un_Int_assoc_eq
thf(fact_784_Un__Int__assoc__eq,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A4 @ B ) @ C2 )
= ( inf_inf_set_v @ A4 @ ( sup_sup_set_v @ B @ C2 ) ) )
= ( ord_less_eq_set_v @ C2 @ A4 ) ) ).
% Un_Int_assoc_eq
thf(fact_785_Diff__partition,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ B @ A4 ) )
= B ) ) ).
% Diff_partition
thf(fact_786_Diff__partition,axiom,
! [A4: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ( sup_sup_set_v @ A4 @ ( minus_minus_set_v @ B @ A4 ) )
= B ) ) ).
% Diff_partition
thf(fact_787_Diff__subset__conv,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ B ) @ C2 )
= ( ord_le7336532860387713383od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_788_Diff__subset__conv,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A4 @ B ) @ C2 )
= ( ord_less_eq_set_v @ A4 @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_789_Un__Diff__Int,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ B ) @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) )
= A4 ) ).
% Un_Diff_Int
thf(fact_790_Un__Diff__Int,axiom,
! [A4: set_v,B: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A4 @ B ) @ ( inf_inf_set_v @ A4 @ B ) )
= A4 ) ).
% Un_Diff_Int
thf(fact_791_Int__Diff__Un,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B ) @ ( minus_4183494784930505774od_v_v @ A4 @ B ) )
= A4 ) ).
% Int_Diff_Un
thf(fact_792_Int__Diff__Un,axiom,
! [A4: set_v,B: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A4 @ B ) @ ( minus_minus_set_v @ A4 @ B ) )
= A4 ) ).
% Int_Diff_Un
thf(fact_793_Diff__Int,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ B ) @ ( minus_4183494784930505774od_v_v @ A4 @ C2 ) ) ) ).
% Diff_Int
thf(fact_794_Diff__Int,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ A4 @ ( inf_inf_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A4 @ B ) @ ( minus_minus_set_v @ A4 @ C2 ) ) ) ).
% Diff_Int
thf(fact_795_Diff__Un,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ B ) @ ( minus_4183494784930505774od_v_v @ A4 @ C2 ) ) ) ).
% Diff_Un
thf(fact_796_Diff__Un,axiom,
! [A4: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ A4 @ ( sup_sup_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A4 @ B ) @ ( minus_minus_set_v @ A4 @ C2 ) ) ) ).
% Diff_Un
thf(fact_797_graph_Ora__add__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ V @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% graph.ra_add_edge
thf(fact_798_finite__has__maximal2,axiom,
! [A4: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( member8406446414694345712od_v_v @ A @ A4 )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A4 )
& ( ord_le7336532860387713383od_v_v @ A @ X3 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_799_finite__has__maximal2,axiom,
! [A4: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( member_set_v @ A @ A4 )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A4 )
& ( ord_less_eq_set_v @ A @ X3 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_800_finite__has__minimal2,axiom,
! [A4: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( member8406446414694345712od_v_v @ A @ A4 )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A4 )
& ( ord_le7336532860387713383od_v_v @ X3 @ A )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_801_finite__has__minimal2,axiom,
! [A4: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( member_set_v @ A @ A4 )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A4 )
& ( ord_less_eq_set_v @ X3 @ A )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_802_graph_Oavoiding__explored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,X: v,Y2: v,E4: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( ~ ( member_v @ Y2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ) ).
% graph.avoiding_explored
thf(fact_803_finite_OemptyI,axiom,
finite3348123685078250256od_v_v @ bot_bo723834152578015283od_v_v ).
% finite.emptyI
thf(fact_804_finite_OemptyI,axiom,
finite_finite_v @ bot_bot_set_v ).
% finite.emptyI
thf(fact_805_infinite__imp__nonempty,axiom,
! [S: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ( S != bot_bo723834152578015283od_v_v ) ) ).
% infinite_imp_nonempty
thf(fact_806_infinite__imp__nonempty,axiom,
! [S: set_v] :
( ~ ( finite_finite_v @ S )
=> ( S != bot_bot_set_v ) ) ).
% infinite_imp_nonempty
thf(fact_807_finite__subset,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ( finite3348123685078250256od_v_v @ B )
=> ( finite3348123685078250256od_v_v @ A4 ) ) ) ).
% finite_subset
thf(fact_808_finite__subset,axiom,
! [A4: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ( finite_finite_v @ B )
=> ( finite_finite_v @ A4 ) ) ) ).
% finite_subset
thf(fact_809_infinite__super,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ S @ T2 )
=> ( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).
% infinite_super
thf(fact_810_infinite__super,axiom,
! [S: set_v,T2: set_v] :
( ( ord_less_eq_set_v @ S @ T2 )
=> ( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ T2 ) ) ) ).
% infinite_super
thf(fact_811_rev__finite__subset,axiom,
! [B: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( finite3348123685078250256od_v_v @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_812_rev__finite__subset,axiom,
! [B: set_v,A4: set_v] :
( ( finite_finite_v @ B )
=> ( ( ord_less_eq_set_v @ A4 @ B )
=> ( finite_finite_v @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_813_finite_OinsertI,axiom,
! [A4: set_Product_prod_v_v,A: product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A4 )
=> ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A @ A4 ) ) ) ).
% finite.insertI
thf(fact_814_finite_OinsertI,axiom,
! [A4: set_v,A: v] :
( ( finite_finite_v @ A4 )
=> ( finite_finite_v @ ( insert_v2 @ A @ A4 ) ) ) ).
% finite.insertI
thf(fact_815_Diff__infinite__finite,axiom,
! [T2: set_v,S: set_v] :
( ( finite_finite_v @ T2 )
=> ( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_816_finite__has__maximal,axiom,
! [A4: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A4 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_817_finite__has__maximal,axiom,
! [A4: set_set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A4 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_v @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_818_finite__has__minimal,axiom,
! [A4: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A4 )
& ! [Xa2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa2 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_819_finite__has__minimal,axiom,
! [A4: set_set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A4 )
& ! [Xa2: set_v] :
( ( member_set_v @ Xa2 @ A4 )
=> ( ( ord_less_eq_set_v @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_820_finite_Ocases,axiom,
! [A: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A )
=> ( ( A != bot_bo723834152578015283od_v_v )
=> ~ ! [A9: set_Product_prod_v_v] :
( ? [A7: product_prod_v_v] :
( A
= ( insert1338601472111419319od_v_v @ A7 @ A9 ) )
=> ~ ( finite3348123685078250256od_v_v @ A9 ) ) ) ) ).
% finite.cases
thf(fact_821_finite_Ocases,axiom,
! [A: set_v] :
( ( finite_finite_v @ A )
=> ( ( A != bot_bot_set_v )
=> ~ ! [A9: set_v] :
( ? [A7: v] :
( A
= ( insert_v2 @ A7 @ A9 ) )
=> ~ ( finite_finite_v @ A9 ) ) ) ) ).
% finite.cases
thf(fact_822_finite_Osimps,axiom,
( finite3348123685078250256od_v_v
= ( ^ [A6: set_Product_prod_v_v] :
( ( A6 = bot_bo723834152578015283od_v_v )
| ? [A8: set_Product_prod_v_v,B4: product_prod_v_v] :
( ( A6
= ( insert1338601472111419319od_v_v @ B4 @ A8 ) )
& ( finite3348123685078250256od_v_v @ A8 ) ) ) ) ) ).
% finite.simps
thf(fact_823_finite_Osimps,axiom,
( finite_finite_v
= ( ^ [A6: set_v] :
( ( A6 = bot_bot_set_v )
| ? [A8: set_v,B4: v] :
( ( A6
= ( insert_v2 @ B4 @ A8 ) )
& ( finite_finite_v @ A8 ) ) ) ) ) ).
% finite.simps
thf(fact_824_finite__induct,axiom,
! [F2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_825_finite__induct,axiom,
! [F2: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_826_finite__ne__induct,axiom,
! [F2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( F2 != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v] : ( P @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( F3 != bot_bo723834152578015283od_v_v )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_827_finite__ne__induct,axiom,
! [F2: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( F2 != bot_bot_set_v )
=> ( ! [X3: v] : ( P @ ( insert_v2 @ X3 @ bot_bot_set_v ) )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( F3 != bot_bot_set_v )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_828_infinite__finite__induct,axiom,
! [P: set_Product_prod_v_v > $o,A4: set_Product_prod_v_v] :
( ! [A9: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_829_infinite__finite__induct,axiom,
! [P: set_v > $o,A4: set_v] :
( ! [A9: set_v] :
( ~ ( finite_finite_v @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_830_finite__subset__induct,axiom,
! [F2: set_Product_prod_v_v,A4: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( ord_le7336532860387713383od_v_v @ F2 @ A4 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A7: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( member7453568604450474000od_v_v @ A7 @ A4 )
=> ( ~ ( member7453568604450474000od_v_v @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ A7 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_831_finite__subset__induct,axiom,
! [F2: set_v,A4: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( ord_less_eq_set_v @ F2 @ A4 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A7: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( member_v @ A7 @ A4 )
=> ( ~ ( member_v @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ A7 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_832_finite__subset__induct_H,axiom,
! [F2: set_Product_prod_v_v,A4: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( ord_le7336532860387713383od_v_v @ F2 @ A4 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A7: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( member7453568604450474000od_v_v @ A7 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ F3 @ A4 )
=> ( ~ ( member7453568604450474000od_v_v @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ A7 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_833_finite__subset__induct_H,axiom,
! [F2: set_v,A4: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( ord_less_eq_set_v @ F2 @ A4 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A7: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( member_v @ A7 @ A4 )
=> ( ( ord_less_eq_set_v @ F3 @ A4 )
=> ( ~ ( member_v @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ A7 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_834_finite__empty__induct,axiom,
! [A4: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ A4 )
=> ( ( P @ A4 )
=> ( ! [A7: product_prod_v_v,A9: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A9 )
=> ( ( member7453568604450474000od_v_v @ A7 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ A7 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
=> ( P @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% finite_empty_induct
thf(fact_835_finite__empty__induct,axiom,
! [A4: set_v,P: set_v > $o] :
( ( finite_finite_v @ A4 )
=> ( ( P @ A4 )
=> ( ! [A7: v,A9: set_v] :
( ( finite_finite_v @ A9 )
=> ( ( member_v @ A7 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ A7 @ bot_bot_set_v ) ) ) ) ) )
=> ( P @ bot_bot_set_v ) ) ) ) ).
% finite_empty_induct
thf(fact_836_infinite__coinduct,axiom,
! [X5: set_Product_prod_v_v > $o,A4: set_Product_prod_v_v] :
( ( X5 @ A4 )
=> ( ! [A9: set_Product_prod_v_v] :
( ( X5 @ A9 )
=> ? [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A9 )
& ( ( X5 @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) )
| ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
=> ~ ( finite3348123685078250256od_v_v @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_837_infinite__coinduct,axiom,
! [X5: set_v > $o,A4: set_v] :
( ( X5 @ A4 )
=> ( ! [A9: set_v] :
( ( X5 @ A9 )
=> ? [X4: v] :
( ( member_v @ X4 @ A9 )
& ( ( X5 @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) )
| ~ ( finite_finite_v @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) ) ) )
=> ~ ( finite_finite_v @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_838_infinite__remove,axiom,
! [S: set_Product_prod_v_v,A: product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ S @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% infinite_remove
thf(fact_839_infinite__remove,axiom,
! [S: set_v,A: v] :
( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S @ ( insert_v2 @ A @ bot_bot_set_v ) ) ) ) ).
% infinite_remove
thf(fact_840_remove__induct,axiom,
! [P: set_Product_prod_v_v > $o,B: set_Product_prod_v_v] :
( ( P @ bot_bo723834152578015283od_v_v )
=> ( ( ~ ( finite3348123685078250256od_v_v @ B )
=> ( P @ B ) )
=> ( ! [A9: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A9 )
=> ( ( A9 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A9 @ B )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A9 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A9 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_841_remove__induct,axiom,
! [P: set_v > $o,B: set_v] :
( ( P @ bot_bot_set_v )
=> ( ( ~ ( finite_finite_v @ B )
=> ( P @ B ) )
=> ( ! [A9: set_v] :
( ( finite_finite_v @ A9 )
=> ( ( A9 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A9 @ B )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A9 )
=> ( P @ ( minus_minus_set_v @ A9 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_842_Collect__empty__eq__bot,axiom,
! [P: set_v > $o] :
( ( ( collect_set_v @ P )
= bot_bot_set_set_v )
= ( P = bot_bot_set_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_843_Collect__empty__eq__bot,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( P = bot_bo8461541820394803818_v_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_844_Collect__empty__eq__bot,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( P = bot_bot_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_845_unite__subscc,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ).
% unite_subscc
thf(fact_846_Field__insert,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ( field_7153129647634986036od_v_v @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R ) )
= ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R ) ) ) ).
% Field_insert
thf(fact_847_Field__insert,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v] :
( ( field_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R ) )
= ( sup_sup_set_v @ ( insert_v2 @ A @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) @ ( field_v @ R ) ) ) ).
% Field_insert
thf(fact_848_refl__on__singleton,axiom,
! [X: product_prod_v_v] : ( refl_o4548774019903118566od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ X @ X ) @ bot_bo3282589961317712691od_v_v ) ) ).
% refl_on_singleton
thf(fact_849_refl__on__singleton,axiom,
! [X: v] : ( refl_on_v @ ( insert_v2 @ X @ bot_bot_set_v ) @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ X @ X ) @ bot_bo723834152578015283od_v_v ) ) ).
% refl_on_singleton
thf(fact_850__092_060open_062v_A_092_060in_062_A_092_060S_062_Ae_A_Ihd_A_Istack_Ae_J_J_092_060close_062,axiom,
member_v @ v2 @ ( sCC_Bl1280885523602775798t_unit @ e @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e ) ) ) ).
% \<open>v \<in> \<S> e (hd (stack e))\<close>
thf(fact_851__092_060open_062v_A_092_060in_062_A_092_060S_062_Ae_H_A_Ihd_A_Istack_Ae_H_J_J_092_060close_062,axiom,
member_v @ v2 @ ( sCC_Bl1280885523602775798t_unit @ e2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% \<open>v \<in> \<S> e' (hd (stack e'))\<close>
thf(fact_852_Field__empty,axiom,
( ( field_7153129647634986036od_v_v @ bot_bo3282589961317712691od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Field_empty
thf(fact_853_Field__empty,axiom,
( ( field_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Field_empty
thf(fact_854_Field__Un,axiom,
! [R: set_Pr2149350503807050951od_v_v,S2: set_Pr2149350503807050951od_v_v] :
( ( field_7153129647634986036od_v_v @ ( sup_su1742609618068805275od_v_v @ R @ S2 ) )
= ( sup_su414716646722978715od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ ( field_7153129647634986036od_v_v @ S2 ) ) ) ).
% Field_Un
thf(fact_855_Field__Un,axiom,
! [R: set_Product_prod_v_v,S2: set_Product_prod_v_v] :
( ( field_v @ ( sup_su414716646722978715od_v_v @ R @ S2 ) )
= ( sup_sup_set_v @ ( field_v @ R ) @ ( field_v @ S2 ) ) ) ).
