TPTP Problem File: SLH0831^1.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 : Stalnaker_Logic/0000_Stalnaker_Logic/prob_00404_015215__6819118_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1618 ( 506 unt; 332 typ; 0 def)
% Number of atoms : 4364 (1085 equ; 0 cnn)
% Maximal formula atoms : 21 ( 3 avg)
% Number of connectives : 13124 ( 371 ~; 72 |; 395 &;10284 @)
% ( 0 <=>;2002 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 38 ( 37 usr)
% Number of type conns : 1204 (1204 >; 0 *; 0 +; 0 <<)
% Number of symbols : 298 ( 295 usr; 20 con; 0-4 aty)
% Number of variables : 4182 ( 636 ^;3442 !; 104 ?;4182 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:12:48.426
%------------------------------------------------------------------------------
% Could-be-implicit typings (37)
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J_J,type,
episte476198014953467260t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J_J,type,
episte7095840741371228412t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__i_Mt__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J_J,type,
episte6041718960478671640t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__i_Mt__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J_J,type,
episte8649862735182028056t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__i_Mt__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J_J,type,
episte5489734138418521880t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Epistemic____Logic__Ofm_Itf__i_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J_J,type,
episte5703418715120784124t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__i_Mt__Epistemic____Logic__Ofm_Itf__i_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J_J,type,
episte4846006725945195288t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Nat__Onat_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
episte4946720254005641710t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mtf__i_Mt__Epistemic____Logic__Okripke__Okripke____ext_Itf__i_Mt__Product____Type__Ounit_J_J,type,
episte7276436686577410172t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J,type,
episte3552787238722255479t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__i_Mt__Nat__Onat_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
episte8354619181726739466t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__i_Mtf__i_Mt__Epistemic____Logic__Okripke__Okripke____ext_Itf__i_Mt__Product____Type__Ounit_J_J,type,
episte2102749610958417048t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte7133455005732253587t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
set_Pr4658907567593863815c_fm_i: $tType ).
thf(ty_n_t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J,type,
episte3891166926823127411t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
set_se7087778041443875454c_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
set_se7339729205154126530c_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
set_Ep6763084006362001438c_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
list_l4360594050116504750c_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
set_li8269783359719514504c_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
set_se3485332733965609186c_fm_i: $tType ).
thf(ty_n_t__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
episte1308613106011496894c_fm_i: $tType ).
thf(ty_n_t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
list_Epistemic_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
set_Epistemic_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Ounit_J,type,
list_Product_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
set_Product_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Ofm_Itf__i_J,type,
epistemic_fm_i: $tType ).
thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__Set__Oset_Itf__i_J,type,
set_i: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__i,type,
i: $tType ).
% Explicit typings (295)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
bNF_Gr8437504169225076033c_fm_i: set_li8269783359719514504c_fm_i > epistemic_fm_i > set_li8269783359719514504c_fm_i ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
bNF_Gr1093487169986386109c_fm_i: set_li8269783359719514504c_fm_i > list_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Cardinal__Order__Relation_Onlists_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
cardin549358664437198940c_fm_i: set_Epistemic_fm_i > nat > set_li8269783359719514504c_fm_i ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Product____Type__Ounit,type,
comple4687483117567863418t_unit: set_Product_unit > product_unit ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
comple8437045806746640111c_fm_i: set_se3485332733965609186c_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Epistemic__Logic_OAK_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte3601224874663357326c_fm_i: ( episte1308613106011496894c_fm_i > $o ) > episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAK_001tf__i,type,
epistemic_AK_i: ( epistemic_fm_i > $o ) > epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAx4_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte4964217057358206253c_fm_i: episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAx4_001tf__i,type,
epistemic_Ax4_i: epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAx5_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte7251237997873073900c_fm_i: episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAx5_001tf__i,type,
epistemic_Ax5_i: epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAxB_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte89022077147250079c_fm_i: episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAxB_001tf__i,type,
epistemic_AxB_i: epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAxT_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte4361910858995764493c_fm_i: episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAxT_001tf__i,type,
epistemic_AxT_i: epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J,type,
episte8462951133645435945t_unit: episte476198014953467260t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__i_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J,type,
episte8713243348624617413t_unit: episte4846006725945195288t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__i_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte7759965962005403845t_unit: episte8649862735182028056t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__i_001t__Nat__Onat_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
episte9066274068200899311t_unit: episte8354619181726739466t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__i_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte6275404236303465413t_unit: episte5489734138418521880t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__i_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J,type,
episte8780254683117438917t_unit: episte6041718960478671640t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__i_001tf__i_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__i_Mt__Product____Type__Ounit_J,type,
episte3699623721066309957t_unit: episte2102749610958417048t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oconsistent_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte704884158563092166c_fm_i: ( episte1308613106011496894c_fm_i > $o ) > set_Ep6763084006362001438c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Oconsistent_001tf__i,type,
episte2285483198712856234tent_i: ( epistemic_fm_i > $o ) > set_Epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Oeval_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte6292435846501656252c_fm_i: ( list_char > $o ) > ( episte1308613106011496894c_fm_i > $o ) > episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Oeval_001tf__i,type,
epistemic_eval_i: ( list_char > $o ) > ( epistemic_fm_i > $o ) > epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Ofm_OCon_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte3685526521632971807c_fm_i: episte1308613106011496894c_fm_i > episte1308613106011496894c_fm_i > episte1308613106011496894c_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_OCon_001tf__i,type,
epistemic_Con_i: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_ODis_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte6088726798904853267c_fm_i: episte1308613106011496894c_fm_i > episte1308613106011496894c_fm_i > episte1308613106011496894c_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_ODis_001tf__i,type,
epistemic_Dis_i: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_OFF_001tf__i,type,
epistemic_FF_i: epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_OImp_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte260752253203357973c_fm_i: episte1308613106011496894c_fm_i > episte1308613106011496894c_fm_i > episte1308613106011496894c_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_OImp_001tf__i,type,
epistemic_Imp_i: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_OK_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte5657488666450005526c_fm_i: epistemic_fm_i > episte1308613106011496894c_fm_i > episte1308613106011496894c_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_OK_001tf__i,type,
epistemic_K_i: i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_OPro_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte3759128500599061780c_fm_i: list_char > episte1308613106011496894c_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_OPro_001tf__i,type,
epistemic_Pro_i: list_char > epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte3445413862081925600c_fm_i: ( epistemic_fm_i > epistemic_fm_i > $o ) > episte1308613106011496894c_fm_i > episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001tf__i_001tf__i,type,
epistemic_rel_fm_i_i: ( i > i > $o ) > epistemic_fm_i > epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte9089240992906287960c_fm_i: episte1308613106011496894c_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001tf__i,type,
epistemic_set_fm_i: epistemic_fm_i > set_i ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060K_062_001tf__i_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J,type,
episte5576233017153220191t_unit: episte4846006725945195288t_unit > i > epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060K_062_001tf__i_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte852950631662840415t_unit: episte5489734138418521880t_unit > i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J,type,
episte5446861122069384015t_unit: episte5703418715120784124t_unit > set_Epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte1684291789543234127t_unit: episte7095840741371228412t_unit > set_li8269783359719514504c_fm_i ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Nat__Onat_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
episte7295652460417134969t_unit: episte4946720254005641710t_unit > set_nat ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J,type,
episte474035306837431247t_unit: episte476198014953467260t_unit > set_se7087778041443875454c_fm_i ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001t__Epistemic____Logic__Ofm_Itf__i_J_001tf__i_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__i_Mt__Product____Type__Ounit_J,type,
episte5422822166207371983t_unit: episte7276436686577410172t_unit > set_i ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001tf__i_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J,type,
episte1297369412406343275t_unit: episte4846006725945195288t_unit > set_Epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001tf__i_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte970406020673866091t_unit: episte8649862735182028056t_unit > set_li8269783359719514504c_fm_i ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001tf__i_001t__Nat__Onat_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
episte8922775215189319317t_unit: episte8354619181726739466t_unit > set_nat ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001tf__i_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte2675268102452679275t_unit: episte5489734138418521880t_unit > set_se3485332733965609186c_fm_i ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001tf__i_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J,type,
episte3312409906628800107t_unit: episte6041718960478671640t_unit > set_se7339729205154126530c_fm_i ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001tf__i_001tf__i_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__i_Mt__Product____Type__Ounit_J,type,
episte8286435292487608811t_unit: episte2102749610958417048t_unit > set_i ).
thf(sy_c_Epistemic__Logic_Oframe_Oframe__ext_001t__Epistemic____Logic__Ofm_Itf__i_J_001tf__i_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J,type,
episte3869211295228886240t_unit: set_Epistemic_fm_i > ( i > epistemic_fm_i > set_Epistemic_fm_i ) > episte3891166926823127411t_unit > episte4846006725945195288t_unit ).
thf(sy_c_Epistemic__Logic_Oframe_Oframe__ext_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J,type,
episte7444838911182875652t_unit: set_se7087778041443875454c_fm_i > ( epistemic_fm_i > set_Ep6763084006362001438c_fm_i > set_se7087778041443875454c_fm_i ) > episte3552787238722255479t_unit > episte476198014953467260t_unit ).
thf(sy_c_Epistemic__Logic_Oframe_Oframe__ext_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001tf__i_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte8397520288747125024t_unit: set_se3485332733965609186c_fm_i > ( i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i ) > episte7133455005732253587t_unit > episte5489734138418521880t_unit ).
thf(sy_c_Epistemic__Logic_Oframe_Omore_001tf__i_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte7135767042998387833t_unit: episte5489734138418521880t_unit > episte7133455005732253587t_unit ).
thf(sy_c_Epistemic__Logic_Oimply_001tf__i,type,
epistemic_imply_i: list_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Okripke_O_092_060pi_062_001tf__i_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Product____Type__Ounit,type,
episte4915868227582937300t_unit: episte4846006725945195288t_unit > epistemic_fm_i > list_char > $o ).
thf(sy_c_Epistemic__Logic_Okripke_O_092_060pi_062_001tf__i_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Product____Type__Ounit,type,
episte6627496282998226676t_unit: episte5489734138418521880t_unit > set_Epistemic_fm_i > list_char > $o ).
thf(sy_c_Epistemic__Logic_Okripke_O_092_060pi_062__update_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001tf__i_001t__Product____Type__Ounit,type,
episte8744582347493487171t_unit: ( ( set_Epistemic_fm_i > list_char > $o ) > set_Epistemic_fm_i > list_char > $o ) > episte5489734138418521880t_unit > episte5489734138418521880t_unit ).
thf(sy_c_Epistemic__Logic_Okripke_Okripke__ext_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Product____Type__Ounit,type,
episte1793992050974410275t_unit: ( epistemic_fm_i > list_char > $o ) > product_unit > episte3891166926823127411t_unit ).
thf(sy_c_Epistemic__Logic_Okripke_Okripke__ext_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Product____Type__Ounit,type,
episte7452740295420953255t_unit: ( set_Ep6763084006362001438c_fm_i > list_char > $o ) > product_unit > episte3552787238722255479t_unit ).
thf(sy_c_Epistemic__Logic_Okripke_Okripke__ext_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Product____Type__Ounit,type,
episte4955834246032687171t_unit: ( set_Epistemic_fm_i > list_char > $o ) > product_unit > episte7133455005732253587t_unit ).
thf(sy_c_Epistemic__Logic_Okripke_Omore_001tf__i_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Product____Type__Ounit,type,
episte4828294246236881914t_unit: episte4846006725945195288t_unit > product_unit ).
thf(sy_c_Epistemic__Logic_Okripke_Omore_001tf__i_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Product____Type__Ounit,type,
episte484680259488342554t_unit: episte5489734138418521880t_unit > product_unit ).
thf(sy_c_Epistemic__Logic_Okripke_Omore__update_001t__Product____Type__Ounit_001tf__i_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
episte4537596280781771625c_fm_i: ( product_unit > product_unit ) > episte5489734138418521880t_unit > episte5489734138418521880t_unit ).
thf(sy_c_Epistemic__Logic_Oreflexive_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J,type,
episte7789846462086006391t_unit: episte5703418715120784124t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oreflexive_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte3051827627171704183t_unit: episte7095840741371228412t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oreflexive_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Nat__Onat_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
episte8283200250080426145t_unit: episte4946720254005641710t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oreflexive_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J,type,
episte2938229627094415095t_unit: episte476198014953467260t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oreflexive_001t__Epistemic____Logic__Ofm_Itf__i_J_001tf__i_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__i_Mt__Product____Type__Ounit_J,type,
episte5368934292378447351t_unit: episte7276436686577410172t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oreflexive_001tf__i_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J,type,
episte540185425077706131t_unit: episte4846006725945195288t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oreflexive_001tf__i_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte6062502518927837331t_unit: episte8649862735182028056t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oreflexive_001tf__i_001t__Nat__Onat_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
episte3073512803368128957t_unit: episte8354619181726739466t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oreflexive_001tf__i_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte251305198166243731t_unit: episte5489734138418521880t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oreflexive_001tf__i_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J,type,
episte777650505068270995t_unit: episte6041718960478671640t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oreflexive_001tf__i_001tf__i_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__i_Mt__Product____Type__Ounit_J,type,
episte707503779239422227t_unit: episte2102749610958417048t_unit > $o ).
thf(sy_c_Epistemic__Logic_Osemantics_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte7800156050638082596c_fm_i: episte5703418715120784124t_unit > epistemic_fm_i > episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Osemantics_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
episte6060530439555422506c_fm_i: episte7095840741371228412t_unit > list_Epistemic_fm_i > episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Osemantics_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Nat__Onat,type,
episte6892841275385687886_i_nat: episte4946720254005641710t_unit > nat > episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Osemantics_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
episte341539128385704480c_fm_i: episte476198014953467260t_unit > set_Ep6763084006362001438c_fm_i > episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Osemantics_001t__Epistemic____Logic__Ofm_Itf__i_J_001tf__i,type,
episte5578572022554684168fm_i_i: episte7276436686577410172t_unit > i > episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Osemantics_001tf__i_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
episte5182706529199770816c_fm_i: episte4846006725945195288t_unit > epistemic_fm_i > epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Osemantics_001tf__i_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
episte5087030060053943750c_fm_i: episte8649862735182028056t_unit > list_Epistemic_fm_i > epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Osemantics_001tf__i_001t__Nat__Onat,type,
episte4571855713382724530_i_nat: episte8354619181726739466t_unit > nat > epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Osemantics_001tf__i_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
episte2498298383968489120c_fm_i: episte5489734138418521880t_unit > set_Epistemic_fm_i > epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Osemantics_001tf__i_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
episte2123465427079433344c_fm_i: episte6041718960478671640t_unit > set_se3485332733965609186c_fm_i > epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Osemantics_001tf__i_001tf__i,type,
episte5666089339915499428cs_i_i: episte2102749610958417048t_unit > i > epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Osymmetric_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J,type,
episte818161904015704450t_unit: episte476198014953467260t_unit > $o ).
thf(sy_c_Epistemic__Logic_Osymmetric_001tf__i_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J,type,
episte6568605431049406366t_unit: episte4846006725945195288t_unit > $o ).
thf(sy_c_Epistemic__Logic_Osymmetric_001tf__i_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte3343910547084193694t_unit: episte8649862735182028056t_unit > $o ).
thf(sy_c_Epistemic__Logic_Osymmetric_001tf__i_001t__Nat__Onat_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
episte1817073897218058824t_unit: episte8354619181726739466t_unit > $o ).
thf(sy_c_Epistemic__Logic_Osymmetric_001tf__i_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte80897895827131294t_unit: episte5489734138418521880t_unit > $o ).
thf(sy_c_Epistemic__Logic_Osymmetric_001tf__i_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J,type,
episte4692175903937384350t_unit: episte6041718960478671640t_unit > $o ).
thf(sy_c_Epistemic__Logic_Osymmetric_001tf__i_001tf__i_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__i_Mt__Product____Type__Ounit_J,type,
episte2180194749693155358t_unit: episte2102749610958417048t_unit > $o ).
thf(sy_c_Epistemic__Logic_Otransitive_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J,type,
episte3656210382014446050t_unit: episte5703418715120784124t_unit > $o ).
thf(sy_c_Epistemic__Logic_Otransitive_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte1901197506451996130t_unit: episte7095840741371228412t_unit > $o ).
thf(sy_c_Epistemic__Logic_Otransitive_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Nat__Onat_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
episte6561722749315164812t_unit: episte4946720254005641710t_unit > $o ).
thf(sy_c_Epistemic__Logic_Otransitive_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J,type,
episte6770155463157591394t_unit: episte476198014953467260t_unit > $o ).
thf(sy_c_Epistemic__Logic_Otransitive_001t__Epistemic____Logic__Ofm_Itf__i_J_001tf__i_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__i_Mt__Product____Type__Ounit_J,type,
episte7780989443469444706t_unit: episte7276436686577410172t_unit > $o ).
thf(sy_c_Epistemic__Logic_Otransitive_001tf__i_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J,type,
episte6394096064484582270t_unit: episte4846006725945195288t_unit > $o ).
thf(sy_c_Epistemic__Logic_Otransitive_001tf__i_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte3140986294363532158t_unit: episte8649862735182028056t_unit > $o ).
thf(sy_c_Epistemic__Logic_Otransitive_001tf__i_001t__Nat__Onat_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
episte7906311534470272552t_unit: episte8354619181726739466t_unit > $o ).
thf(sy_c_Epistemic__Logic_Otransitive_001tf__i_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Product____Type__Ounit_J,type,
episte2966952217288386430t_unit: episte5489734138418521880t_unit > $o ).
thf(sy_c_Epistemic__Logic_Otransitive_001tf__i_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_Mt__Product____Type__Ounit_J,type,
episte5217350381671328638t_unit: episte6041718960478671640t_unit > $o ).
thf(sy_c_Epistemic__Logic_Otransitive_001tf__i_001tf__i_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__i_Mt__Product____Type__Ounit_J,type,
episte6298788599024636926t_unit: episte2102749610958417048t_unit > $o ).
thf(sy_c_Finite__Set_OFpow_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
finite7994277067554142856c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
finite3304564979551393739c_fm_i: set_Epistemic_fm_i > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
finite3106398504192302929c_fm_i: set_li8269783359719514504c_fm_i > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Ounit,type,
finite4290736615968046902t_unit: set_Product_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
finite7933139204641697195c_fm_i: set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
finite5529483035118572448et_nat: ( nat > set_nat > set_nat ) > set_nat > set_nat > set_nat ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite4178521680790401110et_nat: ( nat > set_set_nat > set_set_nat ) > set_set_nat > set_nat > set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_M_Eo_J,type,
minus_6519238007984220724fm_i_o: ( epistemic_fm_i > $o ) > ( epistemic_fm_i > $o ) > epistemic_fm_i > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
minus_5671197887077578836fm_i_o: ( set_Epistemic_fm_i > $o ) > ( set_Epistemic_fm_i > $o ) > set_Epistemic_fm_i > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Ounit,type,
minus_3524152463667985524t_unit: product_unit > product_unit > product_unit ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
minus_1543973939109951465c_fm_i: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Ounit_J,type,
minus_6452836326544984404t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
minus_1073301138667672009c_fm_i: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_HOL_OThe_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
the_Epistemic_fm_i: ( epistemic_fm_i > $o ) > epistemic_fm_i ).
thf(sy_c_HOL_OUniq_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
uniq_Epistemic_fm_i: ( epistemic_fm_i > $o ) > $o ).
thf(sy_c_HOL_Odefault__class_Odefault_001t__Product____Type__Ounit,type,
defaul566961228789861419t_unit: product_unit ).
thf(sy_c_If_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
if_Epistemic_fm_i: $o > epistemic_fm_i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
if_lis3446954105117697518c_fm_i: $o > list_Epistemic_fm_i > list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_M_Eo_J,type,
inf_in8976956063904982253fm_i_o: ( epistemic_fm_i > $o ) > ( epistemic_fm_i > $o ) > epistemic_fm_i > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
inf_in853329969422372621fm_i_o: ( set_Epistemic_fm_i > $o ) > ( set_Epistemic_fm_i > $o ) > set_Epistemic_fm_i > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Product____Type__Ounit,type,
inf_inf_Product_unit: product_unit > product_unit > product_unit ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
inf_in3450601097109690352c_fm_i: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
inf_in161960956874937808c_fm_i: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_M_Eo_J,type,
sup_su2844414564337698183fm_i_o: ( epistemic_fm_i > $o ) > ( epistemic_fm_i > $o ) > epistemic_fm_i > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
sup_su6723579903606884263fm_i_o: ( set_Epistemic_fm_i > $o ) > ( set_Epistemic_fm_i > $o ) > set_Epistemic_fm_i > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Ounit,type,
sup_sup_Product_unit: product_unit > product_unit > product_unit ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
sup_su1936195050962291414c_fm_i: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Ounit_J,type,
sup_su793286257634532545t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
sup_su2582925890723967158c_fm_i: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices__Big_Olinorder_OMax_001t__Nat__Onat,type,
lattices_Max_nat: ( nat > nat > $o ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Nat__Onat,type,
lattic7487310264250698137_i_nat: ( epistemic_fm_i > nat ) > set_Epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Product____Type__Ounit,type,
lattic1263872656861969706t_unit: set_Product_unit > product_unit ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
lattic8495942116038605215c_fm_i: set_se3485332733965609186c_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Product____Type__Ounit,type,
lattic5294303975357428420t_unit: set_Product_unit > product_unit ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
lattic3744900211830597177c_fm_i: set_se3485332733965609186c_fm_i > set_Epistemic_fm_i ).
thf(sy_c_List_Odistinct_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
distin5379319190628067867c_fm_i: list_Epistemic_fm_i > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Ofold_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
fold_P8565935628131843659t_unit: ( product_unit > product_unit > product_unit ) > list_Product_unit > product_unit > product_unit ).
thf(sy_c_List_Oinsert_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
insert177310195918387262c_fm_i: epistemic_fm_i > list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord8961336180081300637at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
cons_Epistemic_fm_i: epistemic_fm_i > list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Ounit,type,
cons_Product_unit: product_unit > list_Product_unit > list_Product_unit ).
thf(sy_c_List_Olist_ONil_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
nil_Epistemic_fm_i: list_Epistemic_fm_i ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Omap_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
map_Ep2755178516647988292c_fm_i: ( epistemic_fm_i > epistemic_fm_i ) > list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_List_Olist_Oset_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
set_Epistemic_fm_i2: list_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
set_li9010496130497932989c_fm_i: list_l4360594050116504750c_fm_i > set_li8269783359719514504c_fm_i ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Ounit,type,
set_Product_unit2: list_Product_unit > set_Product_unit ).
thf(sy_c_List_Olists_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
lists_Epistemic_fm_i: set_Epistemic_fm_i > set_li8269783359719514504c_fm_i ).
thf(sy_c_List_On__lists_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
n_list3185984609264368688c_fm_i: nat > list_Epistemic_fm_i > list_l4360594050116504750c_fm_i ).