% Field_Un
thf(fact_856_FieldI1,axiom,
! [I2: product_prod_v_v,J2: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I2 @ J2 ) @ R3 )
=> ( member7453568604450474000od_v_v @ I2 @ ( field_7153129647634986036od_v_v @ R3 ) ) ) ).
% FieldI1
thf(fact_857_FieldI1,axiom,
! [I2: v,J2: v,R3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I2 @ J2 ) @ R3 )
=> ( member_v @ I2 @ ( field_v @ R3 ) ) ) ).
% FieldI1
thf(fact_858_FieldI2,axiom,
! [I2: product_prod_v_v,J2: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I2 @ J2 ) @ R3 )
=> ( member7453568604450474000od_v_v @ J2 @ ( field_7153129647634986036od_v_v @ R3 ) ) ) ).
% FieldI2
thf(fact_859_FieldI2,axiom,
! [I2: v,J2: v,R3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I2 @ J2 ) @ R3 )
=> ( member_v @ J2 @ ( field_v @ R3 ) ) ) ).
% FieldI2
thf(fact_860_finite__Field,axiom,
! [R: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ R )
=> ( finite_finite_v @ ( field_v @ R ) ) ) ).
% finite_Field
thf(fact_861_refl__onD,axiom,
! [A4: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A4 @ R )
=> ( ( member7453568604450474000od_v_v @ A @ A4 )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_862_refl__onD,axiom,
! [A4: set_v,R: set_Product_prod_v_v,A: v] :
( ( refl_on_v @ A4 @ R )
=> ( ( member_v @ A @ A4 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_863_refl__onD1,axiom,
! [A4: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A4 @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y2 ) @ R )
=> ( member7453568604450474000od_v_v @ X @ A4 ) ) ) ).
% refl_onD1
thf(fact_864_refl__onD1,axiom,
! [A4: set_v,R: set_Product_prod_v_v,X: v,Y2: v] :
( ( refl_on_v @ A4 @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
=> ( member_v @ X @ A4 ) ) ) ).
% refl_onD1
thf(fact_865_refl__onD2,axiom,
! [A4: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A4 @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y2 ) @ R )
=> ( member7453568604450474000od_v_v @ Y2 @ A4 ) ) ) ).
% refl_onD2
thf(fact_866_refl__onD2,axiom,
! [A4: set_v,R: set_Product_prod_v_v,X: v,Y2: v] :
( ( refl_on_v @ A4 @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
=> ( member_v @ Y2 @ A4 ) ) ) ).
% refl_onD2
thf(fact_867_refl__on__Int,axiom,
! [A4: set_v,R: set_Product_prod_v_v,B: set_v,S2: set_Product_prod_v_v] :
( ( refl_on_v @ A4 @ R )
=> ( ( refl_on_v @ B @ S2 )
=> ( refl_on_v @ ( inf_inf_set_v @ A4 @ B ) @ ( inf_in6271465464967711157od_v_v @ R @ S2 ) ) ) ) ).
% refl_on_Int
thf(fact_868_mono__Field,axiom,
! [R: set_Pr2149350503807050951od_v_v,S2: set_Pr2149350503807050951od_v_v] :
( ( ord_le6241436655786843239od_v_v @ R @ S2 )
=> ( ord_le7336532860387713383od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ ( field_7153129647634986036od_v_v @ S2 ) ) ) ).
% mono_Field
thf(fact_869_mono__Field,axiom,
! [R: set_Product_prod_v_v,S2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S2 )
=> ( ord_less_eq_set_v @ ( field_v @ R ) @ ( field_v @ S2 ) ) ) ).
% mono_Field
thf(fact_870_refl__on__empty,axiom,
refl_o4548774019903118566od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% refl_on_empty
thf(fact_871_refl__on__empty,axiom,
refl_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% refl_on_empty
thf(fact_872_refl__on__Un,axiom,
! [A4: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v,S2: set_Pr2149350503807050951od_v_v] :
( ( refl_o4548774019903118566od_v_v @ A4 @ R )
=> ( ( refl_o4548774019903118566od_v_v @ B @ S2 )
=> ( refl_o4548774019903118566od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B ) @ ( sup_su1742609618068805275od_v_v @ R @ S2 ) ) ) ) ).
% refl_on_Un
thf(fact_873_refl__on__Un,axiom,
! [A4: set_v,R: set_Product_prod_v_v,B: set_v,S2: set_Product_prod_v_v] :
( ( refl_on_v @ A4 @ R )
=> ( ( refl_on_v @ B @ S2 )
=> ( refl_on_v @ ( sup_sup_set_v @ A4 @ B ) @ ( sup_su414716646722978715od_v_v @ R @ S2 ) ) ) ) ).
% refl_on_Un
thf(fact_874_graph_Ounite__subscc,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) @ ( hd_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_875_graph_Ounite__subscc,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_876_stack__class,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ).
% stack_class
thf(fact_877_dfs__S__hd__stack_I1_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ).
% dfs_S_hd_stack(1)
thf(fact_878_dfs__S__hd__stack_I2_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ).
% dfs_S_hd_stack(2)
thf(fact_879_visited__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_880__092_060open_062stack_Ae_H_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e2 )
!= nil_v ) ).
% \<open>stack e' \<noteq> []\<close>
thf(fact_881__092_060open_062stack_Ae_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e )
!= nil_v ) ).
% \<open>stack e \<noteq> []\<close>
thf(fact_882_dfs__S__tl__stack_I1_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ).
% dfs_S_tl_stack(1)
thf(fact_883_stack__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ).
% stack_visited
thf(fact_884_stack__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ).
% stack_unexplored
thf(fact_885_graph_Odfs__S__tl__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ).
% graph.dfs_S_tl_stack(1)
thf(fact_886_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ).
% graph.stack_visited
thf(fact_887_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_888_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_889_graph_Odfs__S__hd__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(1)
thf(fact_890_graph_Odfs__S__hd__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(2)
thf(fact_891_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_892_pre__dfs__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ~ ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) ) ) ) ).
% pre_dfs_def
thf(fact_893_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_894_set__empty,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( ( set_Product_prod_v_v2 @ Xs )
= bot_bo723834152578015283od_v_v )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty
thf(fact_895_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_896_set__empty2,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ Xs ) )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty2
thf(fact_897_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_898_List_Ofinite__set,axiom,
! [Xs: list_v] : ( finite_finite_v @ ( set_v2 @ Xs ) ) ).
% List.finite_set
thf(fact_899_dfs__S__tl__stack_I2_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) ) ).
% dfs_S_tl_stack(2)
thf(fact_900_graph_Odfs__S__tl__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) ) ) ).
% graph.dfs_S_tl_stack(2)
thf(fact_901_finite__list,axiom,
! [A4: set_v] :
( ( finite_finite_v @ A4 )
=> ? [Xs2: list_v] :
( ( set_v2 @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_902_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_903_subset__code_I1_J,axiom,
! [Xs: list_v,B: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ( member_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_904_empty__set,axiom,
( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% empty_set
thf(fact_905_empty__set,axiom,
( bot_bot_set_v
= ( set_v2 @ nil_v ) ) ).
% empty_set
thf(fact_906_graph_Ounite__S__tl,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V: product_prod_v_v,N: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( member7453568604450474000od_v_v @ N @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) @ N )
= ( sCC_Bl8440648026628373538t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_907_graph_Ounite__S__tl,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_908_graph_Opre__dfs__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl36166008131615352t_unit @ Successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ~ ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V ) ) ) ) ) ).
% graph.pre_dfs_def
thf(fact_909_post__dfs__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V )
= ( successors @ V ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( ( ( member_v @ V @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
= ( sCC_Bl8828226123343373779t_unit @ E ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) )
| ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfs_def
thf(fact_910_pre__dfss__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V @ Ns ) ) ) ) ).
% pre_dfss_def
thf(fact_911_post__dfss__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl6082031138996704384t_unit @ successors @ V @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V )
= ( successors @ V ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V @ bot_bot_set_v ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ! [X2: v] :
( ( member_v @ X2 @ ( successors @ V ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E2 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) )
& ( ( ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
= V )
=> ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ V @ X2 ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfss_def
thf(fact_912_list_Osimps_I15_J,axiom,
! [X21: product_prod_v_v,X222: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X222 ) )
= ( insert1338601472111419319od_v_v @ X21 @ ( set_Product_prod_v_v2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_913_list_Osimps_I15_J,axiom,
! [X21: v,X222: list_v] :
( ( set_v2 @ ( cons_v @ X21 @ X222 ) )
= ( insert_v2 @ X21 @ ( set_v2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_914_set__subset__Cons,axiom,
! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_915_set__subset__Cons,axiom,
! [Xs: list_v,X: v] : ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ ( cons_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_916_the__elem__set,axiom,
! [X: v] :
( ( the_elem_v @ ( set_v2 @ ( cons_v @ X @ nil_v ) ) )
= X ) ).
% the_elem_set
thf(fact_917_graph_Opre__dfss__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V @ Ns ) ) ) ) ) ).
% graph.pre_dfss_def
thf(fact_918_equality,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R4: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ R )
= ( sCC_Bl1090238580953940555t_unit @ R4 ) )
=> ( ( ( sCC_Bl1280885523602775798t_unit @ R )
= ( sCC_Bl1280885523602775798t_unit @ R4 ) )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R )
= ( sCC_Bl157864678168468314t_unit @ R4 ) )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R )
= ( sCC_Bl4645233313691564917t_unit @ R4 ) )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R )
= ( sCC_Bl3795065053823578884t_unit @ R4 ) )
=> ( ( ( sCC_Bl2536197123907397897t_unit @ R )
= ( sCC_Bl2536197123907397897t_unit @ R4 ) )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ R )
= ( sCC_Bl8828226123343373779t_unit @ R4 ) )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R )
= ( sCC_Bl9201514103433284750t_unit @ R4 ) )
=> ( ( ( sCC_Bl3567736435408124606t_unit @ R )
= ( sCC_Bl3567736435408124606t_unit @ R4 ) )
=> ( R = R4 ) ) ) ) ) ) ) ) ) ) ).
% equality
thf(fact_919_cofinal__def,axiom,
( bNF_Ca4386975739854426340inal_v
= ( ^ [A8: set_v,R2: set_Product_prod_v_v] :
! [X2: v] :
( ( member_v @ X2 @ ( field_v @ R2 ) )
=> ? [Y3: v] :
( ( member_v @ Y3 @ A8 )
& ( X2 != Y3 )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ R2 ) ) ) ) ) ).
% cofinal_def
thf(fact_920_refl__on__domain,axiom,
! [A4: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A4 @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ A4 )
& ( member7453568604450474000od_v_v @ B2 @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_921_refl__on__domain,axiom,
! [A4: set_v,R: set_Product_prod_v_v,A: v,B2: v] :
( ( refl_on_v @ A4 @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( ( member_v @ A @ A4 )
& ( member_v @ B2 @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_922_is__empty__set,axiom,
! [Xs: list_v] :
( ( is_empty_v @ ( set_v2 @ Xs ) )
= ( null_v @ Xs ) ) ).
% is_empty_set
thf(fact_923_surjective,axiom,
! [R: sCC_Bl1394983891496994913t_unit] :
( R
= ( sCC_Bl8064756265740546429t_unit @ ( sCC_Bl1090238580953940555t_unit @ R ) @ ( sCC_Bl1280885523602775798t_unit @ R ) @ ( sCC_Bl157864678168468314t_unit @ R ) @ ( sCC_Bl4645233313691564917t_unit @ R ) @ ( sCC_Bl3795065053823578884t_unit @ R ) @ ( sCC_Bl2536197123907397897t_unit @ R ) @ ( sCC_Bl8828226123343373779t_unit @ R ) @ ( sCC_Bl9201514103433284750t_unit @ R ) @ ( sCC_Bl3567736435408124606t_unit @ R ) ) ) ).
% surjective
thf(fact_924_linear__order__on__singleton,axiom,
! [X: product_prod_v_v] : ( order_6462556390437124636od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ X @ X ) @ bot_bo3282589961317712691od_v_v ) ) ).
% linear_order_on_singleton
thf(fact_925_linear__order__on__singleton,axiom,
! [X: v] : ( order_8768733634509060168r_on_v @ ( insert_v2 @ X @ bot_bot_set_v ) @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ X @ X ) @ bot_bo723834152578015283od_v_v ) ) ).
% linear_order_on_singleton
thf(fact_926_set__removeAll,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( remove481895986417801203od_v_v @ X @ Xs ) )
= ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ).
% set_removeAll
thf(fact_927_set__removeAll,axiom,
! [X: v,Xs: list_v] :
( ( set_v2 @ ( removeAll_v @ X @ Xs ) )
= ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ).
% set_removeAll
thf(fact_928_select__convs_I7_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Stack ) ).
% select_convs(7)
thf(fact_929_select__convs_I4_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl4645233313691564917t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Visited ) ).
% select_convs(4)
thf(fact_930_select__convs_I5_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Vsuccs ) ).
% select_convs(5)
thf(fact_931_select__convs_I3_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl157864678168468314t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Explored ) ).
% select_convs(3)
thf(fact_932_select__convs_I2_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= S6 ) ).