thf(sy_c_List_Oremdups_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
remdup2862625226703050045c_fm_i: list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_List_Oremove1_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
remove8001893566138066622c_fm_i: epistemic_fm_i > list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
remove817376155714978094c_fm_i: epistemic_fm_i > list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_List_Osubseqs_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
subseq859285874047815415c_fm_i: list_Epistemic_fm_i > list_l4360594050116504750c_fm_i ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
maxima1924290493700099598c_fm_i: ( set_Epistemic_fm_i > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_Omaximal_001t__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
maxima2128808098672023021c_fm_i: ( set_Ep6763084006362001438c_fm_i > $o ) > set_Ep6763084006362001438c_fm_i > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_Omaximal_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
maxima3264069618988350929c_fm_i: ( set_Epistemic_fm_i > $o ) > set_Epistemic_fm_i > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS__Lim__Ord_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
maxima8116622286610175645c_fm_i: set_Pr4658907567593863815c_fm_i > ( set_Epistemic_fm_i > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS__Lim__Ord_001t__Nat__Onat,type,
maxima4501515429357695765rd_nat: set_Pr1261947904930325089at_nat > ( set_nat > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS__Lim__Ord_OextendS_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
maxima5458213620894205884c_fm_i: ( set_Epistemic_fm_i > $o ) > set_Epistemic_fm_i > epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
size_s3357407047445144980c_fm_i: list_Epistemic_fm_i > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_M_Eo_J,type,
bot_bo2580527446789968623fm_i_o: epistemic_fm_i > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
bot_bo6089404950617257231fm_i_o: set_Epistemic_fm_i > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Ounit,type,
bot_bot_Product_unit: product_unit ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
bot_bo281379535698878090c_fm_i: set_Ep6763084006362001438c_fm_i ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
bot_bo4194595901900360558c_fm_i: set_Epistemic_fm_i ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
bot_bo6562031733678445556c_fm_i: set_li8269783359719514504c_fm_i ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Ounit_J,type,
bot_bo3957492148770167129t_unit: set_Product_unit ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
bot_bo145720340923748686c_fm_i: set_se3485332733965609186c_fm_i ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
bot_bo4781621276559889198c_fm_i: set_se7339729205154126530c_fm_i ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__i_J,type,
bot_bot_set_i: set_i ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
ord_le2155353306922182815fm_i_o: ( episte1308613106011496894c_fm_i > $o ) > ( episte1308613106011496894c_fm_i > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_M_Eo_J,type,
ord_le190830114487426235fm_i_o: ( epistemic_fm_i > $o ) > ( epistemic_fm_i > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
ord_le204167446249480117fm_i_o: ( list_Epistemic_fm_i > $o ) > ( list_Epistemic_fm_i > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Ounit_M_Eo_J,type,
ord_le2788506356509622416unit_o: ( product_unit > $o ) > ( product_unit > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
ord_le5706303257367236315fm_i_o: ( set_Epistemic_fm_i > $o ) > ( set_Epistemic_fm_i > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_M_Eo_J,type,
ord_le1814827286658909435fm_i_o: ( set_se3485332733965609186c_fm_i > $o ) > ( set_se3485332733965609186c_fm_i > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__i_M_Eo_J,type,
ord_less_eq_i_o: ( i > $o ) > ( i > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Ounit,type,
ord_le3221252021190050221t_unit: product_unit > product_unit > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
ord_le1948573375516223422c_fm_i: set_Ep6763084006362001438c_fm_i > set_Ep6763084006362001438c_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
ord_le3843937902494030498c_fm_i: set_Epistemic_fm_i > set_Epistemic_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
ord_le1209247765174941992c_fm_i: set_li8269783359719514504c_fm_i > set_li8269783359719514504c_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Ounit_J,type,
ord_le3507040750410214029t_unit: set_Product_unit > set_Product_unit > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
ord_le5389487502678872194c_fm_i: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
ord_le2772500130360476258c_fm_i: set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__i_J,type,
ord_less_eq_set_i: set_i > set_i > $o ).
thf(sy_c_Partial__Function_Oflat__lub_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
partia6511039473171299566c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_Set_OCollect_001t__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
collec7849324479490780201c_fm_i: ( episte1308613106011496894c_fm_i > $o ) > set_Ep6763084006362001438c_fm_i ).
thf(sy_c_Set_OCollect_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
collec4904205187116291597c_fm_i: ( epistemic_fm_i > $o ) > set_Epistemic_fm_i ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
collec5759350117362652307c_fm_i: ( list_Epistemic_fm_i > $o ) > set_li8269783359719514504c_fm_i ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Ounit,type,
collect_Product_unit: ( product_unit > $o ) > set_Product_unit ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
collec4512488881437085321c_fm_i: ( set_Ep6763084006362001438c_fm_i > $o ) > set_se7087778041443875454c_fm_i ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
collec3087743281813070829c_fm_i: ( set_Epistemic_fm_i > $o ) > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
collec2761885132556217805c_fm_i: ( set_se3485332733965609186c_fm_i > $o ) > set_se7339729205154126530c_fm_i ).
thf(sy_c_Set_OCollect_001tf__i,type,
collect_i: ( i > $o ) > set_i ).
thf(sy_c_Set_OPow_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
pow_Epistemic_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_Ofilter_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
filter6053540608743173075c_fm_i: ( epistemic_fm_i > $o ) > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > set_nat > set_nat ).
thf(sy_c_Set_Ofilter_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
filter8450032351001387443c_fm_i: ( set_Epistemic_fm_i > $o ) > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oimage_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
image_120052958407486725c_fm_i: ( epistemic_fm_i > epistemic_fm_i ) > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Oimage_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Nat__Onat,type,
image_2291338433772226221_i_nat: ( epistemic_fm_i > nat ) > set_Epistemic_fm_i > set_nat ).
thf(sy_c_Set_Oimage_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Product____Type__Ounit,type,
image_7958789640865678704t_unit: ( epistemic_fm_i > product_unit ) > set_Epistemic_fm_i > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
image_2913337242412145381c_fm_i: ( epistemic_fm_i > set_Epistemic_fm_i ) > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oimage_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5232765980333981283et_nat: ( epistemic_fm_i > set_nat ) > set_Epistemic_fm_i > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
image_4310783338035737183c_fm_i: ( list_Epistemic_fm_i > set_Epistemic_fm_i ) > set_li8269783359719514504c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
image_3894954817185171339c_fm_i: ( nat > epistemic_fm_i ) > set_nat > set_Epistemic_fm_i ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
image_3332579458907315563c_fm_i: ( nat > set_Epistemic_fm_i ) > set_nat > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
image_405062704495631173t_unit: ( product_unit > product_unit ) > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
image_809639422248493861c_fm_i: ( set_Epistemic_fm_i > epistemic_fm_i ) > set_se3485332733965609186c_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Nat__Onat,type,
image_9130930184777177229_i_nat: ( set_Epistemic_fm_i > nat ) > set_se3485332733965609186c_fm_i > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
image_2698589095351095557c_fm_i: ( set_Epistemic_fm_i > set_Epistemic_fm_i ) > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
insert7817948997695205106c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Ounit,type,
insert_Product_unit: product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
insert7698009978809854162c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001tf__i,type,
insert_i: i > set_i > set_i ).
thf(sy_c_Set_Ois__singleton_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
is_sin4923117429174228502c_fm_i: set_Epistemic_fm_i > $o ).
thf(sy_c_Set_Oremove_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
remove5552923401211475591c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Stalnaker__Logic_OAx__2_001tf__i,type,
stalnaker_Ax_2_i: epistemic_fm_i > $o ).
thf(sy_c_Stalnaker__Logic_Oconjunct_001tf__i,type,
stalnaker_conjunct_i: list_Epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Stalnaker__Logic_Oweakly__directed_001tf__i_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
stalna1726574297087873274c_fm_i: episte4846006725945195288t_unit > $o ).
thf(sy_c_Stalnaker__Logic_Oweakly__directed_001tf__i_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
stalna5033156833633940736c_fm_i: episte8649862735182028056t_unit > $o ).
thf(sy_c_Stalnaker__Logic_Oweakly__directed_001tf__i_001t__Nat__Onat,type,
stalna2394636212805329400_i_nat: episte8354619181726739466t_unit > $o ).
thf(sy_c_Stalnaker__Logic_Oweakly__directed_001tf__i_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
stalna4517306826702924506c_fm_i: episte5489734138418521880t_unit > $o ).
thf(sy_c_Stalnaker__Logic_Oweakly__directed_001tf__i_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
stalna1393169086027530426c_fm_i: episte6041718960478671640t_unit > $o ).
thf(sy_c_Stalnaker__Logic_Oweakly__directed_001tf__i_001tf__i,type,
stalna3707719055912537822ed_i_i: episte2102749610958417048t_unit > $o ).
thf(sy_c_Zorn_Ochain__subset_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
chain_7805378461283022331c_fm_i: set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Zorn_Ochains_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
chains830531755891748338c_fm_i: set_se3485332733965609186c_fm_i > set_se7339729205154126530c_fm_i ).
thf(sy_c_fChoice_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
fChoic6950144925765728886c_fm_i: ( epistemic_fm_i > $o ) > epistemic_fm_i ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member1686840210572743143c_fm_i: episte1308613106011496894c_fm_i > set_Ep6763084006362001438c_fm_i > $o ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
member6642669606046002379c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > $o ).
thf(sy_c_member_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member6475149752759350353c_fm_i: list_Epistemic_fm_i > set_li8269783359719514504c_fm_i > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
member6124286717651598407c_fm_i: set_Ep6763084006362001438c_fm_i > set_se7087778041443875454c_fm_i > $o ).
thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member1104366573291651755c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
member1461078328125707403c_fm_i: set_se3485332733965609186c_fm_i > set_se7339729205154126530c_fm_i > $o ).
thf(sy_c_member_001tf__i,type,
member_i: i > set_i > $o ).
thf(sy_v_A,type,
a: epistemic_fm_i > $o ).
% Relevant facts (1277)
thf(fact_0_assms,axiom,
! [P: epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ P )
=> ( a @ P ) ) ).
% assms
thf(fact_1_kripke_Ocases,axiom,
! [R: episte5489734138418521880t_unit] :
~ ! [W: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi: set_Epistemic_fm_i > list_char > $o] :
( R
!= ( episte8397520288747125024t_unit @ W @ K @ ( episte4955834246032687171t_unit @ Pi @ product_Unity ) ) ) ).
% kripke.cases
thf(fact_2_maximal__extension,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ~ ! [W2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ V @ W2 )
=> ( ( episte2285483198712856234tent_i @ A @ W2 )
=> ~ ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W2 ) ) ) ) ).
% maximal_extension
thf(fact_3_kripke_Oext__inject,axiom,
! [Pi2: set_Epistemic_fm_i > list_char > $o,More: product_unit,Pi3: set_Epistemic_fm_i > list_char > $o,More2: product_unit] :
( ( ( episte4955834246032687171t_unit @ Pi2 @ More )
= ( episte4955834246032687171t_unit @ Pi3 @ More2 ) )
= ( ( Pi2 = Pi3 )
& ( More = More2 ) ) ) ).
% kripke.ext_inject
thf(fact_4_frame_Oext__inject,axiom,
! [W3: set_se3485332733965609186c_fm_i,K2: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More: episte7133455005732253587t_unit,W4: set_se3485332733965609186c_fm_i,K3: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More2: episte7133455005732253587t_unit] :
( ( ( episte8397520288747125024t_unit @ W3 @ K2 @ More )
= ( episte8397520288747125024t_unit @ W4 @ K3 @ More2 ) )
= ( ( W3 = W4 )
& ( K2 = K3 )
& ( More = More2 ) ) ) ).
% frame.ext_inject
thf(fact_5_fm_Oinject_I1_J,axiom,
! [X2: list_char,Y2: list_char] :
( ( ( epistemic_Pro_i @ X2 )
= ( epistemic_Pro_i @ Y2 ) )
= ( X2 = Y2 ) ) ).
% fm.inject(1)
thf(fact_6_fm_Oinject_I5_J,axiom,
! [X61: epistemic_fm_i,X62: episte1308613106011496894c_fm_i,Y61: epistemic_fm_i,Y62: episte1308613106011496894c_fm_i] :
( ( ( episte5657488666450005526c_fm_i @ X61 @ X62 )
= ( episte5657488666450005526c_fm_i @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_7_fm_Oinject_I5_J,axiom,
! [X61: i,X62: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
( ( ( epistemic_K_i @ X61 @ X62 )
= ( epistemic_K_i @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_8_kripke_Ocases__scheme,axiom,
! [R: episte5489734138418521880t_unit] :
~ ! [W: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi: set_Epistemic_fm_i > list_char > $o,More3: product_unit] :
( R
!= ( episte8397520288747125024t_unit @ W @ K @ ( episte4955834246032687171t_unit @ Pi @ More3 ) ) ) ).
% kripke.cases_scheme
thf(fact_9_subsetI,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ! [X: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ A )
=> ( member1461078328125707403c_fm_i @ X @ B ) )
=> ( ord_le2772500130360476258c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_10_subsetI,axiom,
! [A: set_i,B: set_i] :
( ! [X: i] :
( ( member_i @ X @ A )
=> ( member_i @ X @ B ) )
=> ( ord_less_eq_set_i @ A @ B ) ) ).
% subsetI
thf(fact_11_subsetI,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ! [X: product_unit] :
( ( member_Product_unit @ X @ A )
=> ( member_Product_unit @ X @ B ) )
=> ( ord_le3507040750410214029t_unit @ A @ B ) ) ).
% subsetI
thf(fact_12_subsetI,axiom,
! [A: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i] :
( ! [X: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X @ A )
=> ( member6475149752759350353c_fm_i @ X @ B ) )
=> ( ord_le1209247765174941992c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_13_subsetI,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ( member1104366573291651755c_fm_i @ X @ B ) )
=> ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_14_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_15_subsetI,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ( member6642669606046002379c_fm_i @ X @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_16_subset__antisym,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ( ord_le3507040750410214029t_unit @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_17_subset__antisym,axiom,
! [A: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A @ B )
=> ( ( ord_le1209247765174941992c_fm_i @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_18_subset__antisym,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_19_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_20_subset__antisym,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_21_Collect__mono__iff,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_22_Collect__mono__iff,axiom,
! [P2: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q ) )
= ( ! [X3: list_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_23_Collect__mono__iff,axiom,
! [P2: product_unit > $o,Q: product_unit > $o] :
( ( ord_le3507040750410214029t_unit @ ( collect_Product_unit @ P2 ) @ ( collect_Product_unit @ Q ) )
= ( ! [X3: product_unit] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_24_Collect__mono__iff,axiom,
! [P2: list_Epistemic_fm_i > $o,Q: list_Epistemic_fm_i > $o] :
( ( ord_le1209247765174941992c_fm_i @ ( collec5759350117362652307c_fm_i @ P2 ) @ ( collec5759350117362652307c_fm_i @ Q ) )
= ( ! [X3: list_Epistemic_fm_i] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_25_Collect__mono__iff,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) @ ( collec3087743281813070829c_fm_i @ Q ) )
= ( ! [X3: set_Epistemic_fm_i] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_26_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_27_Collect__mono__iff,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) @ ( collec4904205187116291597c_fm_i @ Q ) )
= ( ! [X3: epistemic_fm_i] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_28_set__eq__subset,axiom,
( ( ^ [Y: set_Product_unit,Z: set_Product_unit] : ( Y = Z ) )
= ( ^ [A2: set_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ B2 )
& ( ord_le3507040750410214029t_unit @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_29_set__eq__subset,axiom,
( ( ^ [Y: set_li8269783359719514504c_fm_i,Z: set_li8269783359719514504c_fm_i] : ( Y = Z ) )
= ( ^ [A2: set_li8269783359719514504c_fm_i,B2: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A2 @ B2 )
& ( ord_le1209247765174941992c_fm_i @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_30_set__eq__subset,axiom,
( ( ^ [Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( Y = Z ) )
= ( ^ [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B2 )
& ( ord_le5389487502678872194c_fm_i @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_31_set__eq__subset,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_32_set__eq__subset,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z ) )
= ( ^ [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
& ( ord_le3843937902494030498c_fm_i @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_33_subset__trans,axiom,
! [A: set_Product_unit,B: set_Product_unit,C: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ( ord_le3507040750410214029t_unit @ B @ C )
=> ( ord_le3507040750410214029t_unit @ A @ C ) ) ) ).
% subset_trans
thf(fact_34_subset__trans,axiom,
! [A: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i,C: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A @ B )
=> ( ( ord_le1209247765174941992c_fm_i @ B @ C )
=> ( ord_le1209247765174941992c_fm_i @ A @ C ) ) ) ).
% subset_trans
thf(fact_35_subset__trans,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ C )
=> ( ord_le5389487502678872194c_fm_i @ A @ C ) ) ) ).
% subset_trans
thf(fact_36_subset__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_37_subset__trans,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ord_le3843937902494030498c_fm_i @ A @ C ) ) ) ).
% subset_trans
thf(fact_38_Collect__mono,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ! [X: set_nat] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_39_Collect__mono,axiom,
! [P2: list_nat > $o,Q: list_nat > $o] :
( ! [X: list_nat] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_40_Collect__mono,axiom,
! [P2: product_unit > $o,Q: product_unit > $o] :
( ! [X: product_unit] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le3507040750410214029t_unit @ ( collect_Product_unit @ P2 ) @ ( collect_Product_unit @ Q ) ) ) ).
% Collect_mono
thf(fact_41_Collect__mono,axiom,
! [P2: list_Epistemic_fm_i > $o,Q: list_Epistemic_fm_i > $o] :
( ! [X: list_Epistemic_fm_i] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le1209247765174941992c_fm_i @ ( collec5759350117362652307c_fm_i @ P2 ) @ ( collec5759350117362652307c_fm_i @ Q ) ) ) ).
% Collect_mono
thf(fact_42_Collect__mono,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ! [X: set_Epistemic_fm_i] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ).
% Collect_mono
thf(fact_43_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_44_Collect__mono,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ! [X: epistemic_fm_i] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ).
% Collect_mono
thf(fact_45_subset__refl,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ A ) ).
% subset_refl
thf(fact_46_subset__refl,axiom,
! [A: set_li8269783359719514504c_fm_i] : ( ord_le1209247765174941992c_fm_i @ A @ A ) ).
% subset_refl
thf(fact_47_subset__refl,axiom,
! [A: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ A @ A ) ).
% subset_refl
thf(fact_48_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_49_subset__refl,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ A ) ).
% subset_refl
thf(fact_50_subset__iff,axiom,
( ord_le2772500130360476258c_fm_i
= ( ^ [A2: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i] :
! [T: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ T @ A2 )
=> ( member1461078328125707403c_fm_i @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_51_subset__iff,axiom,
( ord_less_eq_set_i
= ( ^ [A2: set_i,B2: set_i] :
! [T: i] :
( ( member_i @ T @ A2 )
=> ( member_i @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_52_subset__iff,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [A2: set_Product_unit,B2: set_Product_unit] :
! [T: product_unit] :
( ( member_Product_unit @ T @ A2 )
=> ( member_Product_unit @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_53_subset__iff,axiom,
( ord_le1209247765174941992c_fm_i
= ( ^ [A2: set_li8269783359719514504c_fm_i,B2: set_li8269783359719514504c_fm_i] :
! [T: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ T @ A2 )
=> ( member6475149752759350353c_fm_i @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_54_subset__iff,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
! [T: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ T @ A2 )
=> ( member1104366573291651755c_fm_i @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_55_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A2 )
=> ( member_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_56_subset__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
! [T: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ T @ A2 )
=> ( member6642669606046002379c_fm_i @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_57_equalityD2,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( A = B )
=> ( ord_le3507040750410214029t_unit @ B @ A ) ) ).
% equalityD2
thf(fact_58_equalityD2,axiom,
! [A: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i] :
( ( A = B )
=> ( ord_le1209247765174941992c_fm_i @ B @ A ) ) ).
% equalityD2
thf(fact_59_equalityD2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( A = B )
=> ( ord_le5389487502678872194c_fm_i @ B @ A ) ) ).
% equalityD2
thf(fact_60_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_61_equalityD2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A = B )
=> ( ord_le3843937902494030498c_fm_i @ B @ A ) ) ).
% equalityD2
thf(fact_62_equalityD1,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( A = B )
=> ( ord_le3507040750410214029t_unit @ A @ B ) ) ).
% equalityD1
thf(fact_63_equalityD1,axiom,
! [A: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i] :
( ( A = B )
=> ( ord_le1209247765174941992c_fm_i @ A @ B ) ) ).
% equalityD1
thf(fact_64_equalityD1,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( A = B )
=> ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ).
% equalityD1
thf(fact_65_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_66_equalityD1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A = B )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% equalityD1
thf(fact_67_subset__eq,axiom,
( ord_le2772500130360476258c_fm_i
= ( ^ [A2: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i] :
! [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ A2 )
=> ( member1461078328125707403c_fm_i @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_68_subset__eq,axiom,
( ord_less_eq_set_i
= ( ^ [A2: set_i,B2: set_i] :
! [X3: i] :
( ( member_i @ X3 @ A2 )
=> ( member_i @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_69_subset__eq,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [A2: set_Product_unit,B2: set_Product_unit] :
! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A2 )
=> ( member_Product_unit @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_70_subset__eq,axiom,
( ord_le1209247765174941992c_fm_i
= ( ^ [A2: set_li8269783359719514504c_fm_i,B2: set_li8269783359719514504c_fm_i] :
! [X3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X3 @ A2 )
=> ( member6475149752759350353c_fm_i @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_71_subset__eq,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A2 )
=> ( member1104366573291651755c_fm_i @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_72_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_73_subset__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A2 )
=> ( member6642669606046002379c_fm_i @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_74_equalityE,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( A = B )
=> ~ ( ( ord_le3507040750410214029t_unit @ A @ B )
=> ~ ( ord_le3507040750410214029t_unit @ B @ A ) ) ) ).
% equalityE
thf(fact_75_equalityE,axiom,
! [A: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i] :
( ( A = B )
=> ~ ( ( ord_le1209247765174941992c_fm_i @ A @ B )
=> ~ ( ord_le1209247765174941992c_fm_i @ B @ A ) ) ) ).
% equalityE
thf(fact_76_equalityE,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( A = B )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ~ ( ord_le5389487502678872194c_fm_i @ B @ A ) ) ) ).
% equalityE
thf(fact_77_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_78_equalityE,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A = B )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ~ ( ord_le3843937902494030498c_fm_i @ B @ A ) ) ) ).
% equalityE
thf(fact_79_subsetD,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A @ B )
=> ( ( member1461078328125707403c_fm_i @ C2 @ A )
=> ( member1461078328125707403c_fm_i @ C2 @ B ) ) ) ).
% subsetD
thf(fact_80_subsetD,axiom,
! [A: set_i,B: set_i,C2: i] :
( ( ord_less_eq_set_i @ A @ B )
=> ( ( member_i @ C2 @ A )
=> ( member_i @ C2 @ B ) ) ) ).
% subsetD
thf(fact_81_subsetD,axiom,
! [A: set_Product_unit,B: set_Product_unit,C2: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ( member_Product_unit @ C2 @ A )
=> ( member_Product_unit @ C2 @ B ) ) ) ).
% subsetD
thf(fact_82_subsetD,axiom,
! [A: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i,C2: list_Epistemic_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A @ B )
=> ( ( member6475149752759350353c_fm_i @ C2 @ A )
=> ( member6475149752759350353c_fm_i @ C2 @ B ) ) ) ).
% subsetD
thf(fact_83_subsetD,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( member1104366573291651755c_fm_i @ C2 @ A )
=> ( member1104366573291651755c_fm_i @ C2 @ B ) ) ) ).
% subsetD
thf(fact_84_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_85_subsetD,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( member6642669606046002379c_fm_i @ C2 @ A )
=> ( member6642669606046002379c_fm_i @ C2 @ B ) ) ) ).
% subsetD
thf(fact_86_in__mono,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A @ B )
=> ( ( member1461078328125707403c_fm_i @ X4 @ A )
=> ( member1461078328125707403c_fm_i @ X4 @ B ) ) ) ).
% in_mono
thf(fact_87_in__mono,axiom,
! [A: set_i,B: set_i,X4: i] :
( ( ord_less_eq_set_i @ A @ B )
=> ( ( member_i @ X4 @ A )
=> ( member_i @ X4 @ B ) ) ) ).
% in_mono
thf(fact_88_in__mono,axiom,
! [A: set_Product_unit,B: set_Product_unit,X4: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ( member_Product_unit @ X4 @ A )
=> ( member_Product_unit @ X4 @ B ) ) ) ).
% in_mono
thf(fact_89_in__mono,axiom,
! [A: set_li8269783359719514504c_fm_i,B: set_li8269783359719514504c_fm_i,X4: list_Epistemic_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A @ B )
=> ( ( member6475149752759350353c_fm_i @ X4 @ A )
=> ( member6475149752759350353c_fm_i @ X4 @ B ) ) ) ).
% in_mono
thf(fact_90_in__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( member1104366573291651755c_fm_i @ X4 @ B ) ) ) ).
% in_mono
thf(fact_91_in__mono,axiom,
! [A: set_nat,B: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_92_in__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X4: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( member6642669606046002379c_fm_i @ X4 @ B ) ) ) ).
% in_mono
thf(fact_93_old_Ounit_Oexhaust,axiom,
! [Y3: product_unit] : ( Y3 = product_Unity ) ).
% old.unit.exhaust
thf(fact_94_frame_Ocases__scheme,axiom,
! [R: episte5489734138418521880t_unit] :
~ ! [W: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More3: episte7133455005732253587t_unit] :
( R
!= ( episte8397520288747125024t_unit @ W @ K @ More3 ) ) ).
% frame.cases_scheme
thf(fact_95_kripke_Oext__induct,axiom,
! [P2: episte7133455005732253587t_unit > $o,R: episte7133455005732253587t_unit] :
( ! [Pi: set_Epistemic_fm_i > list_char > $o,More3: product_unit] : ( P2 @ ( episte4955834246032687171t_unit @ Pi @ More3 ) )
=> ( P2 @ R ) ) ).