% select_convs(2)
thf(fact_933_lnear__order__on__empty,axiom,
order_6462556390437124636od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% lnear_order_on_empty
thf(fact_934_lnear__order__on__empty,axiom,
order_8768733634509060168r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% lnear_order_on_empty
thf(fact_935_remove__code_I1_J,axiom,
! [X: v,Xs: list_v] :
( ( remove_v @ X @ ( set_v2 @ Xs ) )
= ( set_v2 @ ( removeAll_v @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_936_finite__Linear__order__induct,axiom,
! [R: set_Product_prod_v_v,X: v,P: v > $o] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v @ X @ ( field_v @ R ) )
=> ( ( finite3348123685078250256od_v_v @ R )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( field_v @ R ) )
=> ( ! [Y5: v] :
( ( member_v @ Y5 @ ( order_aboveS_v @ R @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% finite_Linear_order_induct
thf(fact_937_finite__Linear__order__induct,axiom,
! [R: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,P: product_prod_v_v > $o] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ X @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( finite5952053201251911184od_v_v @ R )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ! [Y5: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y5 @ ( order_1156346741491923410od_v_v @ R @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% finite_Linear_order_induct
thf(fact_938_Sup__fin_Oremove,axiom,
! [A4: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( member8406446414694345712od_v_v @ X @ A4 )
=> ( ( ( ( minus_7679383599658060814od_v_v @ A4 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ A4 )
= X ) )
& ( ( ( minus_7679383599658060814od_v_v @ A4 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
!= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ A4 )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A4 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_939_Sup__fin_Oremove,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( member_set_v @ X @ A4 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ A4 )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ A4 )
= ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_940_inf__Sup__absorb,axiom,
! [A4: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( member_set_v @ A @ A4 )
=> ( ( inf_inf_set_v @ A @ ( lattic2918178447194608042_set_v @ A4 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_941_Sup__fin_Oinsert,axiom,
! [A4: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A4 ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A4 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_942_Sup__fin_Oinsert,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A4 ) )
= ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ A4 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_943_Sup__fin_OcoboundedI,axiom,
! [A4: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( member8406446414694345712od_v_v @ A @ A4 )
=> ( ord_le7336532860387713383od_v_v @ A @ ( lattic5151207300795964030od_v_v @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_944_Sup__fin_OcoboundedI,axiom,
! [A4: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( member_set_v @ A @ A4 )
=> ( ord_less_eq_set_v @ A @ ( lattic2918178447194608042_set_v @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_945_Sup__fin_Oin__idem,axiom,
! [A4: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( member8406446414694345712od_v_v @ X @ A4 )
=> ( ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A4 ) )
= ( lattic5151207300795964030od_v_v @ A4 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_946_Sup__fin_Oin__idem,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( member_set_v @ X @ A4 )
=> ( ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ A4 ) )
= ( lattic2918178447194608042_set_v @ A4 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_947_Sup__fin_Obounded__iff,axiom,
! [A4: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A4 ) @ X )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A4 )
=> ( ord_le7336532860387713383od_v_v @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_948_Sup__fin_Obounded__iff,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A4 ) @ X )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A4 )
=> ( ord_less_eq_set_v @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_949_Sup__fin_OboundedI,axiom,
! [A4: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ! [A7: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A7 @ A4 )
=> ( ord_le7336532860387713383od_v_v @ A7 @ X ) )
=> ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A4 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_950_Sup__fin_OboundedI,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ! [A7: set_v] :
( ( member_set_v @ A7 @ A4 )
=> ( ord_less_eq_set_v @ A7 @ X ) )
=> ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A4 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_951_Sup__fin_OboundedE,axiom,
! [A4: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A4 ) @ X )
=> ! [A11: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A11 @ A4 )
=> ( ord_le7336532860387713383od_v_v @ A11 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_952_Sup__fin_OboundedE,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A4 ) @ X )
=> ! [A11: set_v] :
( ( member_set_v @ A11 @ A4 )
=> ( ord_less_eq_set_v @ A11 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_953_Sup__fin_Osubset__imp,axiom,
! [A4: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A4 @ B )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A4 ) @ ( lattic5151207300795964030od_v_v @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_954_Sup__fin_Osubset__imp,axiom,
! [A4: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ B )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A4 ) @ ( lattic2918178447194608042_set_v @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_955_Sup__fin_Osubset,axiom,
! [A4: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( B != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le4714265922333009223od_v_v @ B @ A4 )
=> ( ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ B ) @ ( lattic5151207300795964030od_v_v @ A4 ) )
= ( lattic5151207300795964030od_v_v @ A4 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_956_Sup__fin_Osubset,axiom,
! [A4: set_set_v,B: set_set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( B != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ B @ A4 )
=> ( ( sup_sup_set_v @ ( lattic2918178447194608042_set_v @ B ) @ ( lattic2918178447194608042_set_v @ A4 ) )
= ( lattic2918178447194608042_set_v @ A4 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_957_Sup__fin_Oclosed,axiom,
! [A4: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y ) @ ( insert7504383016908236695od_v_v @ X3 @ ( insert7504383016908236695od_v_v @ Y @ bot_bo3497076220358800403od_v_v ) ) )
=> ( member8406446414694345712od_v_v @ ( lattic5151207300795964030od_v_v @ A4 ) @ A4 ) ) ) ) ).
% Sup_fin.closed
thf(fact_958_Sup__fin_Oclosed,axiom,
! [A4: set_set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ! [X3: set_v,Y: set_v] : ( member_set_v @ ( sup_sup_set_v @ X3 @ Y ) @ ( insert_set_v @ X3 @ ( insert_set_v @ Y @ bot_bot_set_set_v ) ) )
=> ( member_set_v @ ( lattic2918178447194608042_set_v @ A4 ) @ A4 ) ) ) ) ).
% Sup_fin.closed
thf(fact_959_Sup__fin_Oinsert__not__elem,axiom,
! [A4: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ~ ( member8406446414694345712od_v_v @ X @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A4 ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A4 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_960_Sup__fin_Oinsert__not__elem,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ~ ( member_set_v @ X @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A4 ) )
= ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ A4 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_961_Sup__fin_Ounion,axiom,
! [A4: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( ( B != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( sup_su335656005089752955od_v_v @ A4 @ B ) )
= ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ A4 ) @ ( lattic5151207300795964030od_v_v @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_962_Sup__fin_Ounion,axiom,
! [A4: set_set_v,B: set_set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ( B != bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( sup_sup_set_set_v @ A4 @ B ) )
= ( sup_sup_set_v @ ( lattic2918178447194608042_set_v @ A4 ) @ ( lattic2918178447194608042_set_v @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_963_Sup__fin_Oinsert__remove,axiom,
! [A4: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( ( ( minus_7679383599658060814od_v_v @ A4 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A4 ) )
= X ) )
& ( ( ( minus_7679383599658060814od_v_v @ A4 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
!= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A4 ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A4 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_964_Sup__fin_Oinsert__remove,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A4 ) )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A4 ) )
= ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_965_Inf__fin_Oinsert__remove,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A4 ) )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A4 ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_966_Inf__fin_Oremove,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( member_set_v @ X @ A4 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ A4 )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ A4 )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_967_finite__Partial__order__induct,axiom,
! [R: set_Product_prod_v_v,X: v,P: v > $o] :
( ( order_5272072345360262664r_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v @ X @ ( field_v @ R ) )
=> ( ( finite3348123685078250256od_v_v @ R )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( field_v @ R ) )
=> ( ! [Y5: v] :
( ( member_v @ Y5 @ ( order_aboveS_v @ R @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% finite_Partial_order_induct
thf(fact_968_finite__Partial__order__induct,axiom,
! [R: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,P: product_prod_v_v > $o] :
( ( order_4212533993404950492od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ X @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( finite5952053201251911184od_v_v @ R )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ! [Y5: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y5 @ ( order_1156346741491923410od_v_v @ R @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% finite_Partial_order_induct
thf(fact_969_sup__Inf__absorb,axiom,
! [A4: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( member8406446414694345712od_v_v @ A @ A4 )
=> ( ( sup_su414716646722978715od_v_v @ ( lattic4767070952889939172od_v_v @ A4 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_970_sup__Inf__absorb,axiom,
! [A4: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( member_set_v @ A @ A4 )
=> ( ( sup_sup_set_v @ ( lattic8209813555532694032_set_v @ A4 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_971_Inf__fin_Oinsert,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A4 ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A4 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_972_Inf__fin_OcoboundedI,axiom,
! [A4: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( member8406446414694345712od_v_v @ A @ A4 )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ A4 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_973_Inf__fin_OcoboundedI,axiom,
! [A4: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( member_set_v @ A @ A4 )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ A4 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_974_Inf__fin_Oin__idem,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( member_set_v @ X @ A4 )
=> ( ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A4 ) )
= ( lattic8209813555532694032_set_v @ A4 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_975_partial__order__on__empty,axiom,
order_4212533993404950492od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% partial_order_on_empty
thf(fact_976_partial__order__on__empty,axiom,
order_5272072345360262664r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% partial_order_on_empty
thf(fact_977_Inf__fin_OboundedE,axiom,
! [A4: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A4 ) )
=> ! [A11: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A11 @ A4 )
=> ( ord_le7336532860387713383od_v_v @ X @ A11 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_978_Inf__fin_OboundedE,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A4 ) )
=> ! [A11: set_v] :
( ( member_set_v @ A11 @ A4 )
=> ( ord_less_eq_set_v @ X @ A11 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_979_Inf__fin_OboundedI,axiom,
! [A4: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ! [A7: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A7 @ A4 )
=> ( ord_le7336532860387713383od_v_v @ X @ A7 ) )
=> ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A4 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_980_Inf__fin_OboundedI,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ! [A7: set_v] :
( ( member_set_v @ A7 @ A4 )
=> ( ord_less_eq_set_v @ X @ A7 ) )
=> ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A4 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_981_Inf__fin_Obounded__iff,axiom,
! [A4: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A4 ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A4 )
=> ( ord_le7336532860387713383od_v_v @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_982_Inf__fin_Obounded__iff,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A4 ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A4 )
=> ( ord_less_eq_set_v @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_983_Inf__fin_Osubset__imp,axiom,
! [A4: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A4 @ B )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ B ) @ ( lattic4767070952889939172od_v_v @ A4 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_984_Inf__fin_Osubset__imp,axiom,
! [A4: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ B )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ B ) @ ( lattic8209813555532694032_set_v @ A4 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_985_Inf__fin_Osubset,axiom,
! [A4: set_set_v,B: set_set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( B != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ B @ A4 )
=> ( ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ B ) @ ( lattic8209813555532694032_set_v @ A4 ) )
= ( lattic8209813555532694032_set_v @ A4 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_986_Inf__fin_Oclosed,axiom,
! [A4: set_set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ! [X3: set_v,Y: set_v] : ( member_set_v @ ( inf_inf_set_v @ X3 @ Y ) @ ( insert_set_v @ X3 @ ( insert_set_v @ Y @ bot_bot_set_set_v ) ) )
=> ( member_set_v @ ( lattic8209813555532694032_set_v @ A4 ) @ A4 ) ) ) ) ).
% Inf_fin.closed
thf(fact_987_Inf__fin_Oinsert__not__elem,axiom,
! [A4: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ~ ( member_set_v @ X @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A4 ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A4 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_988_Inf__fin_Ounion,axiom,
! [A4: set_set_v,B: set_set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ( B != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( sup_sup_set_set_v @ A4 @ B ) )
= ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ A4 ) @ ( lattic8209813555532694032_set_v @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_989_Inf__fin__le__Sup__fin,axiom,
! [A4: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A4 )
=> ( ( A4 != bot_bo3497076220358800403od_v_v )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ A4 ) @ ( lattic5151207300795964030od_v_v @ A4 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_990_Inf__fin__le__Sup__fin,axiom,
! [A4: set_set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ( A4 != bot_bot_set_set_v )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ A4 ) @ ( lattic2918178447194608042_set_v @ A4 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_991_split__list__precedes,axiom,
! [Y2: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ nil_Product_prod_v_v ) ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ Y2 @ X @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_992_split__list__precedes,axiom,
! [Y2: v,Ys: list_v,X: v,Xs: list_v] :
( ( member_v @ Y2 @ ( set_v2 @ ( append_v @ Ys @ ( cons_v @ X @ nil_v ) ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ Y2 @ X @ ( append_v @ Ys @ ( cons_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_993_subset__code_I3_J,axiom,
~ ( ord_le7336532860387713383od_v_v @ ( coset_766761627116920666od_v_v @ nil_Product_prod_v_v ) @ ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% subset_code(3)
thf(fact_994_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_v @ ( coset_v @ nil_v ) @ ( set_v2 @ nil_v ) ) ).
% subset_code(3)
thf(fact_995_precedes__refl,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ X @ Xs )
= ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_996_precedes__refl,axiom,
! [X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ X @ Xs )
= ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_997_precedes__mem_I1_J,axiom,
! [X: product_prod_v_v,Y2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ Xs )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_998_precedes__mem_I1_J,axiom,
! [X: v,Y2: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Xs )
=> ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_999_precedes__mem_I2_J,axiom,
! [X: product_prod_v_v,Y2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ Xs )
=> ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_1000_precedes__mem_I2_J,axiom,
! [X: v,Y2: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Xs )
=> ( member_v @ Y2 @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_1001_head__precedes,axiom,
! [Y2: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_1002_head__precedes,axiom,
! [Y2: v,X: v,Xs: list_v] :
( ( member_v @ Y2 @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ ( cons_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_1003_tail__not__precedes,axiom,
! [Y2: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ Y2 @ X @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( X = Y2 ) ) ) ).
% tail_not_precedes
thf(fact_1004_tail__not__precedes,axiom,
! [Y2: v,X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ Y2 @ X @ ( cons_v @ X @ Xs ) )
=> ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( X = Y2 ) ) ) ).
% tail_not_precedes
thf(fact_1005_precedes__append__left__iff,axiom,
! [X: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,Y2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ ( append2138873909117096322od_v_v @ Ys @ Xs ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_1006_precedes__append__left__iff,axiom,
! [X: v,Ys: list_v,Y2: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ ( append_v @ Ys @ Xs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_1007_precedes__append__right__iff,axiom,
! [Y2: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_1008_precedes__append__right__iff,axiom,
! [Y2: v,Ys: list_v,X: v,Xs: list_v] :
( ~ ( member_v @ Y2 @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ ( append_v @ Xs @ Ys ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_1009_subset__code_I2_J,axiom,
! [A4: set_Product_prod_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( coset_766761627116920666od_v_v @ Ys ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Ys ) )
=> ~ ( member7453568604450474000od_v_v @ X2 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_1010_subset__code_I2_J,axiom,
! [A4: set_v,Ys: list_v] :
( ( ord_less_eq_set_v @ A4 @ ( coset_v @ Ys ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Ys ) )
=> ~ ( member_v @ X2 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_1011_insert__code_I2_J,axiom,
! [X: v,Xs: list_v] :
( ( insert_v2 @ X @ ( coset_v @ Xs ) )
= ( coset_v @ ( removeAll_v @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1012_insert__code_I2_J,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( coset_766761627116920666od_v_v @ Xs ) )
= ( coset_766761627116920666od_v_v @ ( remove481895986417801203od_v_v @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1013_precedes__def,axiom,
( sCC_Bl2026170059108282219od_v_v
= ( ^ [X2: product_prod_v_v,Y3: product_prod_v_v,Xs3: list_P7986770385144383213od_v_v] :
? [L: list_P7986770385144383213od_v_v,R2: list_P7986770385144383213od_v_v] :
( ( Xs3
= ( append2138873909117096322od_v_v @ L @ ( cons_P4120604216776828829od_v_v @ X2 @ R2 ) ) )
& ( member7453568604450474000od_v_v @ Y3 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X2 @ R2 ) ) ) ) ) ) ).
% precedes_def
thf(fact_1014_precedes__def,axiom,
( sCC_Bl4022239298816431255edes_v
= ( ^ [X2: v,Y3: v,Xs3: list_v] :
? [L: list_v,R2: list_v] :
( ( Xs3
= ( append_v @ L @ ( cons_v @ X2 @ R2 ) ) )
& ( member_v @ Y3 @ ( set_v2 @ ( cons_v @ X2 @ R2 ) ) ) ) ) ) ).
% precedes_def
thf(fact_1015_List_Oset__insert,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( insert4539780211034306307od_v_v @ X @ Xs ) )
= ( insert1338601472111419319od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1016_List_Oset__insert,axiom,
! [X: v,Xs: list_v] :
( ( set_v2 @ ( insert_v @ X @ Xs ) )
= ( insert_v2 @ X @ ( set_v2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1017_set__remove1__subset,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ X @ Xs ) ) @ ( set_Product_prod_v_v2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1018_set__remove1__subset,axiom,
! [X: v,Xs: list_v] : ( ord_less_eq_set_v @ ( set_v2 @ ( remove1_v @ X @ Xs ) ) @ ( set_v2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1019_set__remove1__eq,axiom,
! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] :
( ( distin6159370996967099744od_v_v @ Xs )
=> ( ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ X @ Xs ) )
= ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% set_remove1_eq
thf(fact_1020_set__remove1__eq,axiom,
! [Xs: list_v,X: v] :
( ( distinct_v @ Xs )
=> ( ( set_v2 @ ( remove1_v @ X @ Xs ) )
= ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ).
% set_remove1_eq
thf(fact_1021_Linear__order__Well__order__iff,axiom,
! [R: set_Pr2149350503807050951od_v_v] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( order_7541072052284126853od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
= ( ! [A8: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A8 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( A8 != bot_bo723834152578015283od_v_v )
=> ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A8 )
& ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ A8 )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1022_Linear__order__Well__order__iff,axiom,
! [R: set_Product_prod_v_v] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
=> ( ( order_6972113574731384241r_on_v @ ( field_v @ R ) @ R )
= ( ! [A8: set_v] :
( ( ord_less_eq_set_v @ A8 @ ( field_v @ R ) )
=> ( ( A8 != bot_bot_set_v )
=> ? [X2: v] :
( ( member_v @ X2 @ A8 )
& ! [Y3: v] :
( ( member_v @ Y3 @ A8 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1023_min__bot2,axiom,
! [X: set_Product_prod_v_v] :
( ( ord_mi6996445931809003310od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% min_bot2
thf(fact_1024_min__bot2,axiom,
! [X: set_v] :
( ( ord_min_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% min_bot2
thf(fact_1025_min__bot,axiom,
! [X: set_Product_prod_v_v] :
( ( ord_mi6996445931809003310od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% min_bot
thf(fact_1026_min__bot,axiom,
! [X: set_v] :
( ( ord_min_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% min_bot
thf(fact_1027_distinct__append,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( ( distin6159370996967099744od_v_v @ Xs )
& ( distin6159370996967099744od_v_v @ Ys )
& ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% distinct_append
thf(fact_1028_distinct__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( distinct_v @ ( append_v @ Xs @ Ys ) )
= ( ( distinct_v @ Xs )
& ( distinct_v @ Ys )
& ( ( inf_inf_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) )
= bot_bot_set_v ) ) ) ).
% distinct_append
thf(fact_1029_min__absorb2,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( ord_mi6996445931809003310od_v_v @ X @ Y2 )
= Y2 ) ) ).
% min_absorb2
thf(fact_1030_min__absorb2,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( ord_min_set_v @ X @ Y2 )
= Y2 ) ) ).
% min_absorb2
thf(fact_1031_min__absorb1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_mi6996445931809003310od_v_v @ X @ Y2 )
= X ) ) ).
% min_absorb1
thf(fact_1032_min__absorb1,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_min_set_v @ X @ Y2 )
= X ) ) ).
% min_absorb1
thf(fact_1033_min__def,axiom,
( ord_mi6996445931809003310od_v_v
= ( ^ [A6: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( if_set4279007504652509325od_v_v @ ( ord_le7336532860387713383od_v_v @ A6 @ B4 ) @ A6 @ B4 ) ) ) ).
% min_def
thf(fact_1034_min__def,axiom,
( ord_min_set_v
= ( ^ [A6: set_v,B4: set_v] : ( if_set_v @ ( ord_less_eq_set_v @ A6 @ B4 ) @ A6 @ B4 ) ) ) ).
% min_def
thf(fact_1035_well__order__on__domain,axiom,
! [A4: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( order_7541072052284126853od_v_v @ A4 @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ A4 )
& ( member7453568604450474000od_v_v @ B2 @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_1036_well__order__on__domain,axiom,
! [A4: set_v,R: set_Product_prod_v_v,A: v,B2: v] :
( ( order_6972113574731384241r_on_v @ A4 @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( ( member_v @ A @ A4 )
& ( member_v @ B2 @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_1037_finite__distinct__list,axiom,
! [A4: set_v] :
( ( finite_finite_v @ A4 )
=> ? [Xs2: list_v] :
( ( ( set_v2 @ Xs2 )
= A4 )
& ( distinct_v @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_1038_well__order__on__empty,axiom,
order_7541072052284126853od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% well_order_on_empty
thf(fact_1039_well__order__on__empty,axiom,
order_6972113574731384241r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% well_order_on_empty
thf(fact_1040_distinct__concat__iff,axiom,
! [Xs: list_l4795378083388841843od_v_v] :
( ( distin6159370996967099744od_v_v @ ( concat2875663619778446888od_v_v @ Xs ) )
= ( ( distin913317783593574886od_v_v @ ( remove5095778601549809401od_v_v @ nil_Product_prod_v_v @ Xs ) )
& ! [Ys2: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Ys2 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( distin6159370996967099744od_v_v @ Ys2 ) )
& ! [Ys2: list_P7986770385144383213od_v_v,Zs: list_P7986770385144383213od_v_v] :
( ( ( member4190458934886417558od_v_v @ Ys2 @ ( set_li2340707408155270402od_v_v @ Xs ) )
& ( member4190458934886417558od_v_v @ Zs @ ( set_li2340707408155270402od_v_v @ Xs ) )
& ( Ys2 != Zs ) )
=> ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Ys2 ) @ ( set_Product_prod_v_v2 @ Zs ) )
= bot_bo723834152578015283od_v_v ) ) ) ) ).
% distinct_concat_iff
thf(fact_1041_distinct__concat__iff,axiom,
! [Xs: list_list_v] :
( ( distinct_v @ ( concat_v @ Xs ) )
= ( ( distinct_list_v @ ( removeAll_list_v @ nil_v @ Xs ) )
& ! [Ys2: list_v] :
( ( member_list_v @ Ys2 @ ( set_list_v2 @ Xs ) )
=> ( distinct_v @ Ys2 ) )
& ! [Ys2: list_v,Zs: list_v] :
( ( ( member_list_v @ Ys2 @ ( set_list_v2 @ Xs ) )
& ( member_list_v @ Zs @ ( set_list_v2 @ Xs ) )
& ( Ys2 != Zs ) )
=> ( ( inf_inf_set_v @ ( set_v2 @ Ys2 ) @ ( set_v2 @ Zs ) )
= bot_bot_set_v ) ) ) ) ).
% distinct_concat_iff
thf(fact_1042_underS__incl__iff,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( order_5211820470575790509od_v_v @ R @ B2 ) )
= ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1043_underS__incl__iff,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( member_v @ B2 @ ( field_v @ R ) )
=> ( ( ord_less_eq_set_v @ ( order_underS_v @ R @ A ) @ ( order_underS_v @ R @ B2 ) )
= ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1044_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I2: v,R3: set_Product_prod_v_v,J2: v] :
( ( member_v @ I2 @ ( order_underS_v @ R3 @ J2 ) )
=> ( member_v @ I2 @ ( field_v @ R3 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_1045_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I2: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v,J2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ I2 @ ( order_5211820470575790509od_v_v @ R3 @ J2 ) )
=> ( member7453568604450474000od_v_v @ I2 @ ( field_7153129647634986036od_v_v @ R3 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_1046_underS__E,axiom,
! [I2: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v,J2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ I2 @ ( order_5211820470575790509od_v_v @ R3 @ J2 ) )
=> ( ( I2 != J2 )
& ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I2 @ J2 ) @ R3 ) ) ) ).
% underS_E
thf(fact_1047_underS__E,axiom,
! [I2: v,R3: set_Product_prod_v_v,J2: v] :
( ( member_v @ I2 @ ( order_underS_v @ R3 @ J2 ) )
=> ( ( I2 != J2 )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I2 @ J2 ) @ R3 ) ) ) ).
% underS_E
thf(fact_1048_underS__I,axiom,
! [I2: product_prod_v_v,J2: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
( ( I2 != J2 )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I2 @ J2 ) @ R3 )
=> ( member7453568604450474000od_v_v @ I2 @ ( order_5211820470575790509od_v_v @ R3 @ J2 ) ) ) ) ).
% underS_I
thf(fact_1049_underS__I,axiom,
! [I2: v,J2: v,R3: set_Product_prod_v_v] :
( ( I2 != J2 )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I2 @ J2 ) @ R3 )
=> ( member_v @ I2 @ ( order_underS_v @ R3 @ J2 ) ) ) ) ).
% underS_I
thf(fact_1050_underS__empty,axiom,
! [A: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( order_5211820470575790509od_v_v @ R @ A )
= bot_bo723834152578015283od_v_v ) ) ).
% underS_empty
thf(fact_1051_underS__empty,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ~ ( member_v @ A @ ( field_v @ R ) )
=> ( ( order_underS_v @ R @ A )
= bot_bot_set_v ) ) ).
% underS_empty
thf(fact_1052_Order__Relation_OunderS__Field,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( field_7153129647634986036od_v_v @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_1053_Order__Relation_OunderS__Field,axiom,
! [R: set_Product_prod_v_v,A: v] : ( ord_less_eq_set_v @ ( order_underS_v @ R @ A ) @ ( field_v @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_1054_distinct__concat,axiom,
! [Xs: list_l4795378083388841843od_v_v] :
( ( distin913317783593574886od_v_v @ Xs )
=> ( ! [Ys3: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Ys3 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( distin6159370996967099744od_v_v @ Ys3 ) )
=> ( ! [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Ys3 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( ( member4190458934886417558od_v_v @ Zs2 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Ys3 ) @ ( set_Product_prod_v_v2 @ Zs2 ) )
= bot_bo723834152578015283od_v_v ) ) ) )
=> ( distin6159370996967099744od_v_v @ ( concat2875663619778446888od_v_v @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1055_distinct__concat,axiom,
! [Xs: list_list_v] :
( ( distinct_list_v @ Xs )
=> ( ! [Ys3: list_v] :
( ( member_list_v @ Ys3 @ ( set_list_v2 @ Xs ) )
=> ( distinct_v @ Ys3 ) )
=> ( ! [Ys3: list_v,Zs2: list_v] :
( ( member_list_v @ Ys3 @ ( set_list_v2 @ Xs ) )
=> ( ( member_list_v @ Zs2 @ ( set_list_v2 @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_inf_set_v @ ( set_v2 @ Ys3 ) @ ( set_v2 @ Zs2 ) )
= bot_bot_set_v ) ) ) )
=> ( distinct_v @ ( concat_v @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1056_Refl__under__underS,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( order_6892855479609198156od_v_v @ R @ A )
= ( sup_su414716646722978715od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1057_Refl__under__underS,axiom,
! [R: set_Product_prod_v_v,A: v] :
( ( refl_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( order_under_v @ R @ A )
= ( sup_sup_set_v @ ( order_underS_v @ R @ A ) @ ( insert_v2 @ A @ bot_bot_set_v ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1058_Range__insert,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v] :
( ( range_v_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R ) )
= ( insert_v2 @ B2 @ ( range_v_v @ R ) ) ) ).
% Range_insert
thf(fact_1059_Range__empty,axiom,
( ( range_v_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Range_empty
thf(fact_1060_Range__iff,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( range_v_v @ R ) )
= ( ? [Y3: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ A ) @ R ) ) ) ).
% Range_iff
thf(fact_1061_RangeE,axiom,
! [B2: v,R: set_Product_prod_v_v] :
( ( member_v @ B2 @ ( range_v_v @ R ) )
=> ~ ! [A7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ B2 ) @ R ) ) ).
% RangeE
thf(fact_1062_Range_Ointros,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( member_v @ B2 @ ( range_v_v @ R ) ) ) ).
% Range.intros
thf(fact_1063_Range_Osimps,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( range_v_v @ R ) )
= ( ? [A6: v,B4: v] :
( ( A = B4 )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A6 @ B4 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_1064_Range_Ocases,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( range_v_v @ R ) )
=> ~ ! [A7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ A ) @ R ) ) ).
% Range.cases
thf(fact_1065_Range__mono,axiom,
! [R: set_Product_prod_v_v,S2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S2 )
=> ( ord_less_eq_set_v @ ( range_v_v @ R ) @ ( range_v_v @ S2 ) ) ) ).
% Range_mono
thf(fact_1066_Range__empty__iff,axiom,
! [R: set_Product_prod_v_v] :
( ( ( range_v_v @ R )
= bot_bot_set_v )
= ( R = bot_bo723834152578015283od_v_v ) ) ).
% Range_empty_iff
thf(fact_1067_under__Field,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_6892855479609198156od_v_v @ R @ A ) @ ( field_7153129647634986036od_v_v @ R ) ) ).
% under_Field
thf(fact_1068_under__Field,axiom,
! [R: set_Product_prod_v_v,A: v] : ( ord_less_eq_set_v @ ( order_under_v @ R @ A ) @ ( field_v @ R ) ) ).
% under_Field
thf(fact_1069_Range__Un__eq,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( range_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B ) )
= ( sup_sup_set_v @ ( range_v_v @ A4 ) @ ( range_v_v @ B ) ) ) ).
% Range_Un_eq
thf(fact_1070_underS__subset__under,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( order_6892855479609198156od_v_v @ R @ A ) ) ).
% underS_subset_under
thf(fact_1071_underS__subset__under,axiom,
! [R: set_Product_prod_v_v,A: v] : ( ord_less_eq_set_v @ ( order_underS_v @ R @ A ) @ ( order_under_v @ R @ A ) ) ).
% underS_subset_under
thf(fact_1072_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_v,X: v,Ys: list_v,Y2: v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ Xs @ ( cons_v @ X @ nil_v ) ) @ ( append_v @ Ys @ ( cons_v @ Y2 @ nil_v ) ) ) @ ( listrel1_v @ R ) )
= ( ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R ) )
& ( X = Y2 ) )
| ( ( Xs = Ys )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1073_distinct__disjoint__shuffles,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ Xs )
=> ( ( distin6159370996967099744od_v_v @ Ys )
=> ( ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) )
= bot_bo723834152578015283od_v_v )
=> ( ( member4190458934886417558od_v_v @ Zs3 @ ( shuffl71542398924059522od_v_v @ Xs @ Ys ) )
=> ( distin6159370996967099744od_v_v @ Zs3 ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_1074_distinct__disjoint__shuffles,axiom,
! [Xs: list_v,Ys: list_v,Zs3: list_v] :
( ( distinct_v @ Xs )
=> ( ( distinct_v @ Ys )
=> ( ( ( inf_inf_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) )
= bot_bot_set_v )
=> ( ( member_list_v @ Zs3 @ ( shuffles_v @ Xs @ Ys ) )
=> ( distinct_v @ Zs3 ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_1075_Domain__insert,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v] :
( ( domain_v_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R ) )
= ( insert_v2 @ A @ ( domain_v_v @ R ) ) ) ).
% Domain_insert
thf(fact_1076_Domain__empty,axiom,
( ( domain_v_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Domain_empty
thf(fact_1077_Cons__listrel1__Cons,axiom,
! [X: v,Xs: list_v,Y2: v,Ys: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ X @ Xs ) @ ( cons_v @ Y2 @ Ys ) ) @ ( listrel1_v @ R ) )
= ( ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y2 )
& ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_1078_shuffles_Osimps_I2_J,axiom,
! [Xs: list_v] :
( ( shuffles_v @ Xs @ nil_v )
= ( insert_list_v @ Xs @ bot_bot_set_list_v ) ) ).