% kripke.ext_induct
thf(fact_96_Collect__subset,axiom,
! [A: set_se7339729205154126530c_fm_i,P2: set_se3485332733965609186c_fm_i > $o] :
( ord_le2772500130360476258c_fm_i
@ ( collec2761885132556217805c_fm_i
@ ^ [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_97_Collect__subset,axiom,
! [A: set_i,P2: i > $o] :
( ord_less_eq_set_i
@ ( collect_i
@ ^ [X3: i] :
( ( member_i @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_98_Collect__subset,axiom,
! [A: set_set_nat,P2: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_99_Collect__subset,axiom,
! [A: set_list_nat,P2: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_100_Collect__subset,axiom,
! [A: set_Product_unit,P2: product_unit > $o] :
( ord_le3507040750410214029t_unit
@ ( collect_Product_unit
@ ^ [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_101_Collect__subset,axiom,
! [A: set_li8269783359719514504c_fm_i,P2: list_Epistemic_fm_i > $o] :
( ord_le1209247765174941992c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [X3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_102_Collect__subset,axiom,
! [A: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o] :
( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_103_Collect__subset,axiom,
! [A: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_104_Collect__subset,axiom,
! [A: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_105_fm_Odistinct_I17_J,axiom,
! [X2: list_char,X61: epistemic_fm_i,X62: episte1308613106011496894c_fm_i] :
( ( episte3759128500599061780c_fm_i @ X2 )
!= ( episte5657488666450005526c_fm_i @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_106_fm_Odistinct_I17_J,axiom,
! [X2: list_char,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Pro_i @ X2 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_107_consistent__hereditary,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,S2: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( ( ord_le3843937902494030498c_fm_i @ S2 @ S )
=> ( episte2285483198712856234tent_i @ A @ S2 ) ) ) ).
% consistent_hereditary
thf(fact_108_ax__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( A @ P3 )
=> ( member6642669606046002379c_fm_i @ P3 @ V ) ) ) ) ).
% ax_in_maximal
thf(fact_109_truth__lemma,axiom,
! [A: episte1308613106011496894c_fm_i > $o,V: set_Ep6763084006362001438c_fm_i,P3: episte1308613106011496894c_fm_i] :
( ( episte704884158563092166c_fm_i @ A @ V )
=> ( ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ V )
=> ( ( member1686840210572743143c_fm_i @ P3 @ V )
= ( episte341539128385704480c_fm_i
@ ( episte7444838911182875652t_unit
@ ( collec4512488881437085321c_fm_i
@ ^ [W5: set_Ep6763084006362001438c_fm_i] :
( ( episte704884158563092166c_fm_i @ A @ W5 )
& ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ W5 ) ) )
@ ^ [I: epistemic_fm_i,V2: set_Ep6763084006362001438c_fm_i] :
( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte7452740295420953255t_unit
@ ^ [V2: set_Ep6763084006362001438c_fm_i,X3: list_char] : ( member1686840210572743143c_fm_i @ ( episte3759128500599061780c_fm_i @ X3 ) @ V2 )
@ product_Unity ) )
@ V
@ P3 ) ) ) ) ).
% truth_lemma
thf(fact_110_truth__lemma,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ P3 @ V )
= ( episte2498298383968489120c_fm_i
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W5: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W5 )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W5 ) ) )
@ ^ [I: i,V2: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V2: set_Epistemic_fm_i,X3: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X3 ) @ V2 )
@ product_Unity ) )
@ V
@ P3 ) ) ) ) ).
% truth_lemma
thf(fact_111_default__unit__def,axiom,
defaul566961228789861419t_unit = product_Unity ).
% default_unit_def
thf(fact_112_dual__order_Orefl,axiom,
! [A3: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_113_dual__order_Orefl,axiom,
! [A3: set_li8269783359719514504c_fm_i] : ( ord_le1209247765174941992c_fm_i @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_114_dual__order_Orefl,axiom,
! [A3: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_115_dual__order_Orefl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_116_dual__order_Orefl,axiom,
! [A3: epistemic_fm_i > $o] : ( ord_le190830114487426235fm_i_o @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_117_dual__order_Orefl,axiom,
! [A3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_118_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_119_order__refl,axiom,
! [X4: set_Product_unit] : ( ord_le3507040750410214029t_unit @ X4 @ X4 ) ).
% order_refl
thf(fact_120_order__refl,axiom,
! [X4: set_li8269783359719514504c_fm_i] : ( ord_le1209247765174941992c_fm_i @ X4 @ X4 ) ).
% order_refl
thf(fact_121_order__refl,axiom,
! [X4: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ X4 @ X4 ) ).
% order_refl
thf(fact_122_order__refl,axiom,
! [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_123_order__refl,axiom,
! [X4: epistemic_fm_i > $o] : ( ord_le190830114487426235fm_i_o @ X4 @ X4 ) ).
% order_refl
thf(fact_124_order__refl,axiom,
! [X4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X4 @ X4 ) ).
% order_refl
thf(fact_125_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_126_kripke_Oupdate__convs_I2_J,axiom,
! [More2: product_unit > product_unit,W3: set_se3485332733965609186c_fm_i,K2: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi2: set_Epistemic_fm_i > list_char > $o,More: product_unit] :
( ( episte4537596280781771625c_fm_i @ More2 @ ( episte8397520288747125024t_unit @ W3 @ K2 @ ( episte4955834246032687171t_unit @ Pi2 @ More ) ) )
= ( episte8397520288747125024t_unit @ W3 @ K2 @ ( episte4955834246032687171t_unit @ Pi2 @ ( More2 @ More ) ) ) ) ).
% kripke.update_convs(2)
thf(fact_127_kripke_Oselect__convs_I1_J,axiom,
! [W3: set_se3485332733965609186c_fm_i,K2: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi2: set_Epistemic_fm_i > list_char > $o,More: product_unit] :
( ( episte6627496282998226676t_unit @ ( episte8397520288747125024t_unit @ W3 @ K2 @ ( episte4955834246032687171t_unit @ Pi2 @ More ) ) )
= Pi2 ) ).
% kripke.select_convs(1)
thf(fact_128_kripke_Oselect__convs_I2_J,axiom,
! [W3: set_se3485332733965609186c_fm_i,K2: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi2: set_Epistemic_fm_i > list_char > $o,More: product_unit] :
( ( episte484680259488342554t_unit @ ( episte8397520288747125024t_unit @ W3 @ K2 @ ( episte4955834246032687171t_unit @ Pi2 @ More ) ) )
= More ) ).
% kripke.select_convs(2)
thf(fact_129_kripke_Oupdate__convs_I1_J,axiom,
! [Pi3: ( set_Epistemic_fm_i > list_char > $o ) > set_Epistemic_fm_i > list_char > $o,W3: set_se3485332733965609186c_fm_i,K2: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi2: set_Epistemic_fm_i > list_char > $o,More: product_unit] :
( ( episte8744582347493487171t_unit @ Pi3 @ ( episte8397520288747125024t_unit @ W3 @ K2 @ ( episte4955834246032687171t_unit @ Pi2 @ More ) ) )
= ( episte8397520288747125024t_unit @ W3 @ K2 @ ( episte4955834246032687171t_unit @ ( Pi3 @ Pi2 ) @ More ) ) ) ).
% kripke.update_convs(1)
thf(fact_130_mem__Collect__eq,axiom,
! [A3: set_se3485332733965609186c_fm_i,P2: set_se3485332733965609186c_fm_i > $o] :
( ( member1461078328125707403c_fm_i @ A3 @ ( collec2761885132556217805c_fm_i @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_131_mem__Collect__eq,axiom,
! [A3: i,P2: i > $o] :
( ( member_i @ A3 @ ( collect_i @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_132_mem__Collect__eq,axiom,
! [A3: set_nat,P2: set_nat > $o] :
( ( member_set_nat @ A3 @ ( collect_set_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_133_mem__Collect__eq,axiom,
! [A3: list_Epistemic_fm_i,P2: list_Epistemic_fm_i > $o] :
( ( member6475149752759350353c_fm_i @ A3 @ ( collec5759350117362652307c_fm_i @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_134_mem__Collect__eq,axiom,
! [A3: list_nat,P2: list_nat > $o] :
( ( member_list_nat @ A3 @ ( collect_list_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_135_mem__Collect__eq,axiom,
! [A3: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ A3 @ ( collec3087743281813070829c_fm_i @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_136_mem__Collect__eq,axiom,
! [A3: epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ A3 @ ( collec4904205187116291597c_fm_i @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_137_mem__Collect__eq,axiom,
! [A3: nat,P2: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_138_Collect__mem__eq,axiom,
! [A: set_se7339729205154126530c_fm_i] :
( ( collec2761885132556217805c_fm_i
@ ^ [X3: set_se3485332733965609186c_fm_i] : ( member1461078328125707403c_fm_i @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A: set_i] :
( ( collect_i
@ ^ [X3: i] : ( member_i @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_140_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
! [A: set_li8269783359719514504c_fm_i] :
( ( collec5759350117362652307c_fm_i
@ ^ [X3: list_Epistemic_fm_i] : ( member6475149752759350353c_fm_i @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_142_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_143_Collect__mem__eq,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_144_Collect__mem__eq,axiom,
! [A: set_Epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_145_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_146_Collect__cong,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ! [X: set_nat] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collect_set_nat @ P2 )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_147_Collect__cong,axiom,
! [P2: list_Epistemic_fm_i > $o,Q: list_Epistemic_fm_i > $o] :
( ! [X: list_Epistemic_fm_i] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collec5759350117362652307c_fm_i @ P2 )
= ( collec5759350117362652307c_fm_i @ Q ) ) ) ).
% Collect_cong
thf(fact_148_Collect__cong,axiom,
! [P2: list_nat > $o,Q: list_nat > $o] :
( ! [X: list_nat] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collect_list_nat @ P2 )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_149_Collect__cong,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ! [X: set_Epistemic_fm_i] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collec3087743281813070829c_fm_i @ P2 )
= ( collec3087743281813070829c_fm_i @ Q ) ) ) ).
% Collect_cong
thf(fact_150_Collect__cong,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ! [X: epistemic_fm_i] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collec4904205187116291597c_fm_i @ P2 )
= ( collec4904205187116291597c_fm_i @ Q ) ) ) ).
% Collect_cong
thf(fact_151_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_152_subset__Collect__iff,axiom,
! [B: set_se7339729205154126530c_fm_i,A: set_se7339729205154126530c_fm_i,P2: set_se3485332733965609186c_fm_i > $o] :
( ( ord_le2772500130360476258c_fm_i @ B @ A )
=> ( ( ord_le2772500130360476258c_fm_i @ B
@ ( collec2761885132556217805c_fm_i
@ ^ [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) )
= ( ! [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ B )
=> ( P2 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_153_subset__Collect__iff,axiom,
! [B: set_i,A: set_i,P2: i > $o] :
( ( ord_less_eq_set_i @ B @ A )
=> ( ( ord_less_eq_set_i @ B
@ ( collect_i
@ ^ [X3: i] :
( ( member_i @ X3 @ A )
& ( P2 @ X3 ) ) ) )
= ( ! [X3: i] :
( ( member_i @ X3 @ B )
=> ( P2 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_154_subset__Collect__iff,axiom,
! [B: set_set_nat,A: set_set_nat,P2: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ B
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( P2 @ X3 ) ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
=> ( P2 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_155_subset__Collect__iff,axiom,
! [B: set_list_nat,A: set_list_nat,P2: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ B
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( P2 @ X3 ) ) ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ B )
=> ( P2 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_156_subset__Collect__iff,axiom,
! [B: set_Product_unit,A: set_Product_unit,P2: product_unit > $o] :
( ( ord_le3507040750410214029t_unit @ B @ A )
=> ( ( ord_le3507040750410214029t_unit @ B
@ ( collect_Product_unit
@ ^ [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
& ( P2 @ X3 ) ) ) )
= ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ B )
=> ( P2 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_157_subset__Collect__iff,axiom,
! [B: set_li8269783359719514504c_fm_i,A: set_li8269783359719514504c_fm_i,P2: list_Epistemic_fm_i > $o] :
( ( ord_le1209247765174941992c_fm_i @ B @ A )
=> ( ( ord_le1209247765174941992c_fm_i @ B
@ ( collec5759350117362652307c_fm_i
@ ^ [X3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) )
= ( ! [X3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X3 @ B )
=> ( P2 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_158_subset__Collect__iff,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ B
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) )
= ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ B )
=> ( P2 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_159_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P2 @ X3 ) ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( P2 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_160_subset__Collect__iff,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ B
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) )
= ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ B )
=> ( P2 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_161_subset__CollectI,axiom,
! [B: set_se7339729205154126530c_fm_i,A: set_se7339729205154126530c_fm_i,Q: set_se3485332733965609186c_fm_i > $o,P2: set_se3485332733965609186c_fm_i > $o] :
( ( ord_le2772500130360476258c_fm_i @ B @ A )
=> ( ! [X: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ B )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_le2772500130360476258c_fm_i
@ ( collec2761885132556217805c_fm_i
@ ^ [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collec2761885132556217805c_fm_i
@ ^ [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_162_subset__CollectI,axiom,
! [B: set_i,A: set_i,Q: i > $o,P2: i > $o] :
( ( ord_less_eq_set_i @ B @ A )
=> ( ! [X: i] :
( ( member_i @ X @ B )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_less_eq_set_i
@ ( collect_i
@ ^ [X3: i] :
( ( member_i @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_i
@ ^ [X3: i] :
( ( member_i @ X3 @ A )
& ( P2 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_163_subset__CollectI,axiom,
! [B: set_set_nat,A: set_set_nat,Q: set_nat > $o,P2: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ B )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( P2 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_164_subset__CollectI,axiom,
! [B: set_list_nat,A: set_list_nat,Q: list_nat > $o,P2: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ! [X: list_nat] :
( ( member_list_nat @ X @ B )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( P2 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_165_subset__CollectI,axiom,
! [B: set_Product_unit,A: set_Product_unit,Q: product_unit > $o,P2: product_unit > $o] :
( ( ord_le3507040750410214029t_unit @ B @ A )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ B )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_le3507040750410214029t_unit
@ ( collect_Product_unit
@ ^ [X3: product_unit] :
( ( member_Product_unit @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_Product_unit
@ ^ [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
& ( P2 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_166_subset__CollectI,axiom,
! [B: set_li8269783359719514504c_fm_i,A: set_li8269783359719514504c_fm_i,Q: list_Epistemic_fm_i > $o,P2: list_Epistemic_fm_i > $o] :
( ( ord_le1209247765174941992c_fm_i @ B @ A )
=> ( ! [X: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X @ B )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_le1209247765174941992c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [X3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collec5759350117362652307c_fm_i
@ ^ [X3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_167_subset__CollectI,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,Q: set_Epistemic_fm_i > $o,P2: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ B )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_168_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P2 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_169_subset__CollectI,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i,Q: epistemic_fm_i > $o,P2: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_170_Collect__restrict,axiom,
! [X5: set_se7339729205154126530c_fm_i,P2: set_se3485332733965609186c_fm_i > $o] :
( ord_le2772500130360476258c_fm_i
@ ( collec2761885132556217805c_fm_i
@ ^ [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ X5 )
& ( P2 @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_171_Collect__restrict,axiom,
! [X5: set_i,P2: i > $o] :
( ord_less_eq_set_i
@ ( collect_i
@ ^ [X3: i] :
( ( member_i @ X3 @ X5 )
& ( P2 @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_172_Collect__restrict,axiom,
! [X5: set_set_nat,P2: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
& ( P2 @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_173_Collect__restrict,axiom,
! [X5: set_list_nat,P2: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ X5 )
& ( P2 @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_174_Collect__restrict,axiom,
! [X5: set_Product_unit,P2: product_unit > $o] :
( ord_le3507040750410214029t_unit
@ ( collect_Product_unit
@ ^ [X3: product_unit] :
( ( member_Product_unit @ X3 @ X5 )
& ( P2 @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_175_Collect__restrict,axiom,
! [X5: set_li8269783359719514504c_fm_i,P2: list_Epistemic_fm_i > $o] :
( ord_le1209247765174941992c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [X3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X3 @ X5 )
& ( P2 @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_176_Collect__restrict,axiom,
! [X5: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o] :
( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ X5 )
& ( P2 @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_177_Collect__restrict,axiom,
! [X5: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P2 @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_178_Collect__restrict,axiom,
! [X5: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ X5 )
& ( P2 @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_179_kripke_Ofold__congs_I4_J,axiom,
! [R: episte5489734138418521880t_unit,R2: episte5489734138418521880t_unit,V3: product_unit,F: product_unit > product_unit,F2: product_unit > product_unit] :
( ( R = R2 )
=> ( ( ( episte484680259488342554t_unit @ R2 )
= V3 )
=> ( ! [V4: product_unit] :
( ( V3 = V4 )
=> ( ( F @ V4 )
= ( F2 @ V4 ) ) )
=> ( ( episte4537596280781771625c_fm_i @ F @ R )
= ( episte4537596280781771625c_fm_i @ F2 @ R2 ) ) ) ) ) ).
% kripke.fold_congs(4)
thf(fact_180_kripke_Ofold__congs_I3_J,axiom,
! [R: episte5489734138418521880t_unit,R2: episte5489734138418521880t_unit,V3: set_Epistemic_fm_i > list_char > $o,F: ( set_Epistemic_fm_i > list_char > $o ) > set_Epistemic_fm_i > list_char > $o,F2: ( set_Epistemic_fm_i > list_char > $o ) > set_Epistemic_fm_i > list_char > $o] :
( ( R = R2 )
=> ( ( ( episte6627496282998226676t_unit @ R2 )
= V3 )
=> ( ! [V4: set_Epistemic_fm_i > list_char > $o] :
( ( V3 = V4 )
=> ( ( F @ V4 )
= ( F2 @ V4 ) ) )
=> ( ( episte8744582347493487171t_unit @ F @ R )
= ( episte8744582347493487171t_unit @ F2 @ R2 ) ) ) ) ) ).
% kripke.fold_congs(3)
thf(fact_181_kripke_Ounfold__congs_I4_J,axiom,
! [R: episte5489734138418521880t_unit,R2: episte5489734138418521880t_unit,V3: product_unit,F: product_unit > product_unit,F2: product_unit > product_unit] :
( ( R = R2 )
=> ( ( ( episte484680259488342554t_unit @ R2 )
= V3 )
=> ( ! [V4: product_unit] :
( ( V4 = V3 )
=> ( ( F @ V4 )
= ( F2 @ V4 ) ) )
=> ( ( episte4537596280781771625c_fm_i @ F @ R )
= ( episte4537596280781771625c_fm_i @ F2 @ R2 ) ) ) ) ) ).
% kripke.unfold_congs(4)
thf(fact_182_kripke_Ounfold__congs_I3_J,axiom,
! [R: episte5489734138418521880t_unit,R2: episte5489734138418521880t_unit,V3: set_Epistemic_fm_i > list_char > $o,F: ( set_Epistemic_fm_i > list_char > $o ) > set_Epistemic_fm_i > list_char > $o,F2: ( set_Epistemic_fm_i > list_char > $o ) > set_Epistemic_fm_i > list_char > $o] :
( ( R = R2 )
=> ( ( ( episte6627496282998226676t_unit @ R2 )
= V3 )
=> ( ! [V4: set_Epistemic_fm_i > list_char > $o] :
( ( V4 = V3 )
=> ( ( F @ V4 )
= ( F2 @ V4 ) ) )
=> ( ( episte8744582347493487171t_unit @ F @ R )
= ( episte8744582347493487171t_unit @ F2 @ R2 ) ) ) ) ) ).
% kripke.unfold_congs(3)
thf(fact_183_semantics_Osimps_I2_J,axiom,
! [M: episte4846006725945195288t_unit,W6: epistemic_fm_i,X4: list_char] :
( ( episte5182706529199770816c_fm_i @ M @ W6 @ ( epistemic_Pro_i @ X4 ) )
= ( episte4915868227582937300t_unit @ M @ W6 @ X4 ) ) ).
% semantics.simps(2)
thf(fact_184_semantics_Osimps_I2_J,axiom,
! [M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i,X4: list_char] :
( ( episte2498298383968489120c_fm_i @ M @ W6 @ ( epistemic_Pro_i @ X4 ) )
= ( episte6627496282998226676t_unit @ M @ W6 @ X4 ) ) ).