% shuffles.simps(2)
thf(fact_1079_shuffles_Osimps_I1_J,axiom,
! [Ys: list_v] :
( ( shuffles_v @ nil_v @ Ys )
= ( insert_list_v @ Ys @ bot_bot_set_list_v ) ) ).
% shuffles.simps(1)
thf(fact_1080_listrel1__mono,axiom,
! [R: set_Product_prod_v_v,S2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S2 )
=> ( ord_le791731619978752231list_v @ ( listrel1_v @ R ) @ ( listrel1_v @ S2 ) ) ) ).
% listrel1_mono
thf(fact_1081_Domain_Ocases,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( domain_v_v @ R ) )
=> ~ ! [B5: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B5 ) @ R ) ) ).
% Domain.cases
thf(fact_1082_Domain_Osimps,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( domain_v_v @ R ) )
= ( ? [A6: v,B4: v] :
( ( A = A6 )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A6 @ B4 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_1083_Domain_ODomainI,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( member_v @ A @ ( domain_v_v @ R ) ) ) ).
% Domain.DomainI
thf(fact_1084_DomainE,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( domain_v_v @ R ) )
=> ~ ! [B5: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B5 ) @ R ) ) ).
% DomainE
thf(fact_1085_Domain__iff,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v @ A @ ( domain_v_v @ R ) )
= ( ? [Y3: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ Y3 ) @ R ) ) ) ).
% Domain_iff
thf(fact_1086_listrel1I1,axiom,
! [X: v,Y2: v,R: set_Product_prod_v_v,Xs: list_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ X @ Xs ) @ ( cons_v @ Y2 @ Xs ) ) @ ( listrel1_v @ R ) ) ) ).
% listrel1I1
thf(fact_1087_Cons__listrel1E1,axiom,
! [X: v,Xs: list_v,Ys: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ X @ Xs ) @ Ys ) @ ( listrel1_v @ R ) )
=> ( ! [Y: v] :
( ( Ys
= ( cons_v @ Y @ Xs ) )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R ) )
=> ~ ! [Zs2: list_v] :
( ( Ys
= ( cons_v @ X @ Zs2 ) )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Zs2 ) @ ( listrel1_v @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_1088_Cons__listrel1E2,axiom,
! [Xs: list_v,Y2: v,Ys: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ ( cons_v @ Y2 @ Ys ) ) @ ( listrel1_v @ R ) )
=> ( ! [X3: v] :
( ( Xs
= ( cons_v @ X3 @ Ys ) )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y2 ) @ R ) )
=> ~ ! [Zs2: list_v] :
( ( Xs
= ( cons_v @ Y2 @ Zs2 ) )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Zs2 @ Ys ) @ ( listrel1_v @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_1089_Domain__mono,axiom,
! [R: set_Product_prod_v_v,S2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S2 )
=> ( ord_less_eq_set_v @ ( domain_v_v @ R ) @ ( domain_v_v @ S2 ) ) ) ).
% Domain_mono
thf(fact_1090_Domain__empty__iff,axiom,
! [R: set_Product_prod_v_v] :
( ( ( domain_v_v @ R )
= bot_bot_set_v )
= ( R = bot_bo723834152578015283od_v_v ) ) ).
% Domain_empty_iff
thf(fact_1091_Domain__Un__eq,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( domain_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B ) )
= ( sup_sup_set_v @ ( domain_v_v @ A4 ) @ ( domain_v_v @ B ) ) ) ).
% Domain_Un_eq
thf(fact_1092_listrel1E,axiom,
! [Xs: list_v,Ys: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R ) )
=> ~ ! [X3: v,Y: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R )
=> ! [Us: list_v,Vs: list_v] :
( ( Xs
= ( append_v @ Us @ ( cons_v @ X3 @ Vs ) ) )
=> ( Ys
!= ( append_v @ Us @ ( cons_v @ Y @ Vs ) ) ) ) ) ) ).
% listrel1E
thf(fact_1093_listrel1I,axiom,
! [X: v,Y2: v,R: set_Product_prod_v_v,Xs: list_v,Us2: list_v,Vs2: list_v,Ys: list_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ R )
=> ( ( Xs
= ( append_v @ Us2 @ ( cons_v @ X @ Vs2 ) ) )
=> ( ( Ys
= ( append_v @ Us2 @ ( cons_v @ Y2 @ Vs2 ) ) )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( listrel1_v @ R ) ) ) ) ) ).
% listrel1I
thf(fact_1094_Field__def,axiom,
( field_7153129647634986036od_v_v
= ( ^ [R2: set_Pr2149350503807050951od_v_v] : ( sup_su414716646722978715od_v_v @ ( domain6359000466948879308od_v_v @ R2 ) @ ( range_7878975032137371189od_v_v @ R2 ) ) ) ) ).
% Field_def
thf(fact_1095_Field__def,axiom,
( field_v
= ( ^ [R2: set_Product_prod_v_v] : ( sup_sup_set_v @ ( domain_v_v @ R2 ) @ ( range_v_v @ R2 ) ) ) ) ).
% Field_def
thf(fact_1096_Linear__order__in__diff__Id,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R )
= ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ A ) @ ( minus_5255927943254941998od_v_v @ R @ id_Product_prod_v_v ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1097_Linear__order__in__diff__Id,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( member_v @ B2 @ ( field_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
= ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ A ) @ ( minus_4183494784930505774od_v_v @ R @ id_v ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1098_lexord__same__pref__iff,axiom,
! [Xs: list_v,Ys: list_v,Zs3: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ Xs @ Ys ) @ ( append_v @ Xs @ Zs3 ) ) @ ( lexord_v @ R ) )
= ( ? [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ X2 ) @ R ) )
| ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Ys @ Zs3 ) @ ( lexord_v @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_1099_IdI,axiom,
! [A: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ A ) @ id_v ) ).
% IdI
thf(fact_1100_pair__in__Id__conv,axiom,
! [A: v,B2: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ id_v )
= ( A = B2 ) ) ).
% pair_in_Id_conv
thf(fact_1101_lexord__cons__cons,axiom,
! [A: v,X: list_v,B2: v,Y2: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( cons_v @ A @ X ) @ ( cons_v @ B2 @ Y2 ) ) @ ( lexord_v @ R ) )
= ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
| ( ( A = B2 )
& ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ X @ Y2 ) @ ( lexord_v @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_1102_IdE,axiom,
! [P2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ P2 @ id_v )
=> ~ ! [X3: v] :
( P2
!= ( product_Pair_v_v @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_1103_lexord__irreflexive,axiom,
! [R: set_Product_prod_v_v,Xs: list_v] :
( ! [X3: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ X3 ) @ R )
=> ~ ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Xs ) @ ( lexord_v @ R ) ) ) ).
% lexord_irreflexive
thf(fact_1104_lexord__linear,axiom,
! [R: set_Product_prod_v_v,X: list_v,Y2: list_v] :
( ! [A7: v,B5: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ B5 ) @ R )
| ( A7 = B5 )
| ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B5 @ A7 ) @ R ) )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ X @ Y2 ) @ ( lexord_v @ R ) )
| ( X = Y2 )
| ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Y2 @ X ) @ ( lexord_v @ R ) ) ) ) ).
% lexord_linear
thf(fact_1105_lexord__partial__trans,axiom,
! [Xs: list_P7986770385144383213od_v_v,R: set_Pr2149350503807050951od_v_v,Ys: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ! [X3: product_prod_v_v,Y: product_prod_v_v,Z3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Y ) @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ Z3 ) @ R )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Z3 ) @ R ) ) ) )
=> ( ( member6382463057129219728od_v_v @ ( produc674067373767953879od_v_v @ Xs @ Ys ) @ ( lexord8601710409828808922od_v_v @ R ) )
=> ( ( member6382463057129219728od_v_v @ ( produc674067373767953879od_v_v @ Ys @ Zs3 ) @ ( lexord8601710409828808922od_v_v @ R ) )
=> ( member6382463057129219728od_v_v @ ( produc674067373767953879od_v_v @ Xs @ Zs3 ) @ ( lexord8601710409828808922od_v_v @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_1106_lexord__partial__trans,axiom,
! [Xs: list_v,R: set_Product_prod_v_v,Ys: list_v,Zs3: list_v] :
( ! [X3: v,Y: v,Z3: v] :
( ( member_v @ X3 @ ( set_v2 @ Xs ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z3 ) @ R )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Z3 ) @ R ) ) ) )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Ys ) @ ( lexord_v @ R ) )
=> ( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Ys @ Zs3 ) @ ( lexord_v @ R ) )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ Xs @ Zs3 ) @ ( lexord_v @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_1107_lexord__append__leftD,axiom,
! [X: list_v,U: list_v,V: list_v,R: set_Product_prod_v_v] :
( ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ X @ U ) @ ( append_v @ X @ V ) ) @ ( lexord_v @ R ) )
=> ( ! [A7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ A7 ) @ R )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ U @ V ) @ ( lexord_v @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_1108_lexord__append__left__rightI,axiom,
! [A: v,B2: v,R: set_Product_prod_v_v,U: list_v,X: list_v,Y2: list_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( member418487059593946000list_v @ ( produc6795410681906604247list_v @ ( append_v @ U @ ( cons_v @ A @ X ) ) @ ( append_v @ U @ ( cons_v @ B2 @ Y2 ) ) ) @ ( lexord_v @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_1109_listset_Osimps_I1_J,axiom,
( ( listset_v @ nil_set_v )
= ( insert_list_v @ nil_v @ bot_bot_set_list_v ) ) ).
% listset.simps(1)
thf(fact_1110_Total__subset__Id,axiom,
! [R: set_Product_prod_v_v] :
( ( total_on_v @ ( field_v @ R ) @ R )
=> ( ( ord_le7336532860387713383od_v_v @ R @ id_v )
=> ( ( R = bot_bo723834152578015283od_v_v )
| ? [A7: v] :
( R
= ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A7 @ A7 ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1111_total__on__subset,axiom,
! [A4: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
( ( total_9075964390993782123od_v_v @ A4 @ R )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A4 )
=> ( total_9075964390993782123od_v_v @ B @ R ) ) ) ).
% total_on_subset
thf(fact_1112_total__on__subset,axiom,
! [A4: set_v,R: set_Product_prod_v_v,B: set_v] :
( ( total_on_v @ A4 @ R )
=> ( ( ord_less_eq_set_v @ B @ A4 )
=> ( total_on_v @ B @ R ) ) ) ).
% total_on_subset
thf(fact_1113_total__on__empty,axiom,
! [R: set_Pr2149350503807050951od_v_v] : ( total_9075964390993782123od_v_v @ bot_bo723834152578015283od_v_v @ R ) ).
% total_on_empty
thf(fact_1114_total__on__empty,axiom,
! [R: set_Product_prod_v_v] : ( total_on_v @ bot_bot_set_v @ R ) ).
% total_on_empty
thf(fact_1115_total__on__def,axiom,
( total_on_v
= ( ^ [A8: set_v,R2: set_Product_prod_v_v] :
! [X2: v] :
( ( member_v @ X2 @ A8 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ A8 )
=> ( ( X2 != Y3 )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ R2 )
| ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ X2 ) @ R2 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_1116_total__onI,axiom,
! [A4: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ! [X3: product_prod_v_v,Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ( ( member7453568604450474000od_v_v @ Y @ A4 )
=> ( ( X3 != Y )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X3 @ Y ) @ R )
| ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ X3 ) @ R ) ) ) ) )
=> ( total_9075964390993782123od_v_v @ A4 @ R ) ) ).
% total_onI
thf(fact_1117_total__onI,axiom,
! [A4: set_v,R: set_Product_prod_v_v] :
( ! [X3: v,Y: v] :
( ( member_v @ X3 @ A4 )
=> ( ( member_v @ Y @ A4 )
=> ( ( X3 != Y )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R )
| ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ X3 ) @ R ) ) ) ) )
=> ( total_on_v @ A4 @ R ) ) ).
% total_onI
thf(fact_1118_total__on__singleton,axiom,
! [X: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] : ( total_9075964390993782123od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) @ R ) ).
% total_on_singleton
thf(fact_1119_total__on__singleton,axiom,
! [X: v,R: set_Product_prod_v_v] : ( total_on_v @ ( insert_v2 @ X @ bot_bot_set_v ) @ R ) ).
% total_on_singleton
thf(fact_1120_Total__Id__Field,axiom,
! [R: set_Product_prod_v_v] :
( ( total_on_v @ ( field_v @ R ) @ R )
=> ( ~ ( ord_le7336532860387713383od_v_v @ R @ id_v )
=> ( ( field_v @ R )
= ( field_v @ ( minus_4183494784930505774od_v_v @ R @ id_v ) ) ) ) ) ).
% Total_Id_Field
thf(fact_1121_IdD,axiom,
! [A: v,B2: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ id_v )
=> ( A = B2 ) ) ).
% IdD
thf(fact_1122_Linear__order__wf__diff__Id,axiom,
! [R: set_Pr2149350503807050951od_v_v] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( wf_Product_prod_v_v @ ( minus_5255927943254941998od_v_v @ R @ id_Product_prod_v_v ) )
= ( ! [A8: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A8 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( A8 != bot_bo723834152578015283od_v_v )
=> ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A8 )
& ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ A8 )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1123_Linear__order__wf__diff__Id,axiom,
! [R: set_Product_prod_v_v] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
=> ( ( wf_v @ ( minus_4183494784930505774od_v_v @ R @ id_v ) )
= ( ! [A8: set_v] :
( ( ord_less_eq_set_v @ A8 @ ( field_v @ R ) )
=> ( ( A8 != bot_bot_set_v )
=> ? [X2: v] :
( ( member_v @ X2 @ A8 )
& ! [Y3: v] :
( ( member_v @ Y3 @ A8 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1124_wf__eq__minimal2,axiom,
( wf_Product_prod_v_v
= ( ^ [R2: set_Pr2149350503807050951od_v_v] :
! [A8: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ A8 @ ( field_7153129647634986036od_v_v @ R2 ) )
& ( A8 != bot_bo723834152578015283od_v_v ) )
=> ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A8 )
& ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ A8 )
=> ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ X2 ) @ R2 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1125_wf__eq__minimal2,axiom,
( wf_v
= ( ^ [R2: set_Product_prod_v_v] :
! [A8: set_v] :
( ( ( ord_less_eq_set_v @ A8 @ ( field_v @ R2 ) )
& ( A8 != bot_bot_set_v ) )
=> ? [X2: v] :
( ( member_v @ X2 @ A8 )
& ! [Y3: v] :
( ( member_v @ Y3 @ A8 )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ X2 ) @ R2 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_1126_wf__Un,axiom,
! [R: set_Pr2149350503807050951od_v_v,S2: set_Pr2149350503807050951od_v_v] :
( ( wf_Product_prod_v_v @ R )
=> ( ( wf_Product_prod_v_v @ S2 )
=> ( ( ( inf_in6271465464967711157od_v_v @ ( domain6359000466948879308od_v_v @ R ) @ ( range_7878975032137371189od_v_v @ S2 ) )
= bot_bo723834152578015283od_v_v )
=> ( wf_Product_prod_v_v @ ( sup_su1742609618068805275od_v_v @ R @ S2 ) ) ) ) ) ).