% semantics.simps(2)
thf(fact_185_less__eq__set__def,axiom,
( ord_le2772500130360476258c_fm_i
= ( ^ [A2: set_se7339729205154126530c_fm_i,B2: set_se7339729205154126530c_fm_i] :
( ord_le1814827286658909435fm_i_o
@ ^ [X3: set_se3485332733965609186c_fm_i] : ( member1461078328125707403c_fm_i @ X3 @ A2 )
@ ^ [X3: set_se3485332733965609186c_fm_i] : ( member1461078328125707403c_fm_i @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_186_less__eq__set__def,axiom,
( ord_less_eq_set_i
= ( ^ [A2: set_i,B2: set_i] :
( ord_less_eq_i_o
@ ^ [X3: i] : ( member_i @ X3 @ A2 )
@ ^ [X3: i] : ( member_i @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_187_less__eq__set__def,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [A2: set_Product_unit,B2: set_Product_unit] :
( ord_le2788506356509622416unit_o
@ ^ [X3: product_unit] : ( member_Product_unit @ X3 @ A2 )
@ ^ [X3: product_unit] : ( member_Product_unit @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_188_less__eq__set__def,axiom,
( ord_le1209247765174941992c_fm_i
= ( ^ [A2: set_li8269783359719514504c_fm_i,B2: set_li8269783359719514504c_fm_i] :
( ord_le204167446249480117fm_i_o
@ ^ [X3: list_Epistemic_fm_i] : ( member6475149752759350353c_fm_i @ X3 @ A2 )
@ ^ [X3: list_Epistemic_fm_i] : ( member6475149752759350353c_fm_i @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_189_less__eq__set__def,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ord_le5706303257367236315fm_i_o
@ ^ [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ A2 )
@ ^ [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_190_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_191_less__eq__set__def,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ord_le190830114487426235fm_i_o
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ A2 )
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_192_nle__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_193_le__cases3,axiom,
! [X4: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_194_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Product_unit,Z: set_Product_unit] : ( Y = Z ) )
= ( ^ [X3: set_Product_unit,Y4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X3 @ Y4 )
& ( ord_le3507040750410214029t_unit @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_195_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_li8269783359719514504c_fm_i,Z: set_li8269783359719514504c_fm_i] : ( Y = Z ) )
= ( ^ [X3: set_li8269783359719514504c_fm_i,Y4: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ X3 @ Y4 )
& ( ord_le1209247765174941992c_fm_i @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_196_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( Y = Z ) )
= ( ^ [X3: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ Y4 )
& ( ord_le5389487502678872194c_fm_i @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_197_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_198_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: epistemic_fm_i > $o,Z: epistemic_fm_i > $o] : ( Y = Z ) )
= ( ^ [X3: epistemic_fm_i > $o,Y4: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ X3 @ Y4 )
& ( ord_le190830114487426235fm_i_o @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_199_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z ) )
= ( ^ [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y4 )
& ( ord_le3843937902494030498c_fm_i @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_200_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_201_ord__eq__le__trans,axiom,
! [A3: set_Product_unit,B3: set_Product_unit,C2: set_Product_unit] :
( ( A3 = B3 )
=> ( ( ord_le3507040750410214029t_unit @ B3 @ C2 )
=> ( ord_le3507040750410214029t_unit @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_202_ord__eq__le__trans,axiom,
! [A3: set_li8269783359719514504c_fm_i,B3: set_li8269783359719514504c_fm_i,C2: set_li8269783359719514504c_fm_i] :
( ( A3 = B3 )
=> ( ( ord_le1209247765174941992c_fm_i @ B3 @ C2 )
=> ( ord_le1209247765174941992c_fm_i @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_203_ord__eq__le__trans,axiom,
! [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( A3 = B3 )
=> ( ( ord_le5389487502678872194c_fm_i @ B3 @ C2 )
=> ( ord_le5389487502678872194c_fm_i @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_204_ord__eq__le__trans,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_205_ord__eq__le__trans,axiom,
! [A3: epistemic_fm_i > $o,B3: epistemic_fm_i > $o,C2: epistemic_fm_i > $o] :
( ( A3 = B3 )
=> ( ( ord_le190830114487426235fm_i_o @ B3 @ C2 )
=> ( ord_le190830114487426235fm_i_o @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_206_ord__eq__le__trans,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( A3 = B3 )
=> ( ( ord_le3843937902494030498c_fm_i @ B3 @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_207_ord__eq__le__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_208_ord__le__eq__trans,axiom,
! [A3: set_Product_unit,B3: set_Product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le3507040750410214029t_unit @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_209_ord__le__eq__trans,axiom,
! [A3: set_li8269783359719514504c_fm_i,B3: set_li8269783359719514504c_fm_i,C2: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le1209247765174941992c_fm_i @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_210_ord__le__eq__trans,axiom,
! [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le5389487502678872194c_fm_i @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_211_ord__le__eq__trans,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_212_ord__le__eq__trans,axiom,
! [A3: epistemic_fm_i > $o,B3: epistemic_fm_i > $o,C2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le190830114487426235fm_i_o @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_213_ord__le__eq__trans,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le3843937902494030498c_fm_i @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_214_ord__le__eq__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_215_order__antisym,axiom,
! [X4: set_Product_unit,Y3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X4 @ Y3 )
=> ( ( ord_le3507040750410214029t_unit @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_216_order__antisym,axiom,
! [X4: set_li8269783359719514504c_fm_i,Y3: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ X4 @ Y3 )
=> ( ( ord_le1209247765174941992c_fm_i @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_217_order__antisym,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X4 @ Y3 )
=> ( ( ord_le5389487502678872194c_fm_i @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_218_order__antisym,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_219_order__antisym,axiom,
! [X4: epistemic_fm_i > $o,Y3: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ X4 @ Y3 )
=> ( ( ord_le190830114487426235fm_i_o @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_220_order__antisym,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
=> ( ( ord_le3843937902494030498c_fm_i @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_221_order__antisym,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% order_antisym
thf(fact_222_order_Otrans,axiom,
! [A3: set_Product_unit,B3: set_Product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A3 @ B3 )
=> ( ( ord_le3507040750410214029t_unit @ B3 @ C2 )
=> ( ord_le3507040750410214029t_unit @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_223_order_Otrans,axiom,
! [A3: set_li8269783359719514504c_fm_i,B3: set_li8269783359719514504c_fm_i,C2: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A3 @ B3 )
=> ( ( ord_le1209247765174941992c_fm_i @ B3 @ C2 )
=> ( ord_le1209247765174941992c_fm_i @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_224_order_Otrans,axiom,
! [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A3 @ B3 )
=> ( ( ord_le5389487502678872194c_fm_i @ B3 @ C2 )
=> ( ord_le5389487502678872194c_fm_i @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_225_order_Otrans,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_226_order_Otrans,axiom,
! [A3: epistemic_fm_i > $o,B3: epistemic_fm_i > $o,C2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ A3 @ B3 )
=> ( ( ord_le190830114487426235fm_i_o @ B3 @ C2 )
=> ( ord_le190830114487426235fm_i_o @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_227_order_Otrans,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ( ( ord_le3843937902494030498c_fm_i @ B3 @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_228_order_Otrans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_229_order__trans,axiom,
! [X4: set_Product_unit,Y3: set_Product_unit,Z2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X4 @ Y3 )
=> ( ( ord_le3507040750410214029t_unit @ Y3 @ Z2 )
=> ( ord_le3507040750410214029t_unit @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_230_order__trans,axiom,
! [X4: set_li8269783359719514504c_fm_i,Y3: set_li8269783359719514504c_fm_i,Z2: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ X4 @ Y3 )
=> ( ( ord_le1209247765174941992c_fm_i @ Y3 @ Z2 )
=> ( ord_le1209247765174941992c_fm_i @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_231_order__trans,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X4 @ Y3 )
=> ( ( ord_le5389487502678872194c_fm_i @ Y3 @ Z2 )
=> ( ord_le5389487502678872194c_fm_i @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_232_order__trans,axiom,
! [X4: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
=> ( ord_less_eq_set_nat @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_233_order__trans,axiom,
! [X4: epistemic_fm_i > $o,Y3: epistemic_fm_i > $o,Z2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ X4 @ Y3 )
=> ( ( ord_le190830114487426235fm_i_o @ Y3 @ Z2 )
=> ( ord_le190830114487426235fm_i_o @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_234_order__trans,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
=> ( ( ord_le3843937902494030498c_fm_i @ Y3 @ Z2 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_235_order__trans,axiom,
! [X4: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_236_linorder__wlog,axiom,
! [P2: nat > nat > $o,A3: nat,B3: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_237_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Product_unit,Z: set_Product_unit] : ( Y = Z ) )
= ( ^ [A5: set_Product_unit,B5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ B5 @ A5 )
& ( ord_le3507040750410214029t_unit @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_238_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_li8269783359719514504c_fm_i,Z: set_li8269783359719514504c_fm_i] : ( Y = Z ) )
= ( ^ [A5: set_li8269783359719514504c_fm_i,B5: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ B5 @ A5 )
& ( ord_le1209247765174941992c_fm_i @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_239_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( Y = Z ) )
= ( ^ [A5: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B5 @ A5 )
& ( ord_le5389487502678872194c_fm_i @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_240_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_241_dual__order_Oeq__iff,axiom,
( ( ^ [Y: epistemic_fm_i > $o,Z: epistemic_fm_i > $o] : ( Y = Z ) )
= ( ^ [A5: epistemic_fm_i > $o,B5: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ B5 @ A5 )
& ( ord_le190830114487426235fm_i_o @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_242_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z ) )
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B5 @ A5 )
& ( ord_le3843937902494030498c_fm_i @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_243_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_244_dual__order_Oantisym,axiom,
! [B3: set_Product_unit,A3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ B3 @ A3 )
=> ( ( ord_le3507040750410214029t_unit @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_245_dual__order_Oantisym,axiom,
! [B3: set_li8269783359719514504c_fm_i,A3: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ B3 @ A3 )
=> ( ( ord_le1209247765174941992c_fm_i @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_246_dual__order_Oantisym,axiom,
! [B3: set_se3485332733965609186c_fm_i,A3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B3 @ A3 )
=> ( ( ord_le5389487502678872194c_fm_i @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_247_dual__order_Oantisym,axiom,
! [B3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_248_dual__order_Oantisym,axiom,
! [B3: epistemic_fm_i > $o,A3: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ B3 @ A3 )
=> ( ( ord_le190830114487426235fm_i_o @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_249_dual__order_Oantisym,axiom,
! [B3: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B3 @ A3 )
=> ( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_250_dual__order_Oantisym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_251_dual__order_Otrans,axiom,
! [B3: set_Product_unit,A3: set_Product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ B3 @ A3 )
=> ( ( ord_le3507040750410214029t_unit @ C2 @ B3 )
=> ( ord_le3507040750410214029t_unit @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_252_dual__order_Otrans,axiom,
! [B3: set_li8269783359719514504c_fm_i,A3: set_li8269783359719514504c_fm_i,C2: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ B3 @ A3 )
=> ( ( ord_le1209247765174941992c_fm_i @ C2 @ B3 )
=> ( ord_le1209247765174941992c_fm_i @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_253_dual__order_Otrans,axiom,
! [B3: set_se3485332733965609186c_fm_i,A3: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B3 @ A3 )
=> ( ( ord_le5389487502678872194c_fm_i @ C2 @ B3 )
=> ( ord_le5389487502678872194c_fm_i @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_254_dual__order_Otrans,axiom,
! [B3: set_nat,A3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( ( ord_less_eq_set_nat @ C2 @ B3 )
=> ( ord_less_eq_set_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_255_dual__order_Otrans,axiom,
! [B3: epistemic_fm_i > $o,A3: epistemic_fm_i > $o,C2: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ B3 @ A3 )
=> ( ( ord_le190830114487426235fm_i_o @ C2 @ B3 )
=> ( ord_le190830114487426235fm_i_o @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_256_dual__order_Otrans,axiom,
! [B3: set_Epistemic_fm_i,A3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B3 @ A3 )
=> ( ( ord_le3843937902494030498c_fm_i @ C2 @ B3 )
=> ( ord_le3843937902494030498c_fm_i @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_257_dual__order_Otrans,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_258_antisym,axiom,
! [A3: set_Product_unit,B3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A3 @ B3 )
=> ( ( ord_le3507040750410214029t_unit @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_259_antisym,axiom,
! [A3: set_li8269783359719514504c_fm_i,B3: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A3 @ B3 )
=> ( ( ord_le1209247765174941992c_fm_i @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_260_antisym,axiom,
! [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A3 @ B3 )
=> ( ( ord_le5389487502678872194c_fm_i @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_261_antisym,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_262_antisym,axiom,
! [A3: epistemic_fm_i > $o,B3: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ A3 @ B3 )
=> ( ( ord_le190830114487426235fm_i_o @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_263_antisym,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ( ( ord_le3843937902494030498c_fm_i @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_264_antisym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_265_le__funD,axiom,
! [F: epistemic_fm_i > $o,G: epistemic_fm_i > $o,X4: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ F @ G )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).
% le_funD
thf(fact_266_le__funE,axiom,
! [F: epistemic_fm_i > $o,G: epistemic_fm_i > $o,X4: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ F @ G )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).
% le_funE
thf(fact_267_le__funI,axiom,
! [F: epistemic_fm_i > $o,G: epistemic_fm_i > $o] :
( ! [X: epistemic_fm_i] : ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) )
=> ( ord_le190830114487426235fm_i_o @ F @ G ) ) ).
% le_funI
thf(fact_268_le__fun__def,axiom,
( ord_le190830114487426235fm_i_o
= ( ^ [F3: epistemic_fm_i > $o,G2: epistemic_fm_i > $o] :
! [X3: epistemic_fm_i] : ( ord_less_eq_o @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_269_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Product_unit,Z: set_Product_unit] : ( Y = Z ) )
= ( ^ [A5: set_Product_unit,B5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A5 @ B5 )
& ( ord_le3507040750410214029t_unit @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_270_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_li8269783359719514504c_fm_i,Z: set_li8269783359719514504c_fm_i] : ( Y = Z ) )
= ( ^ [A5: set_li8269783359719514504c_fm_i,B5: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A5 @ B5 )
& ( ord_le1209247765174941992c_fm_i @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_271_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( Y = Z ) )
= ( ^ [A5: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A5 @ B5 )
& ( ord_le5389487502678872194c_fm_i @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_272_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_273_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: epistemic_fm_i > $o,Z: epistemic_fm_i > $o] : ( Y = Z ) )
= ( ^ [A5: epistemic_fm_i > $o,B5: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ A5 @ B5 )
& ( ord_le190830114487426235fm_i_o @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_274_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z ) )
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A5 @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_275_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_276_order__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_277_order__subst1,axiom,
! [A3: set_Epistemic_fm_i,F: nat > set_Epistemic_fm_i,B3: nat,C2: nat] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_278_order__subst1,axiom,
! [A3: nat,F: set_Epistemic_fm_i > nat,B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B3 @ C2 )
=> ( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_279_order__subst1,axiom,
! [A3: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ ( F @ B3 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B3 @ C2 )
=> ( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_280_order__subst1,axiom,
! [A3: nat,F: set_Product_unit > nat,B3: set_Product_unit,C2: set_Product_unit] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_le3507040750410214029t_unit @ B3 @ C2 )
=> ( ! [X: set_Product_unit,Y5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_281_order__subst1,axiom,
! [A3: nat,F: set_nat > nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_282_order__subst1,axiom,
! [A3: set_Product_unit,F: nat > set_Product_unit,B3: nat,C2: nat] :
( ( ord_le3507040750410214029t_unit @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3507040750410214029t_unit @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_283_order__subst1,axiom,
! [A3: set_nat,F: nat > set_nat,B3: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_284_order__subst1,axiom,
! [A3: set_Product_unit,F: set_Product_unit > set_Product_unit,B3: set_Product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A3 @ ( F @ B3 ) )
=> ( ( ord_le3507040750410214029t_unit @ B3 @ C2 )
=> ( ! [X: set_Product_unit,Y5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X @ Y5 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3507040750410214029t_unit @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_285_order__subst1,axiom,
! [A3: set_Product_unit,F: set_nat > set_Product_unit,B3: set_nat,C2: set_nat] :
( ( ord_le3507040750410214029t_unit @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3507040750410214029t_unit @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_286_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_287_order__subst2,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,F: set_Epistemic_fm_i > nat,C2: nat] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_288_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F @ B3 ) @ C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_289_order__subst2,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_290_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > set_Product_unit,C2: set_Product_unit] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_le3507040750410214029t_unit @ ( F @ B3 ) @ C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3507040750410214029t_unit @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_291_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_292_order__subst2,axiom,
! [A3: set_Product_unit,B3: set_Product_unit,F: set_Product_unit > nat,C2: nat] :
( ( ord_le3507040750410214029t_unit @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_Product_unit,Y5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_293_order__subst2,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_294_order__subst2,axiom,
! [A3: set_Product_unit,B3: set_Product_unit,F: set_Product_unit > set_Product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A3 @ B3 )
=> ( ( ord_le3507040750410214029t_unit @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_Product_unit,Y5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X @ Y5 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3507040750410214029t_unit @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_295_order__subst2,axiom,
! [A3: set_Product_unit,B3: set_Product_unit,F: set_Product_unit > set_nat,C2: set_nat] :
( ( ord_le3507040750410214029t_unit @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X: set_Product_unit,Y5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_296_order__eq__refl,axiom,
! [X4: set_Product_unit,Y3: set_Product_unit] :
( ( X4 = Y3 )
=> ( ord_le3507040750410214029t_unit @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_297_order__eq__refl,axiom,
! [X4: set_li8269783359719514504c_fm_i,Y3: set_li8269783359719514504c_fm_i] :
( ( X4 = Y3 )
=> ( ord_le1209247765174941992c_fm_i @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_298_order__eq__refl,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y3: set_se3485332733965609186c_fm_i] :
( ( X4 = Y3 )
=> ( ord_le5389487502678872194c_fm_i @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_299_order__eq__refl,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( X4 = Y3 )
=> ( ord_less_eq_set_nat @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_300_order__eq__refl,axiom,
! [X4: epistemic_fm_i > $o,Y3: epistemic_fm_i > $o] :
( ( X4 = Y3 )
=> ( ord_le190830114487426235fm_i_o @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_301_order__eq__refl,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( X4 = Y3 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_302_order__eq__refl,axiom,
! [X4: nat,Y3: nat] :
( ( X4 = Y3 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% order_eq_refl
thf(fact_303_linorder__linear,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% linorder_linear
thf(fact_304_ord__eq__le__subst,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_305_ord__eq__le__subst,axiom,
! [A3: nat,F: set_Epistemic_fm_i > nat,B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B3 @ C2 )
=> ( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_306_ord__eq__le__subst,axiom,
! [A3: set_Epistemic_fm_i,F: nat > set_Epistemic_fm_i,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_307_ord__eq__le__subst,axiom,
! [A3: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B3 @ C2 )
=> ( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_308_ord__eq__le__subst,axiom,
! [A3: set_Product_unit,F: nat > set_Product_unit,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3507040750410214029t_unit @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_309_ord__eq__le__subst,axiom,
! [A3: set_nat,F: nat > set_nat,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_310_ord__eq__le__subst,axiom,
! [A3: nat,F: set_Product_unit > nat,B3: set_Product_unit,C2: set_Product_unit] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le3507040750410214029t_unit @ B3 @ C2 )
=> ( ! [X: set_Product_unit,Y5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_311_ord__eq__le__subst,axiom,
! [A3: nat,F: set_nat > nat,B3: set_nat,C2: set_nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_312_ord__eq__le__subst,axiom,
! [A3: set_Product_unit,F: set_Product_unit > set_Product_unit,B3: set_Product_unit,C2: set_Product_unit] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le3507040750410214029t_unit @ B3 @ C2 )
=> ( ! [X: set_Product_unit,Y5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X @ Y5 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3507040750410214029t_unit @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_313_ord__eq__le__subst,axiom,
! [A3: set_nat,F: set_Product_unit > set_nat,B3: set_Product_unit,C2: set_Product_unit] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le3507040750410214029t_unit @ B3 @ C2 )
=> ( ! [X: set_Product_unit,Y5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_314_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_315_ord__le__eq__subst,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,F: set_Epistemic_fm_i > nat,C2: nat] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_316_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_317_ord__le__eq__subst,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_318_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > set_Product_unit,C2: set_Product_unit] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3507040750410214029t_unit @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_319_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_320_ord__le__eq__subst,axiom,
! [A3: set_Product_unit,B3: set_Product_unit,F: set_Product_unit > nat,C2: nat] :
( ( ord_le3507040750410214029t_unit @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_Product_unit,Y5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_321_ord__le__eq__subst,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_322_ord__le__eq__subst,axiom,
! [A3: set_Product_unit,B3: set_Product_unit,F: set_Product_unit > set_Product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_Product_unit,Y5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X @ Y5 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_le3507040750410214029t_unit @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_323_ord__le__eq__subst,axiom,
! [A3: set_Product_unit,B3: set_Product_unit,F: set_Product_unit > set_nat,C2: set_nat] :
( ( ord_le3507040750410214029t_unit @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: set_Product_unit,Y5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_324_linorder__le__cases,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% linorder_le_cases
thf(fact_325_order__antisym__conv,axiom,
! [Y3: set_Product_unit,X4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ Y3 @ X4 )
=> ( ( ord_le3507040750410214029t_unit @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_326_order__antisym__conv,axiom,
! [Y3: set_li8269783359719514504c_fm_i,X4: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ Y3 @ X4 )
=> ( ( ord_le1209247765174941992c_fm_i @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_327_order__antisym__conv,axiom,
! [Y3: set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ Y3 @ X4 )
=> ( ( ord_le5389487502678872194c_fm_i @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_328_order__antisym__conv,axiom,
! [Y3: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_329_order__antisym__conv,axiom,
! [Y3: epistemic_fm_i > $o,X4: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ Y3 @ X4 )
=> ( ( ord_le190830114487426235fm_i_o @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_330_order__antisym__conv,axiom,
! [Y3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y3 @ X4 )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_331_order__antisym__conv,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_332_prop__restrict,axiom,
! [X4: set_se3485332733965609186c_fm_i,Z3: set_se7339729205154126530c_fm_i,X5: set_se7339729205154126530c_fm_i,P2: set_se3485332733965609186c_fm_i > $o] :
( ( member1461078328125707403c_fm_i @ X4 @ Z3 )
=> ( ( ord_le2772500130360476258c_fm_i @ Z3
@ ( collec2761885132556217805c_fm_i
@ ^ [X3: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X3 @ X5 )
& ( P2 @ X3 ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_333_prop__restrict,axiom,
! [X4: i,Z3: set_i,X5: set_i,P2: i > $o] :
( ( member_i @ X4 @ Z3 )
=> ( ( ord_less_eq_set_i @ Z3
@ ( collect_i
@ ^ [X3: i] :
( ( member_i @ X3 @ X5 )
& ( P2 @ X3 ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_334_prop__restrict,axiom,
! [X4: set_nat,Z3: set_set_nat,X5: set_set_nat,P2: set_nat > $o] :
( ( member_set_nat @ X4 @ Z3 )
=> ( ( ord_le6893508408891458716et_nat @ Z3
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
& ( P2 @ X3 ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_335_prop__restrict,axiom,
! [X4: list_nat,Z3: set_list_nat,X5: set_list_nat,P2: list_nat > $o] :
( ( member_list_nat @ X4 @ Z3 )
=> ( ( ord_le6045566169113846134st_nat @ Z3
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ X5 )
& ( P2 @ X3 ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_336_prop__restrict,axiom,
! [X4: product_unit,Z3: set_Product_unit,X5: set_Product_unit,P2: product_unit > $o] :
( ( member_Product_unit @ X4 @ Z3 )
=> ( ( ord_le3507040750410214029t_unit @ Z3
@ ( collect_Product_unit
@ ^ [X3: product_unit] :
( ( member_Product_unit @ X3 @ X5 )
& ( P2 @ X3 ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_337_prop__restrict,axiom,
! [X4: list_Epistemic_fm_i,Z3: set_li8269783359719514504c_fm_i,X5: set_li8269783359719514504c_fm_i,P2: list_Epistemic_fm_i > $o] :
( ( member6475149752759350353c_fm_i @ X4 @ Z3 )
=> ( ( ord_le1209247765174941992c_fm_i @ Z3
@ ( collec5759350117362652307c_fm_i
@ ^ [X3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X3 @ X5 )
& ( P2 @ X3 ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_338_prop__restrict,axiom,
! [X4: set_Epistemic_fm_i,Z3: set_se3485332733965609186c_fm_i,X5: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ X4 @ Z3 )
=> ( ( ord_le5389487502678872194c_fm_i @ Z3
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ X5 )
& ( P2 @ X3 ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_339_prop__restrict,axiom,
! [X4: nat,Z3: set_nat,X5: set_nat,P2: nat > $o] :
( ( member_nat @ X4 @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P2 @ X3 ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_340_prop__restrict,axiom,
! [X4: epistemic_fm_i,Z3: set_Epistemic_fm_i,X5: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ X4 @ Z3 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z3
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ X5 )
& ( P2 @ X3 ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_341_Euclidean_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [A: episte1308613106011496894c_fm_i > $o] :
( ( ord_le2155353306922182815fm_i_o @ episte7251237997873073900c_fm_i @ A )
=> ( episte8462951133645435945t_unit
@ ( episte7444838911182875652t_unit
@ ( collec4512488881437085321c_fm_i
@ ^ [W5: set_Ep6763084006362001438c_fm_i] :
( ( episte704884158563092166c_fm_i @ A @ W5 )
& ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ W5 ) ) )
@ ^ [I: epistemic_fm_i,V2: set_Ep6763084006362001438c_fm_i] :
( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte7452740295420953255t_unit
@ ^ [V2: set_Ep6763084006362001438c_fm_i,X3: list_char] : ( member1686840210572743143c_fm_i @ ( episte3759128500599061780c_fm_i @ X3 ) @ V2 )
@ product_Unity ) ) ) ) ).
% Euclidean\<^sub>K\<^sub>5
thf(fact_342_Euclidean_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [A: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( episte6275404236303465413t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W5: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W5 )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W5 ) ) )
@ ^ [I: i,V2: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V2: set_Epistemic_fm_i,X3: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X3 ) @ V2 )
@ product_Unity ) ) ) ) ).
% Euclidean\<^sub>K\<^sub>5
thf(fact_343_symmetric_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [A: episte1308613106011496894c_fm_i > $o] :
( ( ord_le2155353306922182815fm_i_o @ episte89022077147250079c_fm_i @ A )
=> ( episte818161904015704450t_unit
@ ( episte7444838911182875652t_unit
@ ( collec4512488881437085321c_fm_i
@ ^ [W5: set_Ep6763084006362001438c_fm_i] :
( ( episte704884158563092166c_fm_i @ A @ W5 )
& ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ W5 ) ) )
@ ^ [I: epistemic_fm_i,V2: set_Ep6763084006362001438c_fm_i] :
( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte7452740295420953255t_unit
@ ^ [V2: set_Ep6763084006362001438c_fm_i,X3: list_char] : ( member1686840210572743143c_fm_i @ ( episte3759128500599061780c_fm_i @ X3 ) @ V2 )
@ product_Unity ) ) ) ) ).
% symmetric\<^sub>K\<^sub>B
thf(fact_344_symmetric_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [A: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( episte80897895827131294t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W5: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W5 )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W5 ) ) )
@ ^ [I: i,V2: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V2: set_Epistemic_fm_i,X3: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X3 ) @ V2 )
@ product_Unity ) ) ) ) ).
% symmetric\<^sub>K\<^sub>B
thf(fact_345_AxB__symmetric_H,axiom,
! [A: episte1308613106011496894c_fm_i > $o,V: set_Ep6763084006362001438c_fm_i,W7: set_Ep6763084006362001438c_fm_i,I2: epistemic_fm_i] :
( ( ord_le2155353306922182815fm_i_o @ episte89022077147250079c_fm_i @ A )
=> ( ( episte704884158563092166c_fm_i @ A @ V )
=> ( ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ V )
=> ( ( episte704884158563092166c_fm_i @ A @ W7 )
=> ( ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ W7 )
=> ( ( member6124286717651598407c_fm_i @ W7
@ ( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P4 ) @ V ) ) ) ) )
=> ( member6124286717651598407c_fm_i @ V
@ ( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P4 ) @ W7 ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_346_AxB__symmetric_H,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,W7: set_Epistemic_fm_i,I2: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ W7 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W7 )
=> ( ( member1104366573291651755c_fm_i @ W7
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P4 ) @ V ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ V
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P4 ) @ W7 ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_347_Ax5__Euclidean,axiom,
! [A: episte1308613106011496894c_fm_i > $o,U: set_Ep6763084006362001438c_fm_i,V: set_Ep6763084006362001438c_fm_i,W7: set_Ep6763084006362001438c_fm_i,I2: epistemic_fm_i] :
( ( ord_le2155353306922182815fm_i_o @ episte7251237997873073900c_fm_i @ A )
=> ( ( episte704884158563092166c_fm_i @ A @ U )
=> ( ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ U )
=> ( ( episte704884158563092166c_fm_i @ A @ V )
=> ( ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ V )
=> ( ( episte704884158563092166c_fm_i @ A @ W7 )
=> ( ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ W7 )
=> ( ( member6124286717651598407c_fm_i @ V
@ ( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P4 ) @ U ) ) ) ) )
=> ( ( member6124286717651598407c_fm_i @ W7
@ ( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P4 ) @ U ) ) ) ) )
=> ( member6124286717651598407c_fm_i @ W7
@ ( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P4 ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_348_Ax5__Euclidean,axiom,
! [A: epistemic_fm_i > $o,U: set_Epistemic_fm_i,V: set_Epistemic_fm_i,W7: set_Epistemic_fm_i,I2: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ U )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ U )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ W7 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W7 )
=> ( ( member1104366573291651755c_fm_i @ V
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P4 ) @ U ) ) ) ) )
=> ( ( member1104366573291651755c_fm_i @ W7
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P4 ) @ U ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ W7
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P4 ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_349_eval__semantics,axiom,
! [Pi4: epistemic_fm_i > list_char > $o,W6: epistemic_fm_i,W7: set_Epistemic_fm_i,R: i > epistemic_fm_i > set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( epistemic_eval_i @ ( Pi4 @ W6 ) @ ( episte5182706529199770816c_fm_i @ ( episte3869211295228886240t_unit @ W7 @ R @ ( episte1793992050974410275t_unit @ Pi4 @ product_Unity ) ) @ W6 ) @ P3 )
= ( episte5182706529199770816c_fm_i @ ( episte3869211295228886240t_unit @ W7 @ R @ ( episte1793992050974410275t_unit @ Pi4 @ product_Unity ) ) @ W6 @ P3 ) ) ).