% wf_Un
thf(fact_1127_wf__Un,axiom,
! [R: set_Product_prod_v_v,S2: set_Product_prod_v_v] :
( ( wf_v @ R )
=> ( ( wf_v @ S2 )
=> ( ( ( inf_inf_set_v @ ( domain_v_v @ R ) @ ( range_v_v @ S2 ) )
= bot_bot_set_v )
=> ( wf_v @ ( sup_su414716646722978715od_v_v @ R @ S2 ) ) ) ) ) ).
% wf_Un
thf(fact_1128_wf__empty,axiom,
wf_v @ bot_bo723834152578015283od_v_v ).
% wf_empty
thf(fact_1129_wf__def,axiom,
( wf_v
= ( ^ [R2: set_Product_prod_v_v] :
! [P3: v > $o] :
( ! [X2: v] :
( ! [Y3: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ X2 ) @ R2 )
=> ( P3 @ Y3 ) )
=> ( P3 @ X2 ) )
=> ! [X6: v] : ( P3 @ X6 ) ) ) ) ).
% wf_def
thf(fact_1130_wfE__min,axiom,
! [R3: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,Q: set_Product_prod_v_v] :
( ( wf_Product_prod_v_v @ R3 )
=> ( ( member7453568604450474000od_v_v @ X @ Q )
=> ~ ! [Z3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Z3 @ Q )
=> ~ ! [Y5: product_prod_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y5 @ Z3 ) @ R3 )
=> ~ ( member7453568604450474000od_v_v @ Y5 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1131_wfE__min,axiom,
! [R3: set_Product_prod_v_v,X: v,Q: set_v] :
( ( wf_v @ R3 )
=> ( ( member_v @ X @ Q )
=> ~ ! [Z3: v] :
( ( member_v @ Z3 @ Q )
=> ~ ! [Y5: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ Z3 ) @ R3 )
=> ~ ( member_v @ Y5 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_1132_wfI__min,axiom,
! [R3: set_Pr2149350503807050951od_v_v] :
( ! [X3: product_prod_v_v,Q2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ Q2 )
=> ? [Xa2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Xa2 @ Q2 )
& ! [Y: product_prod_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ Xa2 ) @ R3 )
=> ~ ( member7453568604450474000od_v_v @ Y @ Q2 ) ) ) )
=> ( wf_Product_prod_v_v @ R3 ) ) ).
% wfI_min
thf(fact_1133_wfI__min,axiom,
! [R3: set_Product_prod_v_v] :
( ! [X3: v,Q2: set_v] :
( ( member_v @ X3 @ Q2 )
=> ? [Xa2: v] :
( ( member_v @ Xa2 @ Q2 )
& ! [Y: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Xa2 ) @ R3 )
=> ~ ( member_v @ Y @ Q2 ) ) ) )
=> ( wf_v @ R3 ) ) ).
% wfI_min
thf(fact_1134_wfUNIVI,axiom,
! [R: set_Product_prod_v_v] :
( ! [P4: v > $o,X3: v] :
( ! [Xa2: v] :
( ! [Y: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Xa2 ) @ R )
=> ( P4 @ Y ) )
=> ( P4 @ Xa2 ) )
=> ( P4 @ X3 ) )
=> ( wf_v @ R ) ) ).
% wfUNIVI
thf(fact_1135_wf__asym,axiom,
! [R: set_Product_prod_v_v,A: v,X: v] :
( ( wf_v @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ X ) @ R )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ A ) @ R ) ) ) ).
% wf_asym
thf(fact_1136_wf__induct,axiom,
! [R: set_Product_prod_v_v,P: v > $o,A: v] :
( ( wf_v @ R )
=> ( ! [X3: v] :
( ! [Y5: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ X3 ) @ R )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ) ).
% wf_induct
thf(fact_1137_wf__irrefl,axiom,
! [R: set_Product_prod_v_v,A: v] :
( ( wf_v @ R )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ A ) @ R ) ) ).
% wf_irrefl
thf(fact_1138_wf__not__sym,axiom,
! [R: set_Product_prod_v_v,A: v,X: v] :
( ( wf_v @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ X ) @ R )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ A ) @ R ) ) ) ).
% wf_not_sym
thf(fact_1139_wf__not__refl,axiom,
! [R: set_Product_prod_v_v,A: v] :
( ( wf_v @ R )
=> ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ A ) @ R ) ) ).
% wf_not_refl
thf(fact_1140_wf__eq__minimal,axiom,
( wf_Product_prod_v_v
= ( ^ [R2: set_Pr2149350503807050951od_v_v] :
! [Q3: set_Product_prod_v_v] :
( ? [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ Q3 )
=> ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ Q3 )
& ! [Y3: product_prod_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ X2 ) @ R2 )
=> ~ ( member7453568604450474000od_v_v @ Y3 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1141_wf__eq__minimal,axiom,
( wf_v
= ( ^ [R2: set_Product_prod_v_v] :
! [Q3: set_v] :
( ? [X2: v] : ( member_v @ X2 @ Q3 )
=> ? [X2: v] :
( ( member_v @ X2 @ Q3 )
& ! [Y3: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ X2 ) @ R2 )
=> ~ ( member_v @ Y3 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_1142_wf__induct__rule,axiom,
! [R: set_Product_prod_v_v,P: v > $o,A: v] :
( ( wf_v @ R )
=> ( ! [X3: v] :
( ! [Y5: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ X3 ) @ R )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ) ).
% wf_induct_rule
thf(fact_1143_wf__if__convertible__to__wf,axiom,
! [S2: set_Product_prod_v_v,R: set_Product_prod_v_v,F: v > v] :
( ( wf_v @ S2 )
=> ( ! [X3: v,Y: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ ( F @ X3 ) @ ( F @ Y ) ) @ S2 ) )
=> ( wf_v @ R ) ) ) ).
% wf_if_convertible_to_wf
thf(fact_1144_wf__subset,axiom,
! [R: set_Product_prod_v_v,P2: set_Product_prod_v_v] :
( ( wf_v @ R )
=> ( ( ord_le7336532860387713383od_v_v @ P2 @ R )
=> ( wf_v @ P2 ) ) ) ).
% wf_subset
thf(fact_1145_wfE__min_H,axiom,
! [R3: set_Pr2149350503807050951od_v_v,Q: set_Product_prod_v_v] :
( ( wf_Product_prod_v_v @ R3 )
=> ( ( Q != bot_bo723834152578015283od_v_v )
=> ~ ! [Z3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Z3 @ Q )
=> ~ ! [Y5: product_prod_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y5 @ Z3 ) @ R3 )
=> ~ ( member7453568604450474000od_v_v @ Y5 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1146_wfE__min_H,axiom,
! [R3: set_Product_prod_v_v,Q: set_v] :
( ( wf_v @ R3 )
=> ( ( Q != bot_bot_set_v )
=> ~ ! [Z3: v] :
( ( member_v @ Z3 @ Q )
=> ~ ! [Y5: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ Z3 ) @ R3 )
=> ~ ( member_v @ Y5 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_1147_lists__empty,axiom,
( ( lists_5865669170805476827od_v_v @ bot_bo723834152578015283od_v_v )
= ( insert4087971119735676093od_v_v @ nil_Product_prod_v_v @ bot_bo54012148666785209od_v_v ) ) ).
% lists_empty
thf(fact_1148_lists__empty,axiom,
( ( lists_v @ bot_bot_set_v )
= ( insert_list_v @ nil_v @ bot_bot_set_list_v ) ) ).
% lists_empty
thf(fact_1149_lists__Int__eq,axiom,
! [A4: set_v,B: set_v] :
( ( lists_v @ ( inf_inf_set_v @ A4 @ B ) )
= ( inf_inf_set_list_v @ ( lists_v @ A4 ) @ ( lists_v @ B ) ) ) ).
% lists_Int_eq
thf(fact_1150_lists__IntI,axiom,
! [L2: list_v,A4: set_v,B: set_v] :
( ( member_list_v @ L2 @ ( lists_v @ A4 ) )
=> ( ( member_list_v @ L2 @ ( lists_v @ B ) )
=> ( member_list_v @ L2 @ ( lists_v @ ( inf_inf_set_v @ A4 @ B ) ) ) ) ) ).
% lists_IntI
thf(fact_1151_lists__mono,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ord_le5393391283775026413od_v_v @ ( lists_5865669170805476827od_v_v @ A4 ) @ ( lists_5865669170805476827od_v_v @ B ) ) ) ).
% lists_mono
thf(fact_1152_lists__mono,axiom,
! [A4: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ord_le1129530298279361049list_v @ ( lists_v @ A4 ) @ ( lists_v @ B ) ) ) ).
% lists_mono
thf(fact_1153_Zorns__po__lemma,axiom,
! [R: set_Product_prod_v_v] :
( ( order_5272072345360262664r_on_v @ ( field_v @ R ) @ R )
=> ( ! [C4: set_v] :
( ( member_set_v @ C4 @ ( chains_v @ R ) )
=> ? [X4: v] :
( ( member_v @ X4 @ ( field_v @ R ) )
& ! [Xa: v] :
( ( member_v @ Xa @ C4 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Xa @ X4 ) @ R ) ) ) )
=> ? [X3: v] :
( ( member_v @ X3 @ ( field_v @ R ) )
& ! [Xa2: v] :
( ( member_v @ Xa2 @ ( field_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Xa2 ) @ R )
=> ( Xa2 = X3 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_1154_wo__rel_Ocases__Total3,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v,Phi: product_prod_v_v > product_prod_v_v > $o] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ ( minus_5255927943254941998od_v_v @ R @ id_Product_prod_v_v ) )
| ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ A ) @ ( minus_5255927943254941998od_v_v @ R @ id_Product_prod_v_v ) ) )
=> ( Phi @ A @ B2 ) )
=> ( ( ( A = B2 )
=> ( Phi @ A @ B2 ) )
=> ( Phi @ A @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1155_wo__rel_Ocases__Total3,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v,Phi: v > v > $o] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( ord_less_eq_set_v @ ( insert_v2 @ A @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) @ ( field_v @ R ) )
=> ( ( ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ ( minus_4183494784930505774od_v_v @ R @ id_v ) )
| ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ A ) @ ( minus_4183494784930505774od_v_v @ R @ id_v ) ) )
=> ( Phi @ A @ B2 ) )
=> ( ( ( A = B2 )
=> ( Phi @ A @ B2 ) )
=> ( Phi @ A @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_1156_mono__Chains,axiom,
! [R: set_Product_prod_v_v,S2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S2 )
=> ( ord_le5216385588623774835_set_v @ ( chains_v @ R ) @ ( chains_v @ S2 ) ) ) ).
% mono_Chains
thf(fact_1157_well__order__induct__imp,axiom,
! [R: set_Pr2149350503807050951od_v_v,P: product_prod_v_v > $o,A: product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ! [X3: product_prod_v_v] :
( ! [Y5: product_prod_v_v] :
( ( ( Y5 != X3 )
& ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y5 @ X3 ) @ R ) )
=> ( ( member7453568604450474000od_v_v @ Y5 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( P @ Y5 ) ) )
=> ( ( member7453568604450474000od_v_v @ X3 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( P @ X3 ) ) )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( P @ A ) ) ) ) ).
% well_order_induct_imp
thf(fact_1158_well__order__induct__imp,axiom,
! [R: set_Product_prod_v_v,P: v > $o,A: v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ! [X3: v] :
( ! [Y5: v] :
( ( ( Y5 != X3 )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ X3 ) @ R ) )
=> ( ( member_v @ Y5 @ ( field_v @ R ) )
=> ( P @ Y5 ) ) )
=> ( ( member_v @ X3 @ ( field_v @ R ) )
=> ( P @ X3 ) ) )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( P @ A ) ) ) ) ).
% well_order_induct_imp
thf(fact_1159_wo__rel_Owell__order__induct,axiom,
! [R: set_Product_prod_v_v,P: v > $o,A: v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ! [X3: v] :
( ! [Y5: v] :
( ( ( Y5 != X3 )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y5 @ X3 ) @ R ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ) ).