% eval_semantics
thf(fact_350_eval__semantics,axiom,
! [Pi4: set_Epistemic_fm_i > list_char > $o,W6: set_Epistemic_fm_i,W7: set_se3485332733965609186c_fm_i,R: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,P3: epistemic_fm_i] :
( ( epistemic_eval_i @ ( Pi4 @ W6 ) @ ( episte2498298383968489120c_fm_i @ ( episte8397520288747125024t_unit @ W7 @ R @ ( episte4955834246032687171t_unit @ Pi4 @ product_Unity ) ) @ W6 ) @ P3 )
= ( episte2498298383968489120c_fm_i @ ( episte8397520288747125024t_unit @ W7 @ R @ ( episte4955834246032687171t_unit @ Pi4 @ product_Unity ) ) @ W6 @ P3 ) ) ).
% eval_semantics
thf(fact_351_transitive_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [A: episte1308613106011496894c_fm_i > $o] :
( ( ord_le2155353306922182815fm_i_o @ episte4964217057358206253c_fm_i @ A )
=> ( episte6770155463157591394t_unit
@ ( episte7444838911182875652t_unit
@ ( collec4512488881437085321c_fm_i
@ ^ [W5: set_Ep6763084006362001438c_fm_i] :
( ( episte704884158563092166c_fm_i @ A @ W5 )
& ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ W5 ) ) )
@ ^ [I: epistemic_fm_i,V2: set_Ep6763084006362001438c_fm_i] :
( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte7452740295420953255t_unit
@ ^ [V2: set_Ep6763084006362001438c_fm_i,X3: list_char] : ( member1686840210572743143c_fm_i @ ( episte3759128500599061780c_fm_i @ X3 ) @ V2 )
@ product_Unity ) ) ) ) ).
% transitive\<^sub>K\<^sub>4
thf(fact_352_transitive_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [A: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( episte2966952217288386430t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W5: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W5 )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W5 ) ) )
@ ^ [I: i,V2: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V2: set_Epistemic_fm_i,X3: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X3 ) @ V2 )
@ product_Unity ) ) ) ) ).
% transitive\<^sub>K\<^sub>4
thf(fact_353_Ax4__transitive,axiom,
! [A: episte1308613106011496894c_fm_i > $o,V: set_Ep6763084006362001438c_fm_i,W7: set_Ep6763084006362001438c_fm_i,I2: epistemic_fm_i,U: set_Ep6763084006362001438c_fm_i] :
( ( ord_le2155353306922182815fm_i_o @ episte4964217057358206253c_fm_i @ A )
=> ( ( episte704884158563092166c_fm_i @ A @ V )
=> ( ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ V )
=> ( ( member6124286717651598407c_fm_i @ W7
@ ( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P4 ) @ V ) ) ) ) )
=> ( ( member6124286717651598407c_fm_i @ U
@ ( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P4 ) @ W7 ) ) ) ) )
=> ( member6124286717651598407c_fm_i @ U
@ ( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P4 ) @ V ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_354_Ax4__transitive,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,W7: set_Epistemic_fm_i,I2: i,U: set_Epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member1104366573291651755c_fm_i @ W7
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P4 ) @ V ) ) ) ) )
=> ( ( member1104366573291651755c_fm_i @ U
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P4 ) @ W7 ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ U
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P4 ) @ V ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_355_reflexive_092_060_094sub_062T,axiom,
! [A: episte1308613106011496894c_fm_i > $o] :
( ( ord_le2155353306922182815fm_i_o @ episte4361910858995764493c_fm_i @ A )
=> ( episte2938229627094415095t_unit
@ ( episte7444838911182875652t_unit
@ ( collec4512488881437085321c_fm_i
@ ^ [W5: set_Ep6763084006362001438c_fm_i] :
( ( episte704884158563092166c_fm_i @ A @ W5 )
& ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ W5 ) ) )
@ ^ [I: epistemic_fm_i,V2: set_Ep6763084006362001438c_fm_i] :
( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte7452740295420953255t_unit
@ ^ [V2: set_Ep6763084006362001438c_fm_i,X3: list_char] : ( member1686840210572743143c_fm_i @ ( episte3759128500599061780c_fm_i @ X3 ) @ V2 )
@ product_Unity ) ) ) ) ).
% reflexive\<^sub>T
thf(fact_356_reflexive_092_060_094sub_062T,axiom,
! [A: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( episte251305198166243731t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W5: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W5 )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W5 ) ) )
@ ^ [I: i,V2: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V2: set_Epistemic_fm_i,X3: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X3 ) @ V2 )
@ product_Unity ) ) ) ) ).
% reflexive\<^sub>T
thf(fact_357_AxT__reflexive,axiom,
! [A: episte1308613106011496894c_fm_i > $o,V: set_Ep6763084006362001438c_fm_i,I2: epistemic_fm_i] :
( ( ord_le2155353306922182815fm_i_o @ episte4361910858995764493c_fm_i @ A )
=> ( ( episte704884158563092166c_fm_i @ A @ V )
=> ( ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ V )
=> ( member6124286717651598407c_fm_i @ V
@ ( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P4 ) @ V ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_358_AxT__reflexive,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,I2: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( member1104366573291651755c_fm_i @ V
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P4 ) @ V ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_359_soundness__Ax__2,axiom,
! [P3: epistemic_fm_i,M: episte8649862735182028056t_unit,W6: list_Epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ P3 )
=> ( ( stalna5033156833633940736c_fm_i @ M )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte970406020673866091t_unit @ M ) )
=> ( episte5087030060053943750c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax_2
thf(fact_360_soundness__Ax__2,axiom,
! [P3: epistemic_fm_i,M: episte6041718960478671640t_unit,W6: set_se3485332733965609186c_fm_i] :
( ( stalnaker_Ax_2_i @ P3 )
=> ( ( stalna1393169086027530426c_fm_i @ M )
=> ( ( member1461078328125707403c_fm_i @ W6 @ ( episte3312409906628800107t_unit @ M ) )
=> ( episte2123465427079433344c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax_2
thf(fact_361_soundness__Ax__2,axiom,
! [P3: epistemic_fm_i,M: episte8354619181726739466t_unit,W6: nat] :
( ( stalnaker_Ax_2_i @ P3 )
=> ( ( stalna2394636212805329400_i_nat @ M )
=> ( ( member_nat @ W6 @ ( episte8922775215189319317t_unit @ M ) )
=> ( episte4571855713382724530_i_nat @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax_2
thf(fact_362_soundness__Ax__2,axiom,
! [P3: epistemic_fm_i,M: episte2102749610958417048t_unit,W6: i] :
( ( stalnaker_Ax_2_i @ P3 )
=> ( ( stalna3707719055912537822ed_i_i @ M )
=> ( ( member_i @ W6 @ ( episte8286435292487608811t_unit @ M ) )
=> ( episte5666089339915499428cs_i_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax_2
thf(fact_363_soundness__Ax__2,axiom,
! [P3: epistemic_fm_i,M: episte4846006725945195288t_unit,W6: epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ P3 )
=> ( ( stalna1726574297087873274c_fm_i @ M )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax_2
thf(fact_364_soundness__Ax__2,axiom,
! [P3: epistemic_fm_i,M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ P3 )
=> ( ( stalna4517306826702924506c_fm_i @ M )
=> ( ( member1104366573291651755c_fm_i @ W6 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax_2
thf(fact_365_pred__subset__eq,axiom,
! [R3: set_se7339729205154126530c_fm_i,S: set_se7339729205154126530c_fm_i] :
( ( ord_le1814827286658909435fm_i_o
@ ^ [X3: set_se3485332733965609186c_fm_i] : ( member1461078328125707403c_fm_i @ X3 @ R3 )
@ ^ [X3: set_se3485332733965609186c_fm_i] : ( member1461078328125707403c_fm_i @ X3 @ S ) )
= ( ord_le2772500130360476258c_fm_i @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_366_pred__subset__eq,axiom,
! [R3: set_i,S: set_i] :
( ( ord_less_eq_i_o
@ ^ [X3: i] : ( member_i @ X3 @ R3 )
@ ^ [X3: i] : ( member_i @ X3 @ S ) )
= ( ord_less_eq_set_i @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_367_pred__subset__eq,axiom,
! [R3: set_Product_unit,S: set_Product_unit] :
( ( ord_le2788506356509622416unit_o
@ ^ [X3: product_unit] : ( member_Product_unit @ X3 @ R3 )
@ ^ [X3: product_unit] : ( member_Product_unit @ X3 @ S ) )
= ( ord_le3507040750410214029t_unit @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_368_pred__subset__eq,axiom,
! [R3: set_li8269783359719514504c_fm_i,S: set_li8269783359719514504c_fm_i] :
( ( ord_le204167446249480117fm_i_o
@ ^ [X3: list_Epistemic_fm_i] : ( member6475149752759350353c_fm_i @ X3 @ R3 )
@ ^ [X3: list_Epistemic_fm_i] : ( member6475149752759350353c_fm_i @ X3 @ S ) )
= ( ord_le1209247765174941992c_fm_i @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_369_pred__subset__eq,axiom,
! [R3: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( ord_le5706303257367236315fm_i_o
@ ^ [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ R3 )
@ ^ [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ S ) )
= ( ord_le5389487502678872194c_fm_i @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_370_pred__subset__eq,axiom,
! [R3: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R3 )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ord_less_eq_set_nat @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_371_pred__subset__eq,axiom,
! [R3: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ R3 )
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ S ) )
= ( ord_le3843937902494030498c_fm_i @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_372_predicate1I,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ! [X: epistemic_fm_i] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le190830114487426235fm_i_o @ P2 @ Q ) ) ).
% predicate1I
thf(fact_373_predicate1D,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o,X4: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ P2 @ Q )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ).
% predicate1D
thf(fact_374_rev__predicate1D,axiom,
! [P2: epistemic_fm_i > $o,X4: epistemic_fm_i,Q: epistemic_fm_i > $o] :
( ( P2 @ X4 )
=> ( ( ord_le190830114487426235fm_i_o @ P2 @ Q )
=> ( Q @ X4 ) ) ) ).
% rev_predicate1D
thf(fact_375_soundness__Ax4,axiom,
! [P3: epistemic_fm_i,M: episte8649862735182028056t_unit,W6: list_Epistemic_fm_i] :
( ( epistemic_Ax4_i @ P3 )
=> ( ( episte3140986294363532158t_unit @ M )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte970406020673866091t_unit @ M ) )
=> ( episte5087030060053943750c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax4
thf(fact_376_soundness__Ax4,axiom,
! [P3: epistemic_fm_i,M: episte6041718960478671640t_unit,W6: set_se3485332733965609186c_fm_i] :
( ( epistemic_Ax4_i @ P3 )
=> ( ( episte5217350381671328638t_unit @ M )
=> ( ( member1461078328125707403c_fm_i @ W6 @ ( episte3312409906628800107t_unit @ M ) )
=> ( episte2123465427079433344c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax4
thf(fact_377_soundness__Ax4,axiom,
! [P3: epistemic_fm_i,M: episte8354619181726739466t_unit,W6: nat] :
( ( epistemic_Ax4_i @ P3 )
=> ( ( episte7906311534470272552t_unit @ M )
=> ( ( member_nat @ W6 @ ( episte8922775215189319317t_unit @ M ) )
=> ( episte4571855713382724530_i_nat @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax4
thf(fact_378_soundness__Ax4,axiom,
! [P3: epistemic_fm_i,M: episte2102749610958417048t_unit,W6: i] :
( ( epistemic_Ax4_i @ P3 )
=> ( ( episte6298788599024636926t_unit @ M )
=> ( ( member_i @ W6 @ ( episte8286435292487608811t_unit @ M ) )
=> ( episte5666089339915499428cs_i_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax4
thf(fact_379_soundness__Ax4,axiom,
! [P3: epistemic_fm_i,M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i] :
( ( epistemic_Ax4_i @ P3 )
=> ( ( episte2966952217288386430t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W6 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax4
thf(fact_380_soundness__Ax4,axiom,
! [P3: epistemic_fm_i,M: episte4846006725945195288t_unit,W6: epistemic_fm_i] :
( ( epistemic_Ax4_i @ P3 )
=> ( ( episte6394096064484582270t_unit @ M )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax4
thf(fact_381_soundness__AxT,axiom,
! [P3: epistemic_fm_i,M: episte8649862735182028056t_unit,W6: list_Epistemic_fm_i] :
( ( epistemic_AxT_i @ P3 )
=> ( ( episte6062502518927837331t_unit @ M )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte970406020673866091t_unit @ M ) )
=> ( episte5087030060053943750c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT
thf(fact_382_soundness__AxT,axiom,
! [P3: epistemic_fm_i,M: episte6041718960478671640t_unit,W6: set_se3485332733965609186c_fm_i] :
( ( epistemic_AxT_i @ P3 )
=> ( ( episte777650505068270995t_unit @ M )
=> ( ( member1461078328125707403c_fm_i @ W6 @ ( episte3312409906628800107t_unit @ M ) )
=> ( episte2123465427079433344c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT
thf(fact_383_soundness__AxT,axiom,
! [P3: epistemic_fm_i,M: episte8354619181726739466t_unit,W6: nat] :
( ( epistemic_AxT_i @ P3 )
=> ( ( episte3073512803368128957t_unit @ M )
=> ( ( member_nat @ W6 @ ( episte8922775215189319317t_unit @ M ) )
=> ( episte4571855713382724530_i_nat @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT
thf(fact_384_soundness__AxT,axiom,
! [P3: epistemic_fm_i,M: episte2102749610958417048t_unit,W6: i] :
( ( epistemic_AxT_i @ P3 )
=> ( ( episte707503779239422227t_unit @ M )
=> ( ( member_i @ W6 @ ( episte8286435292487608811t_unit @ M ) )
=> ( episte5666089339915499428cs_i_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT
thf(fact_385_soundness__AxT,axiom,
! [P3: epistemic_fm_i,M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i] :
( ( epistemic_AxT_i @ P3 )
=> ( ( episte251305198166243731t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W6 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT
thf(fact_386_soundness__AxT,axiom,
! [P3: epistemic_fm_i,M: episte4846006725945195288t_unit,W6: epistemic_fm_i] :
( ( epistemic_AxT_i @ P3 )
=> ( ( episte540185425077706131t_unit @ M )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT
thf(fact_387_soundness__AxT4,axiom,
! [P3: epistemic_fm_i,M: episte8649862735182028056t_unit,W6: list_Epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_Ax4_i @ P3 ) )
=> ( ( ( episte6062502518927837331t_unit @ M )
& ( episte3140986294363532158t_unit @ M ) )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte970406020673866091t_unit @ M ) )
=> ( episte5087030060053943750c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT4
thf(fact_388_soundness__AxT4,axiom,
! [P3: epistemic_fm_i,M: episte6041718960478671640t_unit,W6: set_se3485332733965609186c_fm_i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_Ax4_i @ P3 ) )
=> ( ( ( episte777650505068270995t_unit @ M )
& ( episte5217350381671328638t_unit @ M ) )
=> ( ( member1461078328125707403c_fm_i @ W6 @ ( episte3312409906628800107t_unit @ M ) )
=> ( episte2123465427079433344c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT4
thf(fact_389_soundness__AxT4,axiom,
! [P3: epistemic_fm_i,M: episte8354619181726739466t_unit,W6: nat] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_Ax4_i @ P3 ) )
=> ( ( ( episte3073512803368128957t_unit @ M )
& ( episte7906311534470272552t_unit @ M ) )
=> ( ( member_nat @ W6 @ ( episte8922775215189319317t_unit @ M ) )
=> ( episte4571855713382724530_i_nat @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT4
thf(fact_390_soundness__AxT4,axiom,
! [P3: epistemic_fm_i,M: episte2102749610958417048t_unit,W6: i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_Ax4_i @ P3 ) )
=> ( ( ( episte707503779239422227t_unit @ M )
& ( episte6298788599024636926t_unit @ M ) )
=> ( ( member_i @ W6 @ ( episte8286435292487608811t_unit @ M ) )
=> ( episte5666089339915499428cs_i_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT4
thf(fact_391_soundness__AxT4,axiom,
! [P3: epistemic_fm_i,M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_Ax4_i @ P3 ) )
=> ( ( ( episte251305198166243731t_unit @ M )
& ( episte2966952217288386430t_unit @ M ) )
=> ( ( member1104366573291651755c_fm_i @ W6 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT4
thf(fact_392_soundness__AxT4,axiom,
! [P3: epistemic_fm_i,M: episte4846006725945195288t_unit,W6: epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_Ax4_i @ P3 ) )
=> ( ( ( episte540185425077706131t_unit @ M )
& ( episte6394096064484582270t_unit @ M ) )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT4
thf(fact_393_soundness__AxT5,axiom,
! [P3: epistemic_fm_i,M: episte8649862735182028056t_unit,W6: list_Epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_Ax5_i @ P3 ) )
=> ( ( ( episte6062502518927837331t_unit @ M )
& ( episte3343910547084193694t_unit @ M )
& ( episte3140986294363532158t_unit @ M ) )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte970406020673866091t_unit @ M ) )
=> ( episte5087030060053943750c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT5
thf(fact_394_soundness__AxT5,axiom,
! [P3: epistemic_fm_i,M: episte6041718960478671640t_unit,W6: set_se3485332733965609186c_fm_i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_Ax5_i @ P3 ) )
=> ( ( ( episte777650505068270995t_unit @ M )
& ( episte4692175903937384350t_unit @ M )
& ( episte5217350381671328638t_unit @ M ) )
=> ( ( member1461078328125707403c_fm_i @ W6 @ ( episte3312409906628800107t_unit @ M ) )
=> ( episte2123465427079433344c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT5
thf(fact_395_soundness__AxT5,axiom,
! [P3: epistemic_fm_i,M: episte8354619181726739466t_unit,W6: nat] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_Ax5_i @ P3 ) )
=> ( ( ( episte3073512803368128957t_unit @ M )
& ( episte1817073897218058824t_unit @ M )
& ( episte7906311534470272552t_unit @ M ) )
=> ( ( member_nat @ W6 @ ( episte8922775215189319317t_unit @ M ) )
=> ( episte4571855713382724530_i_nat @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT5
thf(fact_396_soundness__AxT5,axiom,
! [P3: epistemic_fm_i,M: episte2102749610958417048t_unit,W6: i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_Ax5_i @ P3 ) )
=> ( ( ( episte707503779239422227t_unit @ M )
& ( episte2180194749693155358t_unit @ M )
& ( episte6298788599024636926t_unit @ M ) )
=> ( ( member_i @ W6 @ ( episte8286435292487608811t_unit @ M ) )
=> ( episte5666089339915499428cs_i_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT5
thf(fact_397_soundness__AxT5,axiom,
! [P3: epistemic_fm_i,M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_Ax5_i @ P3 ) )
=> ( ( ( episte251305198166243731t_unit @ M )
& ( episte80897895827131294t_unit @ M )
& ( episte2966952217288386430t_unit @ M ) )
=> ( ( member1104366573291651755c_fm_i @ W6 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT5
thf(fact_398_soundness__AxT5,axiom,
! [P3: epistemic_fm_i,M: episte4846006725945195288t_unit,W6: epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_Ax5_i @ P3 ) )
=> ( ( ( episte540185425077706131t_unit @ M )
& ( episte6568605431049406366t_unit @ M )
& ( episte6394096064484582270t_unit @ M ) )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxT5
thf(fact_399_soundness__AxTB4,axiom,
! [P3: epistemic_fm_i,M: episte8649862735182028056t_unit,W6: list_Epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_AxB_i @ P3 )
| ( epistemic_Ax4_i @ P3 ) )
=> ( ( ( episte6062502518927837331t_unit @ M )
& ( episte3343910547084193694t_unit @ M )
& ( episte3140986294363532158t_unit @ M ) )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte970406020673866091t_unit @ M ) )
=> ( episte5087030060053943750c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxTB4
thf(fact_400_soundness__AxTB4,axiom,
! [P3: epistemic_fm_i,M: episte6041718960478671640t_unit,W6: set_se3485332733965609186c_fm_i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_AxB_i @ P3 )
| ( epistemic_Ax4_i @ P3 ) )
=> ( ( ( episte777650505068270995t_unit @ M )
& ( episte4692175903937384350t_unit @ M )
& ( episte5217350381671328638t_unit @ M ) )
=> ( ( member1461078328125707403c_fm_i @ W6 @ ( episte3312409906628800107t_unit @ M ) )
=> ( episte2123465427079433344c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxTB4
thf(fact_401_soundness__AxTB4,axiom,
! [P3: epistemic_fm_i,M: episte8354619181726739466t_unit,W6: nat] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_AxB_i @ P3 )
| ( epistemic_Ax4_i @ P3 ) )
=> ( ( ( episte3073512803368128957t_unit @ M )
& ( episte1817073897218058824t_unit @ M )
& ( episte7906311534470272552t_unit @ M ) )
=> ( ( member_nat @ W6 @ ( episte8922775215189319317t_unit @ M ) )
=> ( episte4571855713382724530_i_nat @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxTB4
thf(fact_402_soundness__AxTB4,axiom,
! [P3: epistemic_fm_i,M: episte2102749610958417048t_unit,W6: i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_AxB_i @ P3 )
| ( epistemic_Ax4_i @ P3 ) )
=> ( ( ( episte707503779239422227t_unit @ M )
& ( episte2180194749693155358t_unit @ M )
& ( episte6298788599024636926t_unit @ M ) )
=> ( ( member_i @ W6 @ ( episte8286435292487608811t_unit @ M ) )
=> ( episte5666089339915499428cs_i_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxTB4
thf(fact_403_soundness__AxTB4,axiom,
! [P3: epistemic_fm_i,M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_AxB_i @ P3 )
| ( epistemic_Ax4_i @ P3 ) )
=> ( ( ( episte251305198166243731t_unit @ M )
& ( episte80897895827131294t_unit @ M )
& ( episte2966952217288386430t_unit @ M ) )
=> ( ( member1104366573291651755c_fm_i @ W6 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxTB4
thf(fact_404_soundness__AxTB4,axiom,
! [P3: epistemic_fm_i,M: episte4846006725945195288t_unit,W6: epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P3 )
| ( epistemic_AxB_i @ P3 )
| ( epistemic_Ax4_i @ P3 ) )
=> ( ( ( episte540185425077706131t_unit @ M )
& ( episte6568605431049406366t_unit @ M )
& ( episte6394096064484582270t_unit @ M ) )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxTB4
thf(fact_405_symm__trans__Euclid,axiom,
! [M: episte5489734138418521880t_unit] :
( ( episte80897895827131294t_unit @ M )
=> ( ( episte2966952217288386430t_unit @ M )
=> ( episte6275404236303465413t_unit @ M ) ) ) ).
% symm_trans_Euclid
thf(fact_406_symm__trans__Euclid,axiom,
! [M: episte4846006725945195288t_unit] :
( ( episte6568605431049406366t_unit @ M )
=> ( ( episte6394096064484582270t_unit @ M )
=> ( episte8713243348624617413t_unit @ M ) ) ) ).
% symm_trans_Euclid
thf(fact_407_refl__Euclid__equiv,axiom,
! [M: episte5489734138418521880t_unit] :
( ( episte251305198166243731t_unit @ M )
=> ( ( episte6275404236303465413t_unit @ M )
=> ( ( episte251305198166243731t_unit @ M )
& ( episte80897895827131294t_unit @ M )
& ( episte2966952217288386430t_unit @ M ) ) ) ) ).
% refl_Euclid_equiv
thf(fact_408_refl__Euclid__equiv,axiom,
! [M: episte4846006725945195288t_unit] :
( ( episte540185425077706131t_unit @ M )
=> ( ( episte8713243348624617413t_unit @ M )
=> ( ( episte540185425077706131t_unit @ M )
& ( episte6568605431049406366t_unit @ M )
& ( episte6394096064484582270t_unit @ M ) ) ) ) ).
% refl_Euclid_equiv
thf(fact_409_soundness__AxB,axiom,
! [P3: epistemic_fm_i,M: episte8649862735182028056t_unit,W6: list_Epistemic_fm_i] :
( ( epistemic_AxB_i @ P3 )
=> ( ( episte3343910547084193694t_unit @ M )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte970406020673866091t_unit @ M ) )
=> ( episte5087030060053943750c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxB
thf(fact_410_soundness__AxB,axiom,
! [P3: epistemic_fm_i,M: episte6041718960478671640t_unit,W6: set_se3485332733965609186c_fm_i] :
( ( epistemic_AxB_i @ P3 )
=> ( ( episte4692175903937384350t_unit @ M )
=> ( ( member1461078328125707403c_fm_i @ W6 @ ( episte3312409906628800107t_unit @ M ) )
=> ( episte2123465427079433344c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxB
thf(fact_411_soundness__AxB,axiom,
! [P3: epistemic_fm_i,M: episte8354619181726739466t_unit,W6: nat] :
( ( epistemic_AxB_i @ P3 )
=> ( ( episte1817073897218058824t_unit @ M )
=> ( ( member_nat @ W6 @ ( episte8922775215189319317t_unit @ M ) )
=> ( episte4571855713382724530_i_nat @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxB
thf(fact_412_soundness__AxB,axiom,
! [P3: epistemic_fm_i,M: episte2102749610958417048t_unit,W6: i] :
( ( epistemic_AxB_i @ P3 )
=> ( ( episte2180194749693155358t_unit @ M )
=> ( ( member_i @ W6 @ ( episte8286435292487608811t_unit @ M ) )
=> ( episte5666089339915499428cs_i_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxB
thf(fact_413_soundness__AxB,axiom,
! [P3: epistemic_fm_i,M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i] :
( ( epistemic_AxB_i @ P3 )
=> ( ( episte80897895827131294t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W6 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxB
thf(fact_414_soundness__AxB,axiom,
! [P3: epistemic_fm_i,M: episte4846006725945195288t_unit,W6: epistemic_fm_i] :
( ( epistemic_AxB_i @ P3 )
=> ( ( episte6568605431049406366t_unit @ M )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_AxB
thf(fact_415_soundness__Ax5,axiom,
! [P3: epistemic_fm_i,M: episte8649862735182028056t_unit,W6: list_Epistemic_fm_i] :
( ( epistemic_Ax5_i @ P3 )
=> ( ( episte7759965962005403845t_unit @ M )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte970406020673866091t_unit @ M ) )
=> ( episte5087030060053943750c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax5
thf(fact_416_soundness__Ax5,axiom,
! [P3: epistemic_fm_i,M: episte6041718960478671640t_unit,W6: set_se3485332733965609186c_fm_i] :
( ( epistemic_Ax5_i @ P3 )
=> ( ( episte8780254683117438917t_unit @ M )
=> ( ( member1461078328125707403c_fm_i @ W6 @ ( episte3312409906628800107t_unit @ M ) )
=> ( episte2123465427079433344c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax5
thf(fact_417_soundness__Ax5,axiom,
! [P3: epistemic_fm_i,M: episte8354619181726739466t_unit,W6: nat] :
( ( epistemic_Ax5_i @ P3 )
=> ( ( episte9066274068200899311t_unit @ M )
=> ( ( member_nat @ W6 @ ( episte8922775215189319317t_unit @ M ) )
=> ( episte4571855713382724530_i_nat @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax5
thf(fact_418_soundness__Ax5,axiom,
! [P3: epistemic_fm_i,M: episte2102749610958417048t_unit,W6: i] :
( ( epistemic_Ax5_i @ P3 )
=> ( ( episte3699623721066309957t_unit @ M )
=> ( ( member_i @ W6 @ ( episte8286435292487608811t_unit @ M ) )
=> ( episte5666089339915499428cs_i_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax5
thf(fact_419_soundness__Ax5,axiom,
! [P3: epistemic_fm_i,M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i] :
( ( epistemic_Ax5_i @ P3 )
=> ( ( episte6275404236303465413t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W6 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax5
thf(fact_420_soundness__Ax5,axiom,
! [P3: epistemic_fm_i,M: episte4846006725945195288t_unit,W6: epistemic_fm_i] :
( ( epistemic_Ax5_i @ P3 )
=> ( ( episte8713243348624617413t_unit @ M )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ P3 ) ) ) ) ).
% soundness_Ax5
thf(fact_421_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H: episte1308613106011496894c_fm_i > $o,I2: epistemic_fm_i,P3: episte1308613106011496894c_fm_i] :
( ( episte6292435846501656252c_fm_i @ Ux @ H @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) )
= ( H @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) ) ) ).
% eval.simps(6)
thf(fact_422_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i] :
( ( epistemic_eval_i @ Ux @ H @ ( epistemic_K_i @ I2 @ P3 ) )
= ( H @ ( epistemic_K_i @ I2 @ P3 ) ) ) ).
% eval.simps(6)
thf(fact_423_tautology,axiom,
! [P3: epistemic_fm_i,M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i] :
( ! [G3: list_char > $o,H2: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H2 @ P3 )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ P3 ) ) ).
% tautology
thf(fact_424_tautology,axiom,
! [P3: epistemic_fm_i,M: episte4846006725945195288t_unit,W6: epistemic_fm_i] :
( ! [G3: list_char > $o,H2: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H2 @ P3 )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ P3 ) ) ).
% tautology
thf(fact_425_eval_Osimps_I2_J,axiom,
! [G: list_char > $o,Uw: epistemic_fm_i > $o,X4: list_char] :
( ( epistemic_eval_i @ G @ Uw @ ( epistemic_Pro_i @ X4 ) )
= ( G @ X4 ) ) ).
% eval.simps(2)
thf(fact_426_frame_Oselect__convs_I1_J,axiom,
! [W3: set_Epistemic_fm_i,K2: i > epistemic_fm_i > set_Epistemic_fm_i,More: episte3891166926823127411t_unit] :
( ( episte1297369412406343275t_unit @ ( episte3869211295228886240t_unit @ W3 @ K2 @ More ) )
= W3 ) ).
% frame.select_convs(1)
thf(fact_427_frame_Oselect__convs_I1_J,axiom,
! [W3: set_se3485332733965609186c_fm_i,K2: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More: episte7133455005732253587t_unit] :
( ( episte2675268102452679275t_unit @ ( episte8397520288747125024t_unit @ W3 @ K2 @ More ) )
= W3 ) ).
% frame.select_convs(1)
thf(fact_428_generalization,axiom,
! [P3: epistemic_fm_i,W6: epistemic_fm_i,M: episte4846006725945195288t_unit,I2: i] :
( ! [M2: episte4846006725945195288t_unit,X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ X @ P3 ) )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ).
% generalization
thf(fact_429_generalization,axiom,
! [P3: epistemic_fm_i,W6: nat,M: episte8354619181726739466t_unit,I2: i] :
( ! [M2: episte8354619181726739466t_unit,X: nat] :
( ( member_nat @ X @ ( episte8922775215189319317t_unit @ M2 ) )
=> ( episte4571855713382724530_i_nat @ M2 @ X @ P3 ) )
=> ( ( member_nat @ W6 @ ( episte8922775215189319317t_unit @ M ) )
=> ( episte4571855713382724530_i_nat @ M @ W6 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ).
% generalization
thf(fact_430_generalization,axiom,
! [P3: epistemic_fm_i,W6: i,M: episte2102749610958417048t_unit,I2: i] :
( ! [M2: episte2102749610958417048t_unit,X: i] :
( ( member_i @ X @ ( episte8286435292487608811t_unit @ M2 ) )
=> ( episte5666089339915499428cs_i_i @ M2 @ X @ P3 ) )
=> ( ( member_i @ W6 @ ( episte8286435292487608811t_unit @ M ) )
=> ( episte5666089339915499428cs_i_i @ M @ W6 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ).
% generalization
thf(fact_431_generalization,axiom,
! [P3: episte1308613106011496894c_fm_i,W6: nat,M: episte4946720254005641710t_unit,I2: epistemic_fm_i] :
( ! [M2: episte4946720254005641710t_unit,X: nat] :
( ( member_nat @ X @ ( episte7295652460417134969t_unit @ M2 ) )
=> ( episte6892841275385687886_i_nat @ M2 @ X @ P3 ) )
=> ( ( member_nat @ W6 @ ( episte7295652460417134969t_unit @ M ) )
=> ( episte6892841275385687886_i_nat @ M @ W6 @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) ) ) ) ).
% generalization
thf(fact_432_generalization,axiom,
! [P3: episte1308613106011496894c_fm_i,W6: i,M: episte7276436686577410172t_unit,I2: epistemic_fm_i] :
( ! [M2: episte7276436686577410172t_unit,X: i] :
( ( member_i @ X @ ( episte5422822166207371983t_unit @ M2 ) )
=> ( episte5578572022554684168fm_i_i @ M2 @ X @ P3 ) )
=> ( ( member_i @ W6 @ ( episte5422822166207371983t_unit @ M ) )
=> ( episte5578572022554684168fm_i_i @ M @ W6 @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) ) ) ) ).
% generalization
thf(fact_433_generalization,axiom,
! [P3: epistemic_fm_i,W6: list_Epistemic_fm_i,M: episte8649862735182028056t_unit,I2: i] :
( ! [M2: episte8649862735182028056t_unit,X: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X @ ( episte970406020673866091t_unit @ M2 ) )
=> ( episte5087030060053943750c_fm_i @ M2 @ X @ P3 ) )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte970406020673866091t_unit @ M ) )
=> ( episte5087030060053943750c_fm_i @ M @ W6 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ).
% generalization
thf(fact_434_generalization,axiom,
! [P3: episte1308613106011496894c_fm_i,W6: epistemic_fm_i,M: episte5703418715120784124t_unit,I2: epistemic_fm_i] :
( ! [M2: episte5703418715120784124t_unit,X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( episte5446861122069384015t_unit @ M2 ) )
=> ( episte7800156050638082596c_fm_i @ M2 @ X @ P3 ) )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte5446861122069384015t_unit @ M ) )
=> ( episte7800156050638082596c_fm_i @ M @ W6 @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) ) ) ) ).
% generalization
thf(fact_435_generalization,axiom,
! [P3: epistemic_fm_i,W6: set_Epistemic_fm_i,M: episte5489734138418521880t_unit,I2: i] :
( ! [M2: episte5489734138418521880t_unit,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ P3 ) )
=> ( ( member1104366573291651755c_fm_i @ W6 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ).
% generalization
thf(fact_436_generalization,axiom,
! [P3: epistemic_fm_i,W6: set_se3485332733965609186c_fm_i,M: episte6041718960478671640t_unit,I2: i] :
( ! [M2: episte6041718960478671640t_unit,X: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ ( episte3312409906628800107t_unit @ M2 ) )
=> ( episte2123465427079433344c_fm_i @ M2 @ X @ P3 ) )
=> ( ( member1461078328125707403c_fm_i @ W6 @ ( episte3312409906628800107t_unit @ M ) )
=> ( episte2123465427079433344c_fm_i @ M @ W6 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ).
% generalization
thf(fact_437_generalization,axiom,
! [P3: episte1308613106011496894c_fm_i,W6: list_Epistemic_fm_i,M: episte7095840741371228412t_unit,I2: epistemic_fm_i] :
( ! [M2: episte7095840741371228412t_unit,X: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ X @ ( episte1684291789543234127t_unit @ M2 ) )
=> ( episte6060530439555422506c_fm_i @ M2 @ X @ P3 ) )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte1684291789543234127t_unit @ M ) )
=> ( episte6060530439555422506c_fm_i @ M @ W6 @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) ) ) ) ).
% generalization
thf(fact_438_kripke_Osurjective,axiom,
! [R: episte4846006725945195288t_unit] :
( R
= ( episte3869211295228886240t_unit @ ( episte1297369412406343275t_unit @ R ) @ ( episte5576233017153220191t_unit @ R ) @ ( episte1793992050974410275t_unit @ ( episte4915868227582937300t_unit @ R ) @ ( episte4828294246236881914t_unit @ R ) ) ) ) ).
% kripke.surjective
thf(fact_439_kripke_Osurjective,axiom,
! [R: episte5489734138418521880t_unit] :
( R
= ( episte8397520288747125024t_unit @ ( episte2675268102452679275t_unit @ R ) @ ( episte852950631662840415t_unit @ R ) @ ( episte4955834246032687171t_unit @ ( episte6627496282998226676t_unit @ R ) @ ( episte484680259488342554t_unit @ R ) ) ) ) ).
% kripke.surjective
thf(fact_440_kripke_Oequality,axiom,
! [R: episte4846006725945195288t_unit,R2: episte4846006725945195288t_unit] :
( ( ( episte1297369412406343275t_unit @ R )
= ( episte1297369412406343275t_unit @ R2 ) )
=> ( ( ( episte5576233017153220191t_unit @ R )
= ( episte5576233017153220191t_unit @ R2 ) )
=> ( ( ( episte4915868227582937300t_unit @ R )
= ( episte4915868227582937300t_unit @ R2 ) )
=> ( ( ( episte4828294246236881914t_unit @ R )
= ( episte4828294246236881914t_unit @ R2 ) )
=> ( R = R2 ) ) ) ) ) ).
% kripke.equality
thf(fact_441_kripke_Oequality,axiom,
! [R: episte5489734138418521880t_unit,R2: episte5489734138418521880t_unit] :
( ( ( episte2675268102452679275t_unit @ R )
= ( episte2675268102452679275t_unit @ R2 ) )
=> ( ( ( episte852950631662840415t_unit @ R )
= ( episte852950631662840415t_unit @ R2 ) )
=> ( ( ( episte6627496282998226676t_unit @ R )
= ( episte6627496282998226676t_unit @ R2 ) )
=> ( ( ( episte484680259488342554t_unit @ R )
= ( episte484680259488342554t_unit @ R2 ) )
=> ( R = R2 ) ) ) ) ) ).
% kripke.equality
thf(fact_442_completeness,axiom,
! [P2: episte476198014953467260t_unit > $o,P3: episte1308613106011496894c_fm_i,A: episte1308613106011496894c_fm_i > $o] :
( ! [M2: episte476198014953467260t_unit] :
( ( P2 @ M2 )
=> ! [X: set_Ep6763084006362001438c_fm_i] :
( ( member6124286717651598407c_fm_i @ X @ ( episte474035306837431247t_unit @ M2 ) )
=> ( ! [Xa: episte1308613106011496894c_fm_i] :
( ( member1686840210572743143c_fm_i @ Xa @ bot_bo281379535698878090c_fm_i )
=> ( episte341539128385704480c_fm_i @ M2 @ X @ Xa ) )
=> ( episte341539128385704480c_fm_i @ M2 @ X @ P3 ) ) ) )
=> ( ( P2
@ ( episte7444838911182875652t_unit
@ ( collec4512488881437085321c_fm_i
@ ^ [W5: set_Ep6763084006362001438c_fm_i] :
( ( episte704884158563092166c_fm_i @ A @ W5 )
& ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ W5 ) ) )
@ ^ [I: epistemic_fm_i,V2: set_Ep6763084006362001438c_fm_i] :
( collec4512488881437085321c_fm_i
@ ( ord_le1948573375516223422c_fm_i
@ ( collec7849324479490780201c_fm_i
@ ^ [P4: episte1308613106011496894c_fm_i] : ( member1686840210572743143c_fm_i @ ( episte5657488666450005526c_fm_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte7452740295420953255t_unit
@ ^ [V2: set_Ep6763084006362001438c_fm_i,X3: list_char] : ( member1686840210572743143c_fm_i @ ( episte3759128500599061780c_fm_i @ X3 ) @ V2 )
@ product_Unity ) ) )
=> ( episte3601224874663357326c_fm_i @ A @ P3 ) ) ) ).
% completeness
thf(fact_443_completeness,axiom,
! [P2: episte5489734138418521880t_unit > $o,P3: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [M2: episte5489734138418521880t_unit] :
( ( P2 @ M2 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ bot_bo4194595901900360558c_fm_i )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ P3 ) ) ) )
=> ( ( P2
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W5: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W5 )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W5 ) ) )
@ ^ [I: i,V2: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V2: set_Epistemic_fm_i,X3: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X3 ) @ V2 )
@ product_Unity ) ) )
=> ( epistemic_AK_i @ A @ P3 ) ) ) ).
% completeness
thf(fact_444_S5__S5_H,axiom,
! [P3: epistemic_fm_i] :
( ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_AxB_i @ P4 )
| ( epistemic_Ax4_i @ P4 ) )
@ P3 )
=> ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_Ax5_i @ P4 ) )
@ P3 ) ) ).
% S5_S5'
thf(fact_445_S5_H__S5,axiom,
! [P3: epistemic_fm_i] :
( ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_Ax5_i @ P4 ) )
@ P3 )
=> ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_AxB_i @ P4 )
| ( epistemic_Ax4_i @ P4 ) )
@ P3 ) ) ).
% S5'_S5
thf(fact_446_conj__subset__def,axiom,
! [A: set_set_nat,P2: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A
@ ( collect_set_nat
@ ^ [X3: set_nat] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le6893508408891458716et_nat @ A @ ( collect_set_nat @ P2 ) )
& ( ord_le6893508408891458716et_nat @ A @ ( collect_set_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_447_conj__subset__def,axiom,
! [A: set_list_nat,P2: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ A
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le6045566169113846134st_nat @ A @ ( collect_list_nat @ P2 ) )
& ( ord_le6045566169113846134st_nat @ A @ ( collect_list_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_448_conj__subset__def,axiom,
! [A: set_Product_unit,P2: product_unit > $o,Q: product_unit > $o] :
( ( ord_le3507040750410214029t_unit @ A
@ ( collect_Product_unit
@ ^ [X3: product_unit] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le3507040750410214029t_unit @ A @ ( collect_Product_unit @ P2 ) )
& ( ord_le3507040750410214029t_unit @ A @ ( collect_Product_unit @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_449_conj__subset__def,axiom,
! [A: set_li8269783359719514504c_fm_i,P2: list_Epistemic_fm_i > $o,Q: list_Epistemic_fm_i > $o] :
( ( ord_le1209247765174941992c_fm_i @ A
@ ( collec5759350117362652307c_fm_i
@ ^ [X3: list_Epistemic_fm_i] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le1209247765174941992c_fm_i @ A @ ( collec5759350117362652307c_fm_i @ P2 ) )
& ( ord_le1209247765174941992c_fm_i @ A @ ( collec5759350117362652307c_fm_i @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_450_conj__subset__def,axiom,
! [A: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ A
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le5389487502678872194c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P2 ) )
& ( ord_le5389487502678872194c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_451_conj__subset__def,axiom,
! [A: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X3: nat] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P2 ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_452_conj__subset__def,axiom,
! [A: set_Epistemic_fm_i,P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ A
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le3843937902494030498c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P2 ) )
& ( ord_le3843937902494030498c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_453_truth,axiom,
! [M: episte4846006725945195288t_unit,W6: epistemic_fm_i,I2: i,P3: epistemic_fm_i] :
( ( episte540185425077706131t_unit @ M )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ P3 ) ) ) ) ).
% truth
thf(fact_454_truth,axiom,
! [M: episte8354619181726739466t_unit,W6: nat,I2: i,P3: epistemic_fm_i] :
( ( episte3073512803368128957t_unit @ M )
=> ( ( member_nat @ W6 @ ( episte8922775215189319317t_unit @ M ) )
=> ( episte4571855713382724530_i_nat @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ P3 ) ) ) ) ).
% truth
thf(fact_455_truth,axiom,
! [M: episte2102749610958417048t_unit,W6: i,I2: i,P3: epistemic_fm_i] :
( ( episte707503779239422227t_unit @ M )
=> ( ( member_i @ W6 @ ( episte8286435292487608811t_unit @ M ) )
=> ( episte5666089339915499428cs_i_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ P3 ) ) ) ) ).
% truth
thf(fact_456_truth,axiom,
! [M: episte4946720254005641710t_unit,W6: nat,I2: epistemic_fm_i,P3: episte1308613106011496894c_fm_i] :
( ( episte8283200250080426145t_unit @ M )
=> ( ( member_nat @ W6 @ ( episte7295652460417134969t_unit @ M ) )
=> ( episte6892841275385687886_i_nat @ M @ W6 @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) @ P3 ) ) ) ) ).
% truth
thf(fact_457_truth,axiom,
! [M: episte7276436686577410172t_unit,W6: i,I2: epistemic_fm_i,P3: episte1308613106011496894c_fm_i] :
( ( episte5368934292378447351t_unit @ M )
=> ( ( member_i @ W6 @ ( episte5422822166207371983t_unit @ M ) )
=> ( episte5578572022554684168fm_i_i @ M @ W6 @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) @ P3 ) ) ) ) ).
% truth
thf(fact_458_truth,axiom,
! [M: episte8649862735182028056t_unit,W6: list_Epistemic_fm_i,I2: i,P3: epistemic_fm_i] :
( ( episte6062502518927837331t_unit @ M )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte970406020673866091t_unit @ M ) )
=> ( episte5087030060053943750c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ P3 ) ) ) ) ).
% truth
thf(fact_459_truth,axiom,
! [M: episte5703418715120784124t_unit,W6: epistemic_fm_i,I2: epistemic_fm_i,P3: episte1308613106011496894c_fm_i] :
( ( episte7789846462086006391t_unit @ M )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte5446861122069384015t_unit @ M ) )
=> ( episte7800156050638082596c_fm_i @ M @ W6 @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) @ P3 ) ) ) ) ).
% truth
thf(fact_460_truth,axiom,
! [M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i,I2: i,P3: epistemic_fm_i] :
( ( episte251305198166243731t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W6 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ P3 ) ) ) ) ).
% truth
thf(fact_461_truth,axiom,
! [M: episte6041718960478671640t_unit,W6: set_se3485332733965609186c_fm_i,I2: i,P3: epistemic_fm_i] :
( ( episte777650505068270995t_unit @ M )
=> ( ( member1461078328125707403c_fm_i @ W6 @ ( episte3312409906628800107t_unit @ M ) )
=> ( episte2123465427079433344c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ P3 ) ) ) ) ).
% truth
thf(fact_462_truth,axiom,
! [M: episte7095840741371228412t_unit,W6: list_Epistemic_fm_i,I2: epistemic_fm_i,P3: episte1308613106011496894c_fm_i] :
( ( episte3051827627171704183t_unit @ M )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte1684291789543234127t_unit @ M ) )
=> ( episte6060530439555422506c_fm_i @ M @ W6 @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) @ P3 ) ) ) ) ).
% truth
thf(fact_463_pos__introspection,axiom,
! [M: episte4846006725945195288t_unit,W6: epistemic_fm_i,I2: i,P3: epistemic_fm_i] :
( ( episte6394096064484582270t_unit @ M )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_464_pos__introspection,axiom,
! [M: episte8354619181726739466t_unit,W6: nat,I2: i,P3: epistemic_fm_i] :
( ( episte7906311534470272552t_unit @ M )
=> ( ( member_nat @ W6 @ ( episte8922775215189319317t_unit @ M ) )
=> ( episte4571855713382724530_i_nat @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_465_pos__introspection,axiom,
! [M: episte2102749610958417048t_unit,W6: i,I2: i,P3: epistemic_fm_i] :
( ( episte6298788599024636926t_unit @ M )
=> ( ( member_i @ W6 @ ( episte8286435292487608811t_unit @ M ) )
=> ( episte5666089339915499428cs_i_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_466_pos__introspection,axiom,
! [M: episte4946720254005641710t_unit,W6: nat,I2: epistemic_fm_i,P3: episte1308613106011496894c_fm_i] :
( ( episte6561722749315164812t_unit @ M )
=> ( ( member_nat @ W6 @ ( episte7295652460417134969t_unit @ M ) )
=> ( episte6892841275385687886_i_nat @ M @ W6 @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) @ ( episte5657488666450005526c_fm_i @ I2 @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_467_pos__introspection,axiom,
! [M: episte7276436686577410172t_unit,W6: i,I2: epistemic_fm_i,P3: episte1308613106011496894c_fm_i] :
( ( episte7780989443469444706t_unit @ M )
=> ( ( member_i @ W6 @ ( episte5422822166207371983t_unit @ M ) )
=> ( episte5578572022554684168fm_i_i @ M @ W6 @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) @ ( episte5657488666450005526c_fm_i @ I2 @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_468_pos__introspection,axiom,
! [M: episte8649862735182028056t_unit,W6: list_Epistemic_fm_i,I2: i,P3: epistemic_fm_i] :
( ( episte3140986294363532158t_unit @ M )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte970406020673866091t_unit @ M ) )
=> ( episte5087030060053943750c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_469_pos__introspection,axiom,
! [M: episte5703418715120784124t_unit,W6: epistemic_fm_i,I2: epistemic_fm_i,P3: episte1308613106011496894c_fm_i] :
( ( episte3656210382014446050t_unit @ M )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte5446861122069384015t_unit @ M ) )
=> ( episte7800156050638082596c_fm_i @ M @ W6 @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) @ ( episte5657488666450005526c_fm_i @ I2 @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_470_pos__introspection,axiom,
! [M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i,I2: i,P3: epistemic_fm_i] :
( ( episte2966952217288386430t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W6 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_471_pos__introspection,axiom,
! [M: episte6041718960478671640t_unit,W6: set_se3485332733965609186c_fm_i,I2: i,P3: epistemic_fm_i] :
( ( episte5217350381671328638t_unit @ M )
=> ( ( member1461078328125707403c_fm_i @ W6 @ ( episte3312409906628800107t_unit @ M ) )
=> ( episte2123465427079433344c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_472_pos__introspection,axiom,
! [M: episte7095840741371228412t_unit,W6: list_Epistemic_fm_i,I2: epistemic_fm_i,P3: episte1308613106011496894c_fm_i] :
( ( episte1901197506451996130t_unit @ M )
=> ( ( member6475149752759350353c_fm_i @ W6 @ ( episte1684291789543234127t_unit @ M ) )
=> ( episte6060530439555422506c_fm_i @ M @ W6 @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) @ ( episte5657488666450005526c_fm_i @ I2 @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_473_Euclidean__def,axiom,
( episte6275404236303465413t_unit
= ( ^ [M3: episte5489734138418521880t_unit] :
! [I: i,X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ! [Y4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ! [Z4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ( member1104366573291651755c_fm_i @ Y4 @ ( episte852950631662840415t_unit @ M3 @ I @ X3 ) )
=> ( ( member1104366573291651755c_fm_i @ Z4 @ ( episte852950631662840415t_unit @ M3 @ I @ X3 ) )
=> ( member1104366573291651755c_fm_i @ Z4 @ ( episte852950631662840415t_unit @ M3 @ I @ Y4 ) ) ) ) ) ) ) ) ) ).