% wo_rel.well_order_induct
thf(fact_1160_wo__rel_Oin__notinI,axiom,
! [R: set_Pr2149350503807050951od_v_v,J2: product_prod_v_v,I2: product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ J2 @ I2 ) @ R )
| ( J2 = I2 ) )
=> ( ( member7453568604450474000od_v_v @ I2 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ J2 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I2 @ J2 ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1161_wo__rel_Oin__notinI,axiom,
! [R: set_Product_prod_v_v,J2: v,I2: v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ J2 @ I2 ) @ R )
| ( J2 = I2 ) )
=> ( ( member_v @ I2 @ ( field_v @ R ) )
=> ( ( member_v @ J2 @ ( field_v @ R ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I2 @ J2 ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1162_wo__rel_OTOTALS,axiom,
! [R: set_Product_prod_v_v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ! [X4: v] :
( ( member_v @ X4 @ ( field_v @ R ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( field_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X4 @ Xa2 ) @ R )
| ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Xa2 @ X4 ) @ R ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_1163_wo__rel_Ocases__Total,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v,Phi: product_prod_v_v > product_prod_v_v > $o] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R )
=> ( Phi @ A @ B2 ) )
=> ( ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ A ) @ R )
=> ( Phi @ A @ B2 ) )
=> ( Phi @ A @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1164_wo__rel_Ocases__Total,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v,Phi: v > v > $o] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( ord_less_eq_set_v @ ( insert_v2 @ A @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) @ ( field_v @ R ) )
=> ( ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( Phi @ A @ B2 ) )
=> ( ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ A ) @ R )
=> ( Phi @ A @ B2 ) )
=> ( Phi @ A @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1165_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( B != bot_bo723834152578015283od_v_v )
=> ? [X_1: product_prod_v_v] : ( bNF_We6235008509051751325od_v_v @ R @ B @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_1166_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Product_prod_v_v,B: set_v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
=> ( ( B != bot_bot_set_v )
=> ? [X_1: v] : ( bNF_We6697304935525757641inim_v @ R @ B @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_1167_wo__rel_Ominim__in,axiom,
! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( B != bot_bo723834152578015283od_v_v )
=> ( member7453568604450474000od_v_v @ ( bNF_We5492458111348578227od_v_v @ R @ B ) @ B ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1168_wo__rel_Ominim__in,axiom,
! [R: set_Product_prod_v_v,B: set_v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
=> ( ( B != bot_bot_set_v )
=> ( member_v @ ( bNF_We5615626441682584799inim_v @ R @ B ) @ B ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1169_wo__rel_Ominim__isMinim,axiom,
! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( B != bot_bo723834152578015283od_v_v )
=> ( bNF_We6235008509051751325od_v_v @ R @ B @ ( bNF_We5492458111348578227od_v_v @ R @ B ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_1170_wo__rel_Ominim__isMinim,axiom,
! [R: set_Product_prod_v_v,B: set_v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
=> ( ( B != bot_bot_set_v )
=> ( bNF_We6697304935525757641inim_v @ R @ B @ ( bNF_We5615626441682584799inim_v @ R @ B ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_1171_wo__rel_OisMinim__def,axiom,
! [R: set_Pr2149350503807050951od_v_v,A4: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( bNF_We6235008509051751325od_v_v @ R @ A4 @ B2 )
= ( ( member7453568604450474000od_v_v @ B2 @ A4 )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ X2 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1172_wo__rel_OisMinim__def,axiom,
! [R: set_Product_prod_v_v,A4: set_v,B2: v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( bNF_We6697304935525757641inim_v @ R @ A4 @ B2 )
= ( ( member_v @ B2 @ A4 )
& ! [X2: v] :
( ( member_v @ X2 @ A4 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ X2 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1173_wo__rel_Oequals__minim,axiom,
! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v,A: product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ A @ B )
=> ( ! [B5: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B5 @ B )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B5 ) @ R ) )
=> ( A
= ( bNF_We5492458111348578227od_v_v @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1174_wo__rel_Oequals__minim,axiom,
! [R: set_Product_prod_v_v,B: set_v,A: v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
=> ( ( member_v @ A @ B )
=> ( ! [B5: v] :
( ( member_v @ B5 @ B )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B5 ) @ R ) )
=> ( A
= ( bNF_We5615626441682584799inim_v @ R @ B ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1175_wo__rel_Ominim__least,axiom,
! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B2 @ B )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ ( bNF_We5492458111348578227od_v_v @ R @ B ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1176_wo__rel_Ominim__least,axiom,
! [R: set_Product_prod_v_v,B: set_v,B2: v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
=> ( ( member_v @ B2 @ B )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ ( bNF_We5615626441682584799inim_v @ R @ B ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1177_wo__rel_Ominim__inField,axiom,
! [R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( ord_le7336532860387713383od_v_v @ B @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( B != bot_bo723834152578015283od_v_v )
=> ( member7453568604450474000od_v_v @ ( bNF_We5492458111348578227od_v_v @ R @ B ) @ ( field_7153129647634986036od_v_v @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1178_wo__rel_Ominim__inField,axiom,
! [R: set_Product_prod_v_v,B: set_v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( ord_less_eq_set_v @ B @ ( field_v @ R ) )
=> ( ( B != bot_bot_set_v )
=> ( member_v @ ( bNF_We5615626441682584799inim_v @ R @ B ) @ ( field_v @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1179_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ ( bNF_We3854103423653685557od_v_v @ R @ A @ B2 ) ) @ R )
& ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ ( bNF_We3854103423653685557od_v_v @ R @ A @ B2 ) ) @ R )
& ( member7453568604450474000od_v_v @ ( bNF_We3854103423653685557od_v_v @ R @ A @ B2 ) @ ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1180_wo__rel_Omax2__greater__among,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( member_v @ B2 @ ( field_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ ( bNF_We3763454674811381857max2_v @ R @ A @ B2 ) ) @ R )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ ( bNF_We3763454674811381857max2_v @ R @ A @ B2 ) ) @ R )
& ( member_v @ ( bNF_We3763454674811381857max2_v @ R @ A @ B2 ) @ ( insert_v2 @ A @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1181_wo__rel_Omax2__among,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( member7453568604450474000od_v_v @ ( bNF_We3854103423653685557od_v_v @ R @ A @ B2 ) @ ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_1182_wo__rel_Omax2__among,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( member_v @ B2 @ ( field_v @ R ) )
=> ( member_v @ ( bNF_We3763454674811381857max2_v @ R @ A @ B2 ) @ ( insert_v2 @ A @ ( insert_v2 @ B2 @ bot_bot_set_v ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_1183_wo__rel_Omax2__def,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( ( bNF_We3763454674811381857max2_v @ R @ A @ B2 )
= B2 ) )
& ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R )
=> ( ( bNF_We3763454674811381857max2_v @ R @ A @ B2 )
= A ) ) ) ) ).
% wo_rel.max2_def
thf(fact_1184_wo__rel_Omax2__greater,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ ( bNF_We3854103423653685557od_v_v @ R @ A @ B2 ) ) @ R )
& ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ ( bNF_We3854103423653685557od_v_v @ R @ A @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1185_wo__rel_Omax2__greater,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( member_v @ B2 @ ( field_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ ( bNF_We3763454674811381857max2_v @ R @ A @ B2 ) ) @ R )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ ( bNF_We3763454674811381857max2_v @ R @ A @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1186_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( ( bNF_We3854103423653685557od_v_v @ R @ A @ B2 )
= B2 )
= ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1187_wo__rel_Omax2__equals2,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( member_v @ B2 @ ( field_v @ R ) )
=> ( ( ( bNF_We3763454674811381857max2_v @ R @ A @ B2 )
= B2 )
= ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1188_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B2: product_prod_v_v] :
( ( bNF_We8684021896157364691od_v_v @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B2 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( ( bNF_We3854103423653685557od_v_v @ R @ A @ B2 )
= A )
= ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B2 @ A ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1189_wo__rel_Omax2__equals1,axiom,
! [R: set_Product_prod_v_v,A: v,B2: v] :
( ( bNF_We1162827675446710015_rel_v @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( member_v @ B2 @ ( field_v @ R ) )
=> ( ( ( bNF_We3763454674811381857max2_v @ R @ A @ B2 )
= A )
= ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B2 @ A ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1190_shuffles_Oelims,axiom,
! [X: list_v,Xa3: list_v,Y2: set_list_v] :
( ( ( shuffles_v @ X @ Xa3 )
= Y2 )
=> ( ( ( X = nil_v )
=> ( Y2
!= ( insert_list_v @ Xa3 @ bot_bot_set_list_v ) ) )
=> ( ( ( Xa3 = nil_v )
=> ( Y2
!= ( insert_list_v @ X @ bot_bot_set_list_v ) ) )
=> ~ ! [X3: v,Xs2: list_v] :
( ( X
= ( cons_v @ X3 @ Xs2 ) )
=> ! [Y: v,Ys3: list_v] :
( ( Xa3
= ( cons_v @ Y @ Ys3 ) )
=> ( Y2
!= ( sup_sup_set_list_v @ ( image_list_v_list_v @ ( cons_v @ X3 ) @ ( shuffles_v @ Xs2 @ ( cons_v @ Y @ Ys3 ) ) ) @ ( image_list_v_list_v @ ( cons_v @ Y ) @ ( shuffles_v @ ( cons_v @ X3 @ Xs2 ) @ Ys3 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_1191_wf__Union,axiom,
! [R3: set_se5707775751431548583od_v_v] :
( ! [X3: set_Pr2149350503807050951od_v_v] :
( ( member2865299526245254384od_v_v @ X3 @ R3 )
=> ( wf_Product_prod_v_v @ X3 ) )
=> ( ! [X3: set_Pr2149350503807050951od_v_v] :
( ( member2865299526245254384od_v_v @ X3 @ R3 )
=> ! [Xa: set_Pr2149350503807050951od_v_v] :
( ( member2865299526245254384od_v_v @ Xa @ R3 )
=> ( ( X3 != Xa )
=> ( ( inf_in6271465464967711157od_v_v @ ( domain6359000466948879308od_v_v @ X3 ) @ ( range_7878975032137371189od_v_v @ Xa ) )
= bot_bo723834152578015283od_v_v ) ) ) )
=> ( wf_Product_prod_v_v @ ( comple514088740646613812od_v_v @ R3 ) ) ) ) ).
% wf_Union
thf(fact_1192_wf__Union,axiom,
! [R3: set_se8455005133513928103od_v_v] :
( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ R3 )
=> ( wf_v @ X3 ) )
=> ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ R3 )
=> ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ R3 )
=> ( ( X3 != Xa )
=> ( ( inf_inf_set_v @ ( domain_v_v @ X3 ) @ ( range_v_v @ Xa ) )
= bot_bot_set_v ) ) ) )
=> ( wf_v @ ( comple5788137035815166516od_v_v @ R3 ) ) ) ) ).
% wf_Union
thf(fact_1193_image__eqI,axiom,
! [B2: v,F: v > v,X: v,A4: set_v] :
( ( B2
= ( F @ X ) )
=> ( ( member_v @ X @ A4 )
=> ( member_v @ B2 @ ( image_v_v @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_1194_image__eqI,axiom,
! [B2: product_prod_v_v,F: v > product_prod_v_v,X: v,A4: set_v] :
( ( B2
= ( F @ X ) )
=> ( ( member_v @ X @ A4 )
=> ( member7453568604450474000od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_1195_image__eqI,axiom,
! [B2: v,F: product_prod_v_v > v,X: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( B2
= ( F @ X ) )
=> ( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( member_v @ B2 @ ( image_6152814753742948081_v_v_v @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_1196_image__eqI,axiom,
! [B2: product_prod_v_v,F: product_prod_v_v > product_prod_v_v,X: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( B2
= ( F @ X ) )
=> ( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( member7453568604450474000od_v_v @ B2 @ ( image_781944334261467077od_v_v @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_1197_image__empty,axiom,
! [F: product_prod_v_v > product_prod_v_v] :
( ( image_781944334261467077od_v_v @ F @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% image_empty
thf(fact_1198_image__empty,axiom,
! [F: product_prod_v_v > v] :
( ( image_6152814753742948081_v_v_v @ F @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% image_empty
thf(fact_1199_image__empty,axiom,
! [F: v > product_prod_v_v] :
( ( image_9222788639401671577od_v_v @ F @ bot_bot_set_v )
= bot_bo723834152578015283od_v_v ) ).
% image_empty
thf(fact_1200_image__empty,axiom,
! [F: v > v] :
( ( image_v_v @ F @ bot_bot_set_v )
= bot_bot_set_v ) ).
% image_empty
thf(fact_1201_empty__is__image,axiom,
! [F: product_prod_v_v > product_prod_v_v,A4: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( image_781944334261467077od_v_v @ F @ A4 ) )
= ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% empty_is_image
thf(fact_1202_empty__is__image,axiom,
! [F: v > product_prod_v_v,A4: set_v] :
( ( bot_bo723834152578015283od_v_v
= ( image_9222788639401671577od_v_v @ F @ A4 ) )
= ( A4 = bot_bot_set_v ) ) ).
% empty_is_image
thf(fact_1203_empty__is__image,axiom,
! [F: product_prod_v_v > v,A4: set_Product_prod_v_v] :
( ( bot_bot_set_v
= ( image_6152814753742948081_v_v_v @ F @ A4 ) )
= ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% empty_is_image
thf(fact_1204_empty__is__image,axiom,
! [F: v > v,A4: set_v] :
( ( bot_bot_set_v
= ( image_v_v @ F @ A4 ) )
= ( A4 = bot_bot_set_v ) ) ).
% empty_is_image
thf(fact_1205_image__is__empty,axiom,
! [F: product_prod_v_v > product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ( image_781944334261467077od_v_v @ F @ A4 )
= bot_bo723834152578015283od_v_v )
= ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% image_is_empty
thf(fact_1206_image__is__empty,axiom,
! [F: v > product_prod_v_v,A4: set_v] :
( ( ( image_9222788639401671577od_v_v @ F @ A4 )
= bot_bo723834152578015283od_v_v )
= ( A4 = bot_bot_set_v ) ) ).
% image_is_empty
thf(fact_1207_image__is__empty,axiom,
! [F: product_prod_v_v > v,A4: set_Product_prod_v_v] :
( ( ( image_6152814753742948081_v_v_v @ F @ A4 )
= bot_bot_set_v )
= ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% image_is_empty
thf(fact_1208_image__is__empty,axiom,
! [F: v > v,A4: set_v] :
( ( ( image_v_v @ F @ A4 )
= bot_bot_set_v )
= ( A4 = bot_bot_set_v ) ) ).
% image_is_empty
thf(fact_1209_finite__imageI,axiom,
! [F2: set_v,H: v > v] :
( ( finite_finite_v @ F2 )
=> ( finite_finite_v @ ( image_v_v @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_1210_insert__image,axiom,
! [X: v,A4: set_v,F: v > v] :
( ( member_v @ X @ A4 )
=> ( ( insert_v2 @ ( F @ X ) @ ( image_v_v @ F @ A4 ) )
= ( image_v_v @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1211_insert__image,axiom,
! [X: v,A4: set_v,F: v > product_prod_v_v] :
( ( member_v @ X @ A4 )
=> ( ( insert1338601472111419319od_v_v @ ( F @ X ) @ ( image_9222788639401671577od_v_v @ F @ A4 ) )
= ( image_9222788639401671577od_v_v @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1212_insert__image,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,F: product_prod_v_v > v] :
( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ( insert_v2 @ ( F @ X ) @ ( image_6152814753742948081_v_v_v @ F @ A4 ) )
= ( image_6152814753742948081_v_v_v @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1213_insert__image,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ( insert1338601472111419319od_v_v @ ( F @ X ) @ ( image_781944334261467077od_v_v @ F @ A4 ) )
= ( image_781944334261467077od_v_v @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1214_image__insert,axiom,
! [F: v > v,A: v,B: set_v] :
( ( image_v_v @ F @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ ( F @ A ) @ ( image_v_v @ F @ B ) ) ) ).
% image_insert
thf(fact_1215_image__insert,axiom,
! [F: v > product_prod_v_v,A: v,B: set_v] :
( ( image_9222788639401671577od_v_v @ F @ ( insert_v2 @ A @ B ) )
= ( insert1338601472111419319od_v_v @ ( F @ A ) @ ( image_9222788639401671577od_v_v @ F @ B ) ) ) ).
% image_insert
thf(fact_1216_image__insert,axiom,
! [F: product_prod_v_v > v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( image_6152814753742948081_v_v_v @ F @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert_v2 @ ( F @ A ) @ ( image_6152814753742948081_v_v_v @ F @ B ) ) ) ).
% image_insert
thf(fact_1217_image__insert,axiom,
! [F: product_prod_v_v > product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( image_781944334261467077od_v_v @ F @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ ( F @ A ) @ ( image_781944334261467077od_v_v @ F @ B ) ) ) ).
% image_insert
thf(fact_1218_Sup__bot__conv_I1_J,axiom,
! [A4: set_se8455005133513928103od_v_v] :
( ( ( comple5788137035815166516od_v_v @ A4 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A4 )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1219_Sup__bot__conv_I1_J,axiom,
! [A4: set_set_v] :
( ( ( comple2307003700295860064_set_v @ A4 )
= bot_bot_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A4 )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1220_Sup__bot__conv_I2_J,axiom,
! [A4: set_se8455005133513928103od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( comple5788137035815166516od_v_v @ A4 ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A4 )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1221_Sup__bot__conv_I2_J,axiom,
! [A4: set_set_v] :
( ( bot_bot_set_v
= ( comple2307003700295860064_set_v @ A4 ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A4 )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1222_finite__Union,axiom,
! [A4: set_set_v] :
( ( finite_finite_set_v @ A4 )
=> ( ! [M2: set_v] :
( ( member_set_v @ M2 @ A4 )
=> ( finite_finite_v @ M2 ) )
=> ( finite_finite_v @ ( comple2307003700295860064_set_v @ A4 ) ) ) ) ).