% Euclidean_def
thf(fact_474_Euclidean__def,axiom,
( episte8713243348624617413t_unit
= ( ^ [M3: episte4846006725945195288t_unit] :
! [I: i,X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( episte1297369412406343275t_unit @ M3 ) )
=> ! [Y4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y4 @ ( episte1297369412406343275t_unit @ M3 ) )
=> ! [Z4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z4 @ ( episte1297369412406343275t_unit @ M3 ) )
=> ( ( member6642669606046002379c_fm_i @ Y4 @ ( episte5576233017153220191t_unit @ M3 @ I @ X3 ) )
=> ( ( member6642669606046002379c_fm_i @ Z4 @ ( episte5576233017153220191t_unit @ M3 @ I @ X3 ) )
=> ( member6642669606046002379c_fm_i @ Z4 @ ( episte5576233017153220191t_unit @ M3 @ I @ Y4 ) ) ) ) ) ) ) ) ) ).
% Euclidean_def
thf(fact_475_weakly__directed__def,axiom,
( stalna1726574297087873274c_fm_i
= ( ^ [M3: episte4846006725945195288t_unit] :
! [I: i,X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( episte1297369412406343275t_unit @ M3 ) )
=> ! [Y4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y4 @ ( episte1297369412406343275t_unit @ M3 ) )
=> ! [Z4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z4 @ ( episte1297369412406343275t_unit @ M3 ) )
=> ( ( ( member6642669606046002379c_fm_i @ Z4 @ ( episte5576233017153220191t_unit @ M3 @ I @ X3 ) )
& ( member6642669606046002379c_fm_i @ Y4 @ ( episte5576233017153220191t_unit @ M3 @ I @ X3 ) ) )
=> ? [Aa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Aa @ ( episte1297369412406343275t_unit @ M3 ) )
& ( member6642669606046002379c_fm_i @ Aa @ ( episte5576233017153220191t_unit @ M3 @ I @ Z4 ) )
& ( member6642669606046002379c_fm_i @ Aa @ ( episte5576233017153220191t_unit @ M3 @ I @ Y4 ) ) ) ) ) ) ) ) ) ).
% weakly_directed_def
thf(fact_476_weakly__directed__def,axiom,
( stalna4517306826702924506c_fm_i
= ( ^ [M3: episte5489734138418521880t_unit] :
! [I: i,X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ! [Y4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ! [Z4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z4 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ( ( member1104366573291651755c_fm_i @ Z4 @ ( episte852950631662840415t_unit @ M3 @ I @ X3 ) )
& ( member1104366573291651755c_fm_i @ Y4 @ ( episte852950631662840415t_unit @ M3 @ I @ X3 ) ) )
=> ? [Aa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Aa @ ( episte2675268102452679275t_unit @ M3 ) )
& ( member1104366573291651755c_fm_i @ Aa @ ( episte852950631662840415t_unit @ M3 @ I @ Z4 ) )
& ( member1104366573291651755c_fm_i @ Aa @ ( episte852950631662840415t_unit @ M3 @ I @ Y4 ) ) ) ) ) ) ) ) ) ).
% weakly_directed_def
thf(fact_477_bot__apply,axiom,
( bot_bo2580527446789968623fm_i_o
= ( ^ [X3: epistemic_fm_i] : bot_bot_o ) ) ).
% bot_apply
thf(fact_478_bot__apply,axiom,
( bot_bo6089404950617257231fm_i_o
= ( ^ [X3: set_Epistemic_fm_i] : bot_bot_o ) ) ).
% bot_apply
thf(fact_479_bot__apply,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_480_empty__iff,axiom,
! [C2: list_Epistemic_fm_i] :
~ ( member6475149752759350353c_fm_i @ C2 @ bot_bo6562031733678445556c_fm_i ) ).
% empty_iff
thf(fact_481_empty__iff,axiom,
! [C2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ C2 @ bot_bo145720340923748686c_fm_i ) ).
% empty_iff
thf(fact_482_empty__iff,axiom,
! [C2: set_se3485332733965609186c_fm_i] :
~ ( member1461078328125707403c_fm_i @ C2 @ bot_bo4781621276559889198c_fm_i ) ).
% empty_iff
thf(fact_483_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_484_empty__iff,axiom,
! [C2: i] :
~ ( member_i @ C2 @ bot_bot_set_i ) ).
% empty_iff
thf(fact_485_empty__iff,axiom,
! [C2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ C2 @ bot_bo4194595901900360558c_fm_i ) ).
% empty_iff
thf(fact_486_all__not__in__conv,axiom,
! [A: set_li8269783359719514504c_fm_i] :
( ( ! [X3: list_Epistemic_fm_i] :
~ ( member6475149752759350353c_fm_i @ X3 @ A ) )
= ( A = bot_bo6562031733678445556c_fm_i ) ) ).
% all_not_in_conv
thf(fact_487_all__not__in__conv,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ! [X3: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X3 @ A ) )
= ( A = bot_bo145720340923748686c_fm_i ) ) ).
% all_not_in_conv
thf(fact_488_all__not__in__conv,axiom,
! [A: set_se7339729205154126530c_fm_i] :
( ( ! [X3: set_se3485332733965609186c_fm_i] :
~ ( member1461078328125707403c_fm_i @ X3 @ A ) )
= ( A = bot_bo4781621276559889198c_fm_i ) ) ).
% all_not_in_conv
thf(fact_489_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_490_all__not__in__conv,axiom,
! [A: set_i] :
( ( ! [X3: i] :
~ ( member_i @ X3 @ A ) )
= ( A = bot_bot_set_i ) ) ).
% all_not_in_conv
thf(fact_491_all__not__in__conv,axiom,
! [A: set_Epistemic_fm_i] :
( ( ! [X3: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X3 @ A ) )
= ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% all_not_in_conv
thf(fact_492_Collect__empty__eq,axiom,
! [P2: set_nat > $o] :
( ( ( collect_set_nat @ P2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_493_Collect__empty__eq,axiom,
! [P2: list_Epistemic_fm_i > $o] :
( ( ( collec5759350117362652307c_fm_i @ P2 )
= bot_bo6562031733678445556c_fm_i )
= ( ! [X3: list_Epistemic_fm_i] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_494_Collect__empty__eq,axiom,
! [P2: list_nat > $o] :
( ( ( collect_list_nat @ P2 )
= bot_bot_set_list_nat )
= ( ! [X3: list_nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_495_Collect__empty__eq,axiom,
! [P2: i > $o] :
( ( ( collect_i @ P2 )
= bot_bot_set_i )
= ( ! [X3: i] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_496_Collect__empty__eq,axiom,
! [P2: set_Epistemic_fm_i > $o] :
( ( ( collec3087743281813070829c_fm_i @ P2 )
= bot_bo145720340923748686c_fm_i )
= ( ! [X3: set_Epistemic_fm_i] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_497_Collect__empty__eq,axiom,
! [P2: epistemic_fm_i > $o] :
( ( ( collec4904205187116291597c_fm_i @ P2 )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X3: epistemic_fm_i] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_498_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_499_empty__Collect__eq,axiom,
! [P2: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P2 ) )
= ( ! [X3: set_nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_500_empty__Collect__eq,axiom,
! [P2: list_Epistemic_fm_i > $o] :
( ( bot_bo6562031733678445556c_fm_i
= ( collec5759350117362652307c_fm_i @ P2 ) )
= ( ! [X3: list_Epistemic_fm_i] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_501_empty__Collect__eq,axiom,
! [P2: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P2 ) )
= ( ! [X3: list_nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_502_empty__Collect__eq,axiom,
! [P2: i > $o] :
( ( bot_bot_set_i
= ( collect_i @ P2 ) )
= ( ! [X3: i] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_503_empty__Collect__eq,axiom,
! [P2: set_Epistemic_fm_i > $o] :
( ( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ P2 ) )
= ( ! [X3: set_Epistemic_fm_i] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_504_empty__Collect__eq,axiom,
! [P2: epistemic_fm_i > $o] :
( ( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ P2 ) )
= ( ! [X3: epistemic_fm_i] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_505_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_506_fm_Oinject_I4_J,axiom,
! [X51: epistemic_fm_i,X52: epistemic_fm_i,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
( ( ( epistemic_Imp_i @ X51 @ X52 )
= ( epistemic_Imp_i @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_507_fm_Oinject_I4_J,axiom,
! [X51: episte1308613106011496894c_fm_i,X52: episte1308613106011496894c_fm_i,Y51: episte1308613106011496894c_fm_i,Y52: episte1308613106011496894c_fm_i] :
( ( ( episte260752253203357973c_fm_i @ X51 @ X52 )
= ( episte260752253203357973c_fm_i @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_508_subset__empty,axiom,
! [A: set_i] :
( ( ord_less_eq_set_i @ A @ bot_bot_set_i )
= ( A = bot_bot_set_i ) ) ).
% subset_empty
thf(fact_509_subset__empty,axiom,
! [A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% subset_empty
thf(fact_510_subset__empty,axiom,
! [A: set_li8269783359719514504c_fm_i] :
( ( ord_le1209247765174941992c_fm_i @ A @ bot_bo6562031733678445556c_fm_i )
= ( A = bot_bo6562031733678445556c_fm_i ) ) ).
% subset_empty
thf(fact_511_subset__empty,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ bot_bo145720340923748686c_fm_i )
= ( A = bot_bo145720340923748686c_fm_i ) ) ).
% subset_empty
thf(fact_512_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_513_subset__empty,axiom,
! [A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% subset_empty
thf(fact_514_empty__subsetI,axiom,
! [A: set_i] : ( ord_less_eq_set_i @ bot_bot_set_i @ A ) ).
% empty_subsetI
thf(fact_515_empty__subsetI,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A ) ).
% empty_subsetI
thf(fact_516_empty__subsetI,axiom,
! [A: set_li8269783359719514504c_fm_i] : ( ord_le1209247765174941992c_fm_i @ bot_bo6562031733678445556c_fm_i @ A ) ).
% empty_subsetI
thf(fact_517_empty__subsetI,axiom,
! [A: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ bot_bo145720340923748686c_fm_i @ A ) ).
% empty_subsetI
thf(fact_518_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_519_empty__subsetI,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A ) ).
% empty_subsetI
thf(fact_520_Imp__intro,axiom,
! [M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i,P3: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( ( episte2498298383968489120c_fm_i @ M @ W6 @ P3 )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ Q2 ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ P3 @ Q2 ) ) ) ).
% Imp_intro
thf(fact_521_Imp__intro,axiom,
! [M: episte4846006725945195288t_unit,W6: epistemic_fm_i,P3: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( ( episte5182706529199770816c_fm_i @ M @ W6 @ P3 )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ Q2 ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ P3 @ Q2 ) ) ) ).
% Imp_intro
thf(fact_522_K__A2_H,axiom,
! [A: episte1308613106011496894c_fm_i > $o,I2: epistemic_fm_i,P3: episte1308613106011496894c_fm_i,Q2: episte1308613106011496894c_fm_i] : ( episte3601224874663357326c_fm_i @ A @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ ( episte260752253203357973c_fm_i @ P3 @ Q2 ) ) @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) @ ( episte5657488666450005526c_fm_i @ I2 @ Q2 ) ) ) ) ).
% K_A2'
thf(fact_523_K__A2_H,axiom,
! [A: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i,Q2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ Q2 ) ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ Q2 ) ) ) ) ).
% K_A2'
thf(fact_524_K__map,axiom,
! [A: episte1308613106011496894c_fm_i > $o,P3: episte1308613106011496894c_fm_i,Q2: episte1308613106011496894c_fm_i,I2: epistemic_fm_i] :
( ( episte3601224874663357326c_fm_i @ A @ ( episte260752253203357973c_fm_i @ P3 @ Q2 ) )
=> ( episte3601224874663357326c_fm_i @ A @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P3 ) @ ( episte5657488666450005526c_fm_i @ I2 @ Q2 ) ) ) ) ).
% K_map
thf(fact_525_K__map,axiom,
! [A: epistemic_fm_i > $o,P3: epistemic_fm_i,Q2: epistemic_fm_i,I2: i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P3 @ Q2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ Q2 ) ) ) ) ).
% K_map
thf(fact_526_Ax,axiom,
! [A: epistemic_fm_i > $o,P3: epistemic_fm_i] :
( ( A @ P3 )
=> ( epistemic_AK_i @ A @ P3 ) ) ).
% Ax
thf(fact_527_R1,axiom,
! [A: epistemic_fm_i > $o,P3: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ P3 )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P3 @ Q2 ) )
=> ( epistemic_AK_i @ A @ Q2 ) ) ) ).
% R1
thf(fact_528_R1,axiom,
! [A: episte1308613106011496894c_fm_i > $o,P3: episte1308613106011496894c_fm_i,Q2: episte1308613106011496894c_fm_i] :
( ( episte3601224874663357326c_fm_i @ A @ P3 )
=> ( ( episte3601224874663357326c_fm_i @ A @ ( episte260752253203357973c_fm_i @ P3 @ Q2 ) )
=> ( episte3601224874663357326c_fm_i @ A @ Q2 ) ) ) ).
% R1
thf(fact_529_emptyE,axiom,
! [A3: list_Epistemic_fm_i] :
~ ( member6475149752759350353c_fm_i @ A3 @ bot_bo6562031733678445556c_fm_i ) ).
% emptyE
thf(fact_530_emptyE,axiom,
! [A3: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ A3 @ bot_bo145720340923748686c_fm_i ) ).
% emptyE
thf(fact_531_emptyE,axiom,
! [A3: set_se3485332733965609186c_fm_i] :
~ ( member1461078328125707403c_fm_i @ A3 @ bot_bo4781621276559889198c_fm_i ) ).
% emptyE
thf(fact_532_emptyE,axiom,
! [A3: nat] :
~ ( member_nat @ A3 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_533_emptyE,axiom,
! [A3: i] :
~ ( member_i @ A3 @ bot_bot_set_i ) ).
% emptyE
thf(fact_534_emptyE,axiom,
! [A3: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) ).
% emptyE
thf(fact_535_equals0D,axiom,
! [A: set_li8269783359719514504c_fm_i,A3: list_Epistemic_fm_i] :
( ( A = bot_bo6562031733678445556c_fm_i )
=> ~ ( member6475149752759350353c_fm_i @ A3 @ A ) ) ).
% equals0D
thf(fact_536_equals0D,axiom,
! [A: set_se3485332733965609186c_fm_i,A3: set_Epistemic_fm_i] :
( ( A = bot_bo145720340923748686c_fm_i )
=> ~ ( member1104366573291651755c_fm_i @ A3 @ A ) ) ).
% equals0D
thf(fact_537_equals0D,axiom,
! [A: set_se7339729205154126530c_fm_i,A3: set_se3485332733965609186c_fm_i] :
( ( A = bot_bo4781621276559889198c_fm_i )
=> ~ ( member1461078328125707403c_fm_i @ A3 @ A ) ) ).
% equals0D
thf(fact_538_equals0D,axiom,
! [A: set_nat,A3: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A3 @ A ) ) ).
% equals0D
thf(fact_539_equals0D,axiom,
! [A: set_i,A3: i] :
( ( A = bot_bot_set_i )
=> ~ ( member_i @ A3 @ A ) ) ).
% equals0D
thf(fact_540_equals0D,axiom,
! [A: set_Epistemic_fm_i,A3: epistemic_fm_i] :
( ( A = bot_bo4194595901900360558c_fm_i )
=> ~ ( member6642669606046002379c_fm_i @ A3 @ A ) ) ).
% equals0D
thf(fact_541_equals0I,axiom,
! [A: set_li8269783359719514504c_fm_i] :
( ! [Y5: list_Epistemic_fm_i] :
~ ( member6475149752759350353c_fm_i @ Y5 @ A )
=> ( A = bot_bo6562031733678445556c_fm_i ) ) ).
% equals0I
thf(fact_542_equals0I,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [Y5: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ Y5 @ A )
=> ( A = bot_bo145720340923748686c_fm_i ) ) ).
% equals0I
thf(fact_543_equals0I,axiom,
! [A: set_se7339729205154126530c_fm_i] :
( ! [Y5: set_se3485332733965609186c_fm_i] :
~ ( member1461078328125707403c_fm_i @ Y5 @ A )
=> ( A = bot_bo4781621276559889198c_fm_i ) ) ).
% equals0I
thf(fact_544_equals0I,axiom,
! [A: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_545_equals0I,axiom,
! [A: set_i] :
( ! [Y5: i] :
~ ( member_i @ Y5 @ A )
=> ( A = bot_bot_set_i ) ) ).
% equals0I
thf(fact_546_equals0I,axiom,
! [A: set_Epistemic_fm_i] :
( ! [Y5: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ Y5 @ A )
=> ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% equals0I
thf(fact_547_ex__in__conv,axiom,
! [A: set_i] :
( ( ? [X3: i] : ( member_i @ X3 @ A ) )
= ( A != bot_bot_set_i ) ) ).
% ex_in_conv
thf(fact_548_ex__in__conv,axiom,
! [A: set_Epistemic_fm_i] :
( ( ? [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ A ) )
= ( A != bot_bo4194595901900360558c_fm_i ) ) ).
% ex_in_conv
thf(fact_549_empty__def,axiom,
( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] : $false ) ) ).
% empty_def
thf(fact_550_empty__def,axiom,
( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] : $false ) ) ).
% empty_def
thf(fact_551_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_552_R2,axiom,
! [A: epistemic_fm_i > $o,P3: epistemic_fm_i,I2: i] :
( ( epistemic_AK_i @ A @ P3 )
=> ( epistemic_AK_i @ A @ ( epistemic_K_i @ I2 @ P3 ) ) ) ).
% R2
thf(fact_553_bot_Oextremum,axiom,
! [A3: product_unit] : ( ord_le3221252021190050221t_unit @ bot_bot_Product_unit @ A3 ) ).
% bot.extremum
thf(fact_554_bot_Oextremum,axiom,
! [A3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A3 ) ).
% bot.extremum
thf(fact_555_bot_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A3 ) ).
% bot.extremum
thf(fact_556_bot_Oextremum__unique,axiom,
! [A3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ bot_bot_Product_unit )
= ( A3 = bot_bot_Product_unit ) ) ).
% bot.extremum_unique
thf(fact_557_bot_Oextremum__unique,axiom,
! [A3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i )
= ( A3 = bot_bo4194595901900360558c_fm_i ) ) ).
% bot.extremum_unique
thf(fact_558_bot_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
= ( A3 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_559_bot_Oextremum__uniqueI,axiom,
! [A3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ bot_bot_Product_unit )
=> ( A3 = bot_bot_Product_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_560_bot_Oextremum__uniqueI,axiom,
! [A3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i )
=> ( A3 = bot_bo4194595901900360558c_fm_i ) ) ).
% bot.extremum_uniqueI
thf(fact_561_bot_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
=> ( A3 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_562_fm_Odistinct_I29_J,axiom,
! [X51: epistemic_fm_i,X52: epistemic_fm_i,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Imp_i @ X51 @ X52 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_563_subset__emptyI,axiom,
! [A: set_Epistemic_fm_i] :
( ! [X: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X @ A )
=> ( ord_le3843937902494030498c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) ) ).
% subset_emptyI
thf(fact_564_fm_Odistinct_I15_J,axiom,
! [X2: list_char,X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( epistemic_Pro_i @ X2 )
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_565_deriv__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( epistemic_AK_i @ A @ P3 )
=> ( member6642669606046002379c_fm_i @ P3 @ V ) ) ) ) ).
% deriv_in_maximal
thf(fact_566_frame_Oselect__convs_I2_J,axiom,
! [W3: set_se3485332733965609186c_fm_i,K2: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More: episte7133455005732253587t_unit] :
( ( episte852950631662840415t_unit @ ( episte8397520288747125024t_unit @ W3 @ K2 @ More ) )
= K2 ) ).
% frame.select_convs(2)
thf(fact_567_S5_H__4,axiom,
! [A: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ) ) ).
% S5'_4
thf(fact_568_AxT_Ointros,axiom,
! [I2: i,P3: epistemic_fm_i] : ( epistemic_AxT_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ P3 ) ) ).
% AxT.intros
thf(fact_569_AxT_Osimps,axiom,
( epistemic_AxT_i
= ( ^ [A5: epistemic_fm_i] :
? [I: i,P4: epistemic_fm_i] :
( A5
= ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ P4 ) ) ) ) ).
% AxT.simps
thf(fact_570_AxT_Ocases,axiom,
! [A3: epistemic_fm_i] :
( ( epistemic_AxT_i @ A3 )
=> ~ ! [I3: i,P5: epistemic_fm_i] :
( A3
!= ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P5 ) @ P5 ) ) ) ).
% AxT.cases
thf(fact_571_consequent__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P3: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ P3 @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P3 @ Q2 ) @ V )
=> ( member6642669606046002379c_fm_i @ Q2 @ V ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_572_Ax4_Ointros,axiom,
! [I2: i,P3: epistemic_fm_i] : ( epistemic_Ax4_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ).
% Ax4.intros
thf(fact_573_Ax4_Osimps,axiom,
( epistemic_Ax4_i
= ( ^ [A5: epistemic_fm_i] :
? [I: i,P4: epistemic_fm_i] :
( A5
= ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ ( epistemic_K_i @ I @ P4 ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_574_Ax4_Ocases,axiom,
! [A3: epistemic_fm_i] :
( ( epistemic_Ax4_i @ A3 )
=> ~ ! [I3: i,P5: epistemic_fm_i] :
( A3
!= ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P5 ) @ ( epistemic_K_i @ I3 @ ( epistemic_K_i @ I3 @ P5 ) ) ) ) ) ).
% Ax4.cases
thf(fact_575_frame_Osurjective,axiom,
! [R: episte5489734138418521880t_unit] :
( R
= ( episte8397520288747125024t_unit @ ( episte2675268102452679275t_unit @ R ) @ ( episte852950631662840415t_unit @ R ) @ ( episte7135767042998387833t_unit @ R ) ) ) ).
% frame.surjective
thf(fact_576_KB4__5,axiom,
! [A: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% KB4_5
thf(fact_577_neg__introspection,axiom,
! [M: episte4846006725945195288t_unit,W6: epistemic_fm_i,I2: i,P3: epistemic_fm_i] :
( ( episte6568605431049406366t_unit @ M )
=> ( ( episte6394096064484582270t_unit @ M )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ epistemic_FF_i ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_578_S5_H__B,axiom,
! [A: epistemic_fm_i > $o,P3: epistemic_fm_i,I2: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P3 @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% S5'_B
thf(fact_579_strong__completeness,axiom,
! [P2: episte5489734138418521880t_unit > $o,G4: set_Epistemic_fm_i,P3: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [M2: episte5489734138418521880t_unit] :
( ( P2 @ M2 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G4 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ P3 ) ) ) )
=> ( ( P2
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W5: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W5 )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W5 ) ) )
@ ^ [I: i,V2: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V2: set_Epistemic_fm_i,X3: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X3 ) @ V2 )
@ product_Unity ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G4 )
& ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Qs @ P3 ) ) ) ) ) ).
% strong_completeness
thf(fact_580_strong__completeness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M2 )
& ( episte80897895827131294t_unit @ M2 )
& ( episte2966952217288386430t_unit @ M2 ) )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G4 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ P3 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G4 )
& ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_AxB_i @ P4 )
| ( epistemic_Ax4_i @ P4 ) )
@ ( epistemic_imply_i @ Qs @ P3 ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5
thf(fact_581_main_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M3 )
& ( episte80897895827131294t_unit @ M3 )
& ( episte2966952217288386430t_unit @ M3 ) )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y4 @ G4 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ Y4 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ P3 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G4 )
& ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_AxB_i @ P4 )
| ( epistemic_Ax4_i @ P4 ) )
@ ( epistemic_imply_i @ Qs2 @ P3 ) ) ) ) ) ).