% finite_Union
thf(fact_1223_Sup__empty,axiom,
( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Sup_empty
thf(fact_1224_Sup__empty,axiom,
( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
= bot_bot_set_v ) ).
% Sup_empty
thf(fact_1225_Sup__insert,axiom,
! [A: set_Product_prod_v_v,A4: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A @ A4 ) )
= ( sup_su414716646722978715od_v_v @ A @ ( comple5788137035815166516od_v_v @ A4 ) ) ) ).
% Sup_insert
thf(fact_1226_Sup__insert,axiom,
! [A: set_v,A4: set_set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ A @ A4 ) )
= ( sup_sup_set_v @ A @ ( comple2307003700295860064_set_v @ A4 ) ) ) ).
% Sup_insert
thf(fact_1227_Union__disjoint,axiom,
! [C2: set_se8455005133513928103od_v_v,A4: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ C2 ) @ A4 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ X2 @ A4 )
= bot_bo723834152578015283od_v_v ) ) ) ) ).
% Union_disjoint
thf(fact_1228_Union__disjoint,axiom,
! [C2: set_set_v,A4: set_v] :
( ( ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ C2 ) @ A4 )
= bot_bot_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ C2 )
=> ( ( inf_inf_set_v @ X2 @ A4 )
= bot_bot_set_v ) ) ) ) ).
% Union_disjoint
thf(fact_1229_insert__partition,axiom,
! [X: set_Product_prod_v_v,F2: set_se8455005133513928103od_v_v] :
( ~ ( member8406446414694345712od_v_v @ X @ F2 )
=> ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ ( insert7504383016908236695od_v_v @ X @ F2 ) )
=> ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ ( insert7504383016908236695od_v_v @ X @ F2 ) )
=> ( ( X3 != Xa )
=> ( ( inf_in6271465464967711157od_v_v @ X3 @ Xa )
= bot_bo723834152578015283od_v_v ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ ( comple5788137035815166516od_v_v @ F2 ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_partition
thf(fact_1230_insert__partition,axiom,
! [X: set_v,F2: set_set_v] :
( ~ ( member_set_v @ X @ F2 )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ ( insert_set_v @ X @ F2 ) )
=> ! [Xa: set_v] :
( ( member_set_v @ Xa @ ( insert_set_v @ X @ F2 ) )
=> ( ( X3 != Xa )
=> ( ( inf_inf_set_v @ X3 @ Xa )
= bot_bot_set_v ) ) ) )
=> ( ( inf_inf_set_v @ X @ ( comple2307003700295860064_set_v @ F2 ) )
= bot_bot_set_v ) ) ) ).
% insert_partition
thf(fact_1231_image__Un,axiom,
! [F: product_prod_v_v > product_prod_v_v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( image_781944334261467077od_v_v @ F @ ( sup_su414716646722978715od_v_v @ A4 @ B ) )
= ( sup_su414716646722978715od_v_v @ ( image_781944334261467077od_v_v @ F @ A4 ) @ ( image_781944334261467077od_v_v @ F @ B ) ) ) ).
% image_Un
thf(fact_1232_image__Un,axiom,
! [F: product_prod_v_v > v,A4: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( image_6152814753742948081_v_v_v @ F @ ( sup_su414716646722978715od_v_v @ A4 @ B ) )
= ( sup_sup_set_v @ ( image_6152814753742948081_v_v_v @ F @ A4 ) @ ( image_6152814753742948081_v_v_v @ F @ B ) ) ) ).
% image_Un
thf(fact_1233_image__Un,axiom,
! [F: v > product_prod_v_v,A4: set_v,B: set_v] :
( ( image_9222788639401671577od_v_v @ F @ ( sup_sup_set_v @ A4 @ B ) )
= ( sup_su414716646722978715od_v_v @ ( image_9222788639401671577od_v_v @ F @ A4 ) @ ( image_9222788639401671577od_v_v @ F @ B ) ) ) ).
% image_Un
thf(fact_1234_image__Un,axiom,
! [F: v > v,A4: set_v,B: set_v] :
( ( image_v_v @ F @ ( sup_sup_set_v @ A4 @ B ) )
= ( sup_sup_set_v @ ( image_v_v @ F @ A4 ) @ ( image_v_v @ F @ B ) ) ) ).
% image_Un
thf(fact_1235_rev__image__eqI,axiom,
! [X: v,A4: set_v,B2: v,F: v > v] :
( ( member_v @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member_v @ B2 @ ( image_v_v @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_1236_rev__image__eqI,axiom,
! [X: v,A4: set_v,B2: product_prod_v_v,F: v > product_prod_v_v] :
( ( member_v @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member7453568604450474000od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_1237_rev__image__eqI,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,B2: v,F: product_prod_v_v > v] :
( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member_v @ B2 @ ( image_6152814753742948081_v_v_v @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_1238_rev__image__eqI,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,B2: product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ( B2
= ( F @ X ) )
=> ( member7453568604450474000od_v_v @ B2 @ ( image_781944334261467077od_v_v @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_1239_imageI,axiom,
! [X: v,A4: set_v,F: v > v] :
( ( member_v @ X @ A4 )
=> ( member_v @ ( F @ X ) @ ( image_v_v @ F @ A4 ) ) ) ).
% imageI
thf(fact_1240_imageI,axiom,
! [X: v,A4: set_v,F: v > product_prod_v_v] :
( ( member_v @ X @ A4 )
=> ( member7453568604450474000od_v_v @ ( F @ X ) @ ( image_9222788639401671577od_v_v @ F @ A4 ) ) ) ).
% imageI
thf(fact_1241_imageI,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,F: product_prod_v_v > v] :
( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( member_v @ ( F @ X ) @ ( image_6152814753742948081_v_v_v @ F @ A4 ) ) ) ).
% imageI
thf(fact_1242_imageI,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( member7453568604450474000od_v_v @ ( F @ X ) @ ( image_781944334261467077od_v_v @ F @ A4 ) ) ) ).
% imageI
thf(fact_1243_finite__UnionD,axiom,
! [A4: set_set_v] :
( ( finite_finite_v @ ( comple2307003700295860064_set_v @ A4 ) )
=> ( finite_finite_set_v @ A4 ) ) ).
% finite_UnionD
thf(fact_1244_Union__insert,axiom,
! [A: set_Product_prod_v_v,B: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A @ B ) )
= ( sup_su414716646722978715od_v_v @ A @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_insert
thf(fact_1245_Union__insert,axiom,
! [A: set_v,B: set_set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ A @ B ) )
= ( sup_sup_set_v @ A @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_insert
thf(fact_1246_Union__empty,axiom,
( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Union_empty
thf(fact_1247_Union__empty,axiom,
( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
= bot_bot_set_v ) ).
% Union_empty
thf(fact_1248_Union__empty__conv,axiom,
! [A4: set_se8455005133513928103od_v_v] :
( ( ( comple5788137035815166516od_v_v @ A4 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A4 )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Union_empty_conv
thf(fact_1249_Union__empty__conv,axiom,
! [A4: set_set_v] :
( ( ( comple2307003700295860064_set_v @ A4 )
= bot_bot_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A4 )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% Union_empty_conv
thf(fact_1250_empty__Union__conv,axiom,
! [A4: set_se8455005133513928103od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( comple5788137035815166516od_v_v @ A4 ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A4 )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% empty_Union_conv
thf(fact_1251_empty__Union__conv,axiom,
! [A4: set_set_v] :
( ( bot_bot_set_v
= ( comple2307003700295860064_set_v @ A4 ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A4 )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% empty_Union_conv
thf(fact_1252_SUP__eq__const,axiom,
! [I3: set_Product_prod_v_v,F: product_prod_v_v > set_v,X: set_v] :
( ( I3 != bot_bo723834152578015283od_v_v )
=> ( ! [I4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple2307003700295860064_set_v @ ( image_2529437795422174673_set_v @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_1253_SUP__eq__const,axiom,
! [I3: set_v,F: v > set_v,X: set_v] :
( ( I3 != bot_bot_set_v )
=> ( ! [I4: v] :
( ( member_v @ I4 @ I3 )
=> ( ( F @ I4 )
= X ) )
=> ( ( comple2307003700295860064_set_v @ ( image_v_set_v @ F @ I3 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_1254_image__mono,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( image_781944334261467077od_v_v @ F @ A4 ) @ ( image_781944334261467077od_v_v @ F @ B ) ) ) ).
% image_mono
thf(fact_1255_image__mono,axiom,
! [A4: set_Product_prod_v_v,B: set_Product_prod_v_v,F: product_prod_v_v > v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B )
=> ( ord_less_eq_set_v @ ( image_6152814753742948081_v_v_v @ F @ A4 ) @ ( image_6152814753742948081_v_v_v @ F @ B ) ) ) ).
% image_mono
thf(fact_1256_image__mono,axiom,
! [A4: set_v,B: set_v,F: v > product_prod_v_v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( image_9222788639401671577od_v_v @ F @ A4 ) @ ( image_9222788639401671577od_v_v @ F @ B ) ) ) ).
% image_mono
thf(fact_1257_image__mono,axiom,
! [A4: set_v,B: set_v,F: v > v] :
( ( ord_less_eq_set_v @ A4 @ B )
=> ( ord_less_eq_set_v @ ( image_v_v @ F @ A4 ) @ ( image_v_v @ F @ B ) ) ) ).
% image_mono
thf(fact_1258_image__subsetI,axiom,
! [A4: set_v,F: v > product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A4 )
=> ( member7453568604450474000od_v_v @ ( F @ X3 ) @ B ) )
=> ( ord_le7336532860387713383od_v_v @ ( image_9222788639401671577od_v_v @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_1259_image__subsetI,axiom,
! [A4: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ( member7453568604450474000od_v_v @ ( F @ X3 ) @ B ) )
=> ( ord_le7336532860387713383od_v_v @ ( image_781944334261467077od_v_v @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_1260_image__subsetI,axiom,
! [A4: set_v,F: v > v,B: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A4 )
=> ( member_v @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_v @ ( image_v_v @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_1261_image__subsetI,axiom,
! [A4: set_Product_prod_v_v,F: product_prod_v_v > v,B: set_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ( member_v @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_v @ ( image_6152814753742948081_v_v_v @ F @ A4 ) @ B ) ) ).
% image_subsetI
thf(fact_1262_subset__imageE,axiom,
! [B: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ ( image_781944334261467077od_v_v @ F @ A4 ) )
=> ~ ! [C4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C4 @ A4 )
=> ( B
!= ( image_781944334261467077od_v_v @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1263_subset__imageE,axiom,
! [B: set_Product_prod_v_v,F: v > product_prod_v_v,A4: set_v] :
( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A4 ) )
=> ~ ! [C4: set_v] :
( ( ord_less_eq_set_v @ C4 @ A4 )
=> ( B
!= ( image_9222788639401671577od_v_v @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1264_subset__imageE,axiom,
! [B: set_v,F: product_prod_v_v > v,A4: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A4 ) )
=> ~ ! [C4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C4 @ A4 )
=> ( B
!= ( image_6152814753742948081_v_v_v @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1265_subset__imageE,axiom,
! [B: set_v,F: v > v,A4: set_v] :
( ( ord_less_eq_set_v @ B @ ( image_v_v @ F @ A4 ) )
=> ~ ! [C4: set_v] :
( ( ord_less_eq_set_v @ C4 @ A4 )
=> ( B
!= ( image_v_v @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1266_subset__image__iff,axiom,
! [B: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ ( image_781944334261467077od_v_v @ F @ A4 ) )
= ( ? [AA: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ AA @ A4 )
& ( B
= ( image_781944334261467077od_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1267_subset__image__iff,axiom,
! [B: set_Product_prod_v_v,F: v > product_prod_v_v,A4: set_v] :
( ( ord_le7336532860387713383od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A4 ) )
= ( ? [AA: set_v] :
( ( ord_less_eq_set_v @ AA @ A4 )
& ( B
= ( image_9222788639401671577od_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1268_subset__image__iff,axiom,
! [B: set_v,F: product_prod_v_v > v,A4: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A4 ) )
= ( ? [AA: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ AA @ A4 )
& ( B
= ( image_6152814753742948081_v_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1269_subset__image__iff,axiom,
! [B: set_v,F: v > v,A4: set_v] :
( ( ord_less_eq_set_v @ B @ ( image_v_v @ F @ A4 ) )
= ( ? [AA: set_v] :
( ( ord_less_eq_set_v @ AA @ A4 )
& ( B
= ( image_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1270_all__subset__image,axiom,
! [F: product_prod_v_v > product_prod_v_v,A4: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( ! [B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B6 @ ( image_781944334261467077od_v_v @ F @ A4 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B6 @ A4 )
=> ( P @ ( image_781944334261467077od_v_v @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1271_all__subset__image,axiom,
! [F: v > product_prod_v_v,A4: set_v,P: set_Product_prod_v_v > $o] :
( ( ! [B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B6 @ ( image_9222788639401671577od_v_v @ F @ A4 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_v] :
( ( ord_less_eq_set_v @ B6 @ A4 )
=> ( P @ ( image_9222788639401671577od_v_v @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1272_all__subset__image,axiom,
! [F: product_prod_v_v > v,A4: set_Product_prod_v_v,P: set_v > $o] :
( ( ! [B6: set_v] :
( ( ord_less_eq_set_v @ B6 @ ( image_6152814753742948081_v_v_v @ F @ A4 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B6 @ A4 )
=> ( P @ ( image_6152814753742948081_v_v_v @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1273_all__subset__image,axiom,
! [F: v > v,A4: set_v,P: set_v > $o] :
( ( ! [B6: set_v] :
( ( ord_less_eq_set_v @ B6 @ ( image_v_v @ F @ A4 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set_v] :
( ( ord_less_eq_set_v @ B6 @ A4 )
=> ( P @ ( image_v_v @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_1274_image__Int__subset,axiom,
! [F: v > v,A4: set_v,B: set_v] : ( ord_less_eq_set_v @ ( image_v_v @ F @ ( inf_inf_set_v @ A4 @ B ) ) @ ( inf_inf_set_v @ ( image_v_v @ F @ A4 ) @ ( image_v_v @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1275_unite__S__equal,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N3 ) )
& ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) )
& ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ).
% unite_S_equal
thf(fact_1276_e_H_H__def,axiom,
( e3
= ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = v2 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ e2 @ v2 ) @ ( insert_v2 @ w @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ e2 @ X2 ) )
@ e2 ) ) ).
% e''_def
% Helper facts (5)
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y2: set_v] :
( ( if_set_v @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y2: set_v] :
( ( if_set_v @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( if_set4279007504652509325od_v_v @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( if_set4279007504652509325od_v_v @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
sCC_Bl9196236973127232072t_unit @ successors @ e3 ).
%------------------------------------------------------------------------------