% main\<^sub>S\<^sub>5
thf(fact_582_weakly__directed,axiom,
! [M: episte4846006725945195288t_unit,W6: epistemic_fm_i,I2: i,P3: epistemic_fm_i] :
( ( stalna1726574297087873274c_fm_i @ M )
=> ( ( member6642669606046002379c_fm_i @ W6 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% weakly_directed
thf(fact_583_weakly__directed,axiom,
! [M: episte5489734138418521880t_unit,W6: set_Epistemic_fm_i,I2: i,P3: epistemic_fm_i] :
( ( stalna4517306826702924506c_fm_i @ M )
=> ( ( member1104366573291651755c_fm_i @ W6 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W6 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% weakly_directed
thf(fact_584_consistent__def,axiom,
( episte2285483198712856234tent_i
= ( ^ [A2: epistemic_fm_i > $o,S3: set_Epistemic_fm_i] :
~ ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ S3 )
& ( epistemic_AK_i @ A2 @ ( epistemic_imply_i @ Qs2 @ epistemic_FF_i ) ) ) ) ) ).
% consistent_def
thf(fact_585_bot__set__def,axiom,
( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ bot_bo6089404950617257231fm_i_o ) ) ).
% bot_set_def
thf(fact_586_bot__set__def,axiom,
( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ bot_bo2580527446789968623fm_i_o ) ) ).
% bot_set_def
thf(fact_587_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_588_bot__empty__eq,axiom,
( bot_bo2580527446789968623fm_i_o
= ( ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% bot_empty_eq
thf(fact_589_K__imply__weaken,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,Q2: epistemic_fm_i,Ps2: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ps ) @ ( set_Epistemic_fm_i2 @ Ps2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps2 @ Q2 ) ) ) ) ).
% K_imply_weaken
thf(fact_590_fm_Odistinct_I9_J,axiom,
! [X61: i,X62: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_591_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
thf(fact_592_fm_Odistinct_I1_J,axiom,
! [X2: list_char] :
( epistemic_FF_i
!= ( epistemic_Pro_i @ X2 ) ) ).
% fm.distinct(1)
thf(fact_593_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_i,Qs3: list_Epistemic_fm_i,R: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ps ) @ ( set_Epistemic_fm_i2 @ Qs3 ) )
=> ! [G5: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G5 @ H3 @ ( epistemic_Imp_i @ ( epistemic_imply_i @ Ps @ R ) @ ( epistemic_imply_i @ Qs3 @ R ) ) ) ) ).
% tautology_imply_superset
thf(fact_594_strong__soundness,axiom,
! [A: epistemic_fm_i > $o,P2: episte4846006725945195288t_unit > $o,G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ! [M2: episte4846006725945195288t_unit,W8: epistemic_fm_i,P5: epistemic_fm_i] :
( ( A @ P5 )
=> ( ( P2 @ M2 )
=> ( ( member6642669606046002379c_fm_i @ W8 @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ W8 @ P5 ) ) ) )
=> ( ? [Qs4: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs4 ) @ G4 )
& ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Qs4 @ P3 ) ) )
=> ! [M4: episte4846006725945195288t_unit] :
( ( P2 @ M4 )
=> ! [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ ( episte1297369412406343275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G4 )
=> ( episte5182706529199770816c_fm_i @ M4 @ X6 @ Xa2 ) )
=> ( episte5182706529199770816c_fm_i @ M4 @ X6 @ P3 ) ) ) ) ) ) ).
% strong_soundness
thf(fact_595_strong__completeness_092_060_094sub_062K,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G4 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ P3 ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G4 )
& ( epistemic_AK_i
@ ^ [Uu: epistemic_fm_i] : $false
@ ( epistemic_imply_i @ Qs @ P3 ) ) ) ) ).
% strong_completeness\<^sub>K
thf(fact_596_main_092_060_094sub_062K,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit,X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y4 @ G4 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ Y4 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ P3 ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G4 )
& ( epistemic_AK_i
@ ^ [Uu: epistemic_fm_i] : $false
@ ( epistemic_imply_i @ Qs2 @ P3 ) ) ) ) ) ).
% main\<^sub>K
thf(fact_597_frame_Oselect__convs_I3_J,axiom,
! [W3: set_se3485332733965609186c_fm_i,K2: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More: episte7133455005732253587t_unit] :
( ( episte7135767042998387833t_unit @ ( episte8397520288747125024t_unit @ W3 @ K2 @ More ) )
= More ) ).
% frame.select_convs(3)
thf(fact_598_K__L__dual,axiom,
! [A: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ P3 ) ) ) ).
% K_L_dual
thf(fact_599_K__LK,axiom,
! [A: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ epistemic_FF_i ) ) ) ).
% K_LK
thf(fact_600_exactly__one__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ P3 @ V )
= ( ~ ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) @ V ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_601_duality__taut,axiom,
! [I2: i,P3: epistemic_fm_i,Q2: epistemic_fm_i,G5: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G5 @ H3 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ Q2 @ epistemic_FF_i ) ) ) @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ Q2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ epistemic_FF_i ) ) ) ) ).
% duality_taut
thf(fact_602_Ax5_Ointros,axiom,
! [I2: i,P3: epistemic_fm_i] : ( epistemic_Ax5_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% Ax5.intros
thf(fact_603_Ax5_Osimps,axiom,
( epistemic_Ax5_i
= ( ^ [A5: epistemic_fm_i] :
? [I: i,P4: epistemic_fm_i] :
( A5
= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_604_Ax5_Ocases,axiom,
! [A3: epistemic_fm_i] :
( ( epistemic_Ax5_i @ A3 )
=> ~ ! [I3: i,P5: epistemic_fm_i] :
( A3
!= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P5 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P5 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% Ax5.cases
thf(fact_605_AxB_Ointros,axiom,
! [P3: epistemic_fm_i,I2: i] : ( epistemic_AxB_i @ ( epistemic_Imp_i @ P3 @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% AxB.intros
thf(fact_606_AxB_Osimps,axiom,
( epistemic_AxB_i
= ( ^ [A5: epistemic_fm_i] :
? [P4: epistemic_fm_i,I: i] :
( A5
= ( epistemic_Imp_i @ P4 @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% AxB.simps
thf(fact_607_AxB_Ocases,axiom,
! [A3: epistemic_fm_i] :
( ( epistemic_AxB_i @ A3 )
=> ~ ! [P5: epistemic_fm_i,I3: i] :
( A3
!= ( epistemic_Imp_i @ P5 @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P5 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% AxB.cases
thf(fact_608_main_092_060_094sub_062T,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( episte251305198166243731t_unit @ M3 )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y4 @ G4 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ Y4 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ P3 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G4 )
& ( epistemic_AK_i @ epistemic_AxT_i @ ( epistemic_imply_i @ Qs2 @ P3 ) ) ) ) ) ).
% main\<^sub>T
thf(fact_609_strong__completeness_092_060_094sub_062T,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( episte251305198166243731t_unit @ M2 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G4 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ P3 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G4 )
& ( epistemic_AK_i @ epistemic_AxT_i @ ( epistemic_imply_i @ Qs @ P3 ) ) ) ) ).
% strong_completeness\<^sub>T
thf(fact_610_main_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( episte2966952217288386430t_unit @ M3 )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y4 @ G4 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ Y4 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ P3 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G4 )
& ( epistemic_AK_i @ epistemic_Ax4_i @ ( epistemic_imply_i @ Qs2 @ P3 ) ) ) ) ) ).
% main\<^sub>K\<^sub>4
thf(fact_611_strong__completeness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( episte2966952217288386430t_unit @ M2 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G4 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ P3 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G4 )
& ( epistemic_AK_i @ epistemic_Ax4_i @ ( epistemic_imply_i @ Qs @ P3 ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>4
thf(fact_612_main_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( episte80897895827131294t_unit @ M3 )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y4 @ G4 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ Y4 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ P3 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G4 )
& ( epistemic_AK_i @ epistemic_AxB_i @ ( epistemic_imply_i @ Qs2 @ P3 ) ) ) ) ) ).
% main\<^sub>K\<^sub>B
thf(fact_613_strong__completeness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( episte80897895827131294t_unit @ M2 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G4 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ P3 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G4 )
& ( epistemic_AK_i @ epistemic_AxB_i @ ( epistemic_imply_i @ Qs @ P3 ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>B
thf(fact_614_Ax__2_Ointros,axiom,
! [I2: i,P3: epistemic_fm_i] : ( stalnaker_Ax_2_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% Ax_2.intros
thf(fact_615_Ax__2_Osimps,axiom,
( stalnaker_Ax_2_i
= ( ^ [A5: epistemic_fm_i] :
? [I: i,P4: epistemic_fm_i] :
( A5
= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% Ax_2.simps
thf(fact_616_Ax__2_Ocases,axiom,
! [A3: epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ A3 )
=> ~ ! [I3: i,P5: epistemic_fm_i] :
( A3
!= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P5 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P5 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% Ax_2.cases
thf(fact_617_S5__S5_H__assms,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G4 )
& ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_AxB_i @ P4 )
| ( epistemic_Ax4_i @ P4 ) )
@ ( epistemic_imply_i @ Qs2 @ P3 ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G4 )
& ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_Ax5_i @ P4 ) )
@ ( epistemic_imply_i @ Qs2 @ P3 ) ) ) ) ) ).
% S5_S5'_assms
thf(fact_618_main_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( episte6275404236303465413t_unit @ M3 )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y4 @ G4 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ Y4 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ P3 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G4 )
& ( epistemic_AK_i @ epistemic_Ax5_i @ ( epistemic_imply_i @ Qs2 @ P3 ) ) ) ) ) ).
% main\<^sub>K\<^sub>5
thf(fact_619_strong__completeness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( episte6275404236303465413t_unit @ M2 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G4 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ P3 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G4 )
& ( epistemic_AK_i @ epistemic_Ax5_i @ ( epistemic_imply_i @ Qs @ P3 ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>5
thf(fact_620_main_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M3 )
& ( episte2966952217288386430t_unit @ M3 ) )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y4 @ G4 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ Y4 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ P3 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G4 )
& ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_Ax4_i @ P4 ) )
@ ( epistemic_imply_i @ Qs2 @ P3 ) ) ) ) ) ).
% main\<^sub>S\<^sub>4
thf(fact_621_strong__completeness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M2 )
& ( episte2966952217288386430t_unit @ M2 ) )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G4 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ P3 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G4 )
& ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_Ax4_i @ P4 ) )
@ ( epistemic_imply_i @ Qs @ P3 ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>4
thf(fact_622_T__L,axiom,
! [A: epistemic_fm_i > $o,P3: epistemic_fm_i,I2: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% T_L
thf(fact_623_K4__L,axiom,
! [A: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% K4_L
thf(fact_624_K5__L,axiom,
! [A: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P3 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ P3 ) ) ) ) ).
% K5_L
thf(fact_625_main_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M3 )
& ( episte80897895827131294t_unit @ M3 )
& ( episte2966952217288386430t_unit @ M3 ) )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y4 @ G4 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ Y4 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X3 @ P3 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G4 )
& ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_Ax5_i @ P4 ) )
@ ( epistemic_imply_i @ Qs2 @ P3 ) ) ) ) ) ).
% main\<^sub>S\<^sub>5'
thf(fact_626_strong__completeness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G4: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M2 )
& ( episte80897895827131294t_unit @ M2 )
& ( episte2966952217288386430t_unit @ M2 ) )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G4 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X @ P3 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G4 )
& ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_Ax5_i @ P4 ) )
@ ( epistemic_imply_i @ Qs @ P3 ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5'
thf(fact_627_reach__dualK,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,W7: set_Epistemic_fm_i,I2: i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ W7 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W7 )
=> ( ( member1104366573291651755c_fm_i @ W7
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P4 ) @ V ) ) ) ) )
=> ! [P: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ P @ W7 )
=> ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ V ) ) ) ) ) ) ) ).
% reach_dualK
thf(fact_628_dual__reach,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,I2: i,P3: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ V )
=> ? [W2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P4 ) @ V ) ) ) ) )
& ( member6642669606046002379c_fm_i @ P3 @ W2 ) ) ) ) ) ).
% dual_reach
thf(fact_629_subset__code_I1_J,axiom,
! [Xs: list_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ B )
= ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_630_IntI,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ A )
=> ( ( member6642669606046002379c_fm_i @ C2 @ B )
=> ( member6642669606046002379c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% IntI
thf(fact_631_Int__iff,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
= ( ( member6642669606046002379c_fm_i @ C2 @ A )
& ( member6642669606046002379c_fm_i @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_632_Int__subset__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
= ( ( ord_le3843937902494030498c_fm_i @ C @ A )
& ( ord_le3843937902494030498c_fm_i @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_633_IntE,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C2 @ A )
=> ~ ( member6642669606046002379c_fm_i @ C2 @ B ) ) ) ).
% IntE
thf(fact_634_IntD1,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
=> ( member6642669606046002379c_fm_i @ C2 @ A ) ) ).
% IntD1
thf(fact_635_IntD2,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
=> ( member6642669606046002379c_fm_i @ C2 @ B ) ) ).
% IntD2
thf(fact_636_Int__def,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A2 )
& ( member1104366573291651755c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_637_Int__def,axiom,
( inf_in3450601097109690352c_fm_i
= ( ^ [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A2 )
& ( member6642669606046002379c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_638_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_639_Int__Collect,axiom,
! [X4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P2 ) ) )
= ( ( member1104366573291651755c_fm_i @ X4 @ A )
& ( P2 @ X4 ) ) ) ).
% Int_Collect
thf(fact_640_Int__Collect,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P2 ) ) )
= ( ( member6642669606046002379c_fm_i @ X4 @ A )
& ( P2 @ X4 ) ) ) ).
% Int_Collect
thf(fact_641_Int__Collect,axiom,
! [X4: nat,A: set_nat,P2: nat > $o] :
( ( member_nat @ X4 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P2 ) ) )
= ( ( member_nat @ X4 @ A )
& ( P2 @ X4 ) ) ) ).
% Int_Collect
thf(fact_642_Collect__conj__eq,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) )
= ( inf_in161960956874937808c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_643_Collect__conj__eq,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) )
= ( inf_in3450601097109690352c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_644_Collect__conj__eq,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_645_Int__emptyI,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ~ ( member6642669606046002379c_fm_i @ X @ B ) )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i ) ) ).
% Int_emptyI
thf(fact_646_disjoint__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ~ ( member6642669606046002379c_fm_i @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_647_Int__mono,axiom,
! [A: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ C @ D ) ) ) ) ).
% Int_mono
thf(fact_648_Int__lower1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_649_Int__lower2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_650_Int__absorb1,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_651_Int__absorb2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_652_Int__greatest,axiom,
! [C: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ B )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_653_Int__Collect__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ( ( P2 @ X )
=> ( Q @ X ) ) )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P2 ) ) @ ( inf_in161960956874937808c_fm_i @ B @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_654_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( P2 @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_655_Int__Collect__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ( ( P2 @ X )
=> ( Q @ X ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P2 ) ) @ ( inf_in3450601097109690352c_fm_i @ B @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_656_fm_Odistinct_I27_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Con_i @ X41 @ X42 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_657_fm_Odistinct_I13_J,axiom,
! [X2: list_char,X41: epistemic_fm_i,X42: epistemic_fm_i] :
( ( epistemic_Pro_i @ X2 )
!= ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_658_mcs__conjunction,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P3: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( ( member6642669606046002379c_fm_i @ P3 @ V )
& ( member6642669606046002379c_fm_i @ Q2 @ V ) )
=> ( member6642669606046002379c_fm_i @ ( epistemic_Con_i @ P3 @ Q2 ) @ V ) ) ) ) ).
% mcs_conjunction
thf(fact_659_A2,axiom,
! [A: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i,Q2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P3 @ Q2 ) ) ) @ ( epistemic_K_i @ I2 @ Q2 ) ) ) ).
% A2
thf(fact_660_K__conjunction__in,axiom,
! [A: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i,Q2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Con_i @ P3 @ Q2 ) ) @ ( epistemic_Con_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ Q2 ) ) ) ) ).
% K_conjunction_in
thf(fact_661_K__conjunction__out,axiom,
! [A: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i,Q2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ Q2 ) ) @ ( epistemic_K_i @ I2 @ ( epistemic_Con_i @ P3 @ Q2 ) ) ) ) ).
% K_conjunction_out
thf(fact_662_K__conj__imply__factor,axiom,
! [A: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i,Q2: epistemic_fm_i,R: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_K_i @ I2 @ Q2 ) ) @ R ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Con_i @ P3 @ Q2 ) ) @ R ) ) ) ).
% K_conj_imply_factor
thf(fact_663_K__thm,axiom,
! [A: epistemic_fm_i > $o,I2: i,P3: epistemic_fm_i,Q2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I2 @ P3 ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ Q2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_Con_i @ P3 @ Q2 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ).
% K_thm
thf(fact_664_AK_Ocases,axiom,
! [A: epistemic_fm_i > $o,A3: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ A3 )
=> ( ~ ! [G5: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G5 @ H3 @ A3 )
=> ( ! [I3: i,P5: epistemic_fm_i,Q3: epistemic_fm_i] :
( A3
!= ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P5 ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P5 @ Q3 ) ) ) @ ( epistemic_K_i @ I3 @ Q3 ) ) )
=> ( ~ ( A @ A3 )
=> ( ! [P5: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ P5 )
=> ~ ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P5 @ A3 ) ) )
=> ~ ! [P5: epistemic_fm_i] :
( ? [I3: i] :
( A3
= ( epistemic_K_i @ I3 @ P5 ) )
=> ~ ( epistemic_AK_i @ A @ P5 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_665_AK_Osimps,axiom,
( epistemic_AK_i
= ( ^ [A2: epistemic_fm_i > $o,A5: epistemic_fm_i] :
( ? [P4: epistemic_fm_i] :
( ( A5 = P4 )
& ! [G2: list_char > $o,H4: epistemic_fm_i > $o] : ( epistemic_eval_i @ G2 @ H4 @ P4 ) )
| ? [I: i,P4: epistemic_fm_i,Q4: epistemic_fm_i] :
( A5
= ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ Q4 ) ) ) @ ( epistemic_K_i @ I @ Q4 ) ) )
| ? [P4: epistemic_fm_i] :
( ( A5 = P4 )
& ( A2 @ P4 ) )
| ? [P4: epistemic_fm_i,Q4: epistemic_fm_i] :
( ( A5 = Q4 )
& ( epistemic_AK_i @ A2 @ P4 )
& ( epistemic_AK_i @ A2 @ ( epistemic_Imp_i @ P4 @ Q4 ) ) )
| ? [P4: epistemic_fm_i,I: i] :
( ( A5
= ( epistemic_K_i @ I @ P4 ) )
& ( epistemic_AK_i @ A2 @ P4 ) ) ) ) ) ).
% AK.simps
thf(fact_666_inf__bot__left,axiom,
! [X4: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X4 )
= bot_bot_Product_unit ) ).
% inf_bot_left
thf(fact_667_inf__bot__right,axiom,
! [X4: product_unit] :
( ( inf_inf_Product_unit @ X4 @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% inf_bot_right
thf(fact_668_boolean__algebra_Oconj__zero__left,axiom,
! [X4: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X4 )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_left
thf(fact_669_boolean__algebra_Oconj__zero__right,axiom,
! [X4: product_unit] :
( ( inf_inf_Product_unit @ X4 @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_right
thf(fact_670_inf_Obounded__iff,axiom,
! [A3: product_unit,B3: product_unit,C2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ ( inf_inf_Product_unit @ B3 @ C2 ) )
= ( ( ord_le3221252021190050221t_unit @ A3 @ B3 )
& ( ord_le3221252021190050221t_unit @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_671_inf_Obounded__iff,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ ( inf_in3450601097109690352c_fm_i @ B3 @ C2 ) )
= ( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
& ( ord_le3843937902494030498c_fm_i @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_672_inf_Obounded__iff,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B3 @ C2 ) )
= ( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_673_le__inf__iff,axiom,
! [X4: product_unit,Y3: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X4 @ ( inf_inf_Product_unit @ Y3 @ Z2 ) )
= ( ( ord_le3221252021190050221t_unit @ X4 @ Y3 )
& ( ord_le3221252021190050221t_unit @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_674_le__inf__iff,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ Y3 @ Z2 ) )
= ( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
& ( ord_le3843937902494030498c_fm_i @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_675_le__inf__iff,axiom,
! [X4: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y3 @ Z2 ) )
= ( ( ord_less_eq_nat @ X4 @ Y3 )
& ( ord_less_eq_nat @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_676_Collect__empty__eq__bot,axiom,
! [P2: set_Epistemic_fm_i > $o] :
( ( ( collec3087743281813070829c_fm_i @ P2 )
= bot_bo145720340923748686c_fm_i )
= ( P2 = bot_bo6089404950617257231fm_i_o ) ) ).
% Collect_empty_eq_bot
thf(fact_677_Collect__empty__eq__bot,axiom,
! [P2: epistemic_fm_i > $o] :
( ( ( collec4904205187116291597c_fm_i @ P2 )
= bot_bo4194595901900360558c_fm_i )
= ( P2 = bot_bo2580527446789968623fm_i_o ) ) ).
% Collect_empty_eq_bot
thf(fact_678_Collect__empty__eq__bot,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( P2 = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_679_inf__unit__def,axiom,
( inf_inf_Product_unit
= ( ^ [Uu2: product_unit,Uv: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_680_inf__set__def,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ( inf_in853329969422372621fm_i_o
@ ^ [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ A2 )
@ ^ [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_681_inf__set__def,axiom,
( inf_in3450601097109690352c_fm_i
= ( ^ [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( inf_in8976956063904982253fm_i_o
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ A2 )
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_682_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_683_inf__Int__eq,axiom,
! [R3: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( inf_in8976956063904982253fm_i_o
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ R3 )
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ S ) )
= ( ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ R3 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_684_inf__sup__ord_I2_J,axiom,
! [X4: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X4 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_685_inf__sup__ord_I2_J,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_686_inf__sup__ord_I2_J,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_687_inf__sup__ord_I1_J,axiom,
! [X4: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X4 @ Y3 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_688_inf__sup__ord_I1_J,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y3 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_689_inf__sup__ord_I1_J,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y3 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_690_inf__le1,axiom,
! [X4: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X4 @ Y3 ) @ X4 ) ).
% inf_le1
thf(fact_691_inf__le1,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y3 ) @ X4 ) ).
% inf_le1
thf(fact_692_inf__le1,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y3 ) @ X4 ) ).
% inf_le1
thf(fact_693_inf__le2,axiom,
! [X4: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X4 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_694_inf__le2,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_695_inf__le2,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_696_le__infE,axiom,
! [X4: product_unit,A3: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X4 @ ( inf_inf_Product_unit @ A3 @ B3 ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ X4 @ A3 )
=> ~ ( ord_le3221252021190050221t_unit @ X4 @ B3 ) ) ) ).
% le_infE
thf(fact_697_le__infE,axiom,
! [X4: set_Epistemic_fm_i,A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ A3 @ B3 ) )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ X4 @ A3 )
=> ~ ( ord_le3843937902494030498c_fm_i @ X4 @ B3 ) ) ) ).
% le_infE
thf(fact_698_le__infE,axiom,
! [X4: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq_nat @ X4 @ A3 )
=> ~ ( ord_less_eq_nat @ X4 @ B3 ) ) ) ).
% le_infE
thf(fact_699_le__infI,axiom,
! [X4: product_unit,A3: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X4 @ A3 )
=> ( ( ord_le3221252021190050221t_unit @ X4 @ B3 )
=> ( ord_le3221252021190050221t_unit @ X4 @ ( inf_inf_Product_unit @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_700_le__infI,axiom,
! [X4: set_Epistemic_fm_i,A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ A3 )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ B3 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_701_le__infI,axiom,
! [X4: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ X4 @ A3 )
=> ( ( ord_less_eq_nat @ X4 @ B3 )
=> ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_702_inf__mono,axiom,
! [A3: product_unit,C2: product_unit,B3: product_unit,D2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ C2 )
=> ( ( ord_le3221252021190050221t_unit @ B3 @ D2 )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A3 @ B3 ) @ ( inf_inf_Product_unit @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_703_inf__mono,axiom,
! [A3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B3 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A3 @ B3 ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_704_inf__mono,axiom,
! [A3: nat,C2: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_705_le__infI1,axiom,
! [A3: product_unit,X4: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ X4 )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A3 @ B3 ) @ X4 ) ) ).
% le_infI1
thf(fact_706_le__infI1,axiom,
! [A3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ X4 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A3 @ B3 ) @ X4 ) ) ).
% le_infI1
thf(fact_707_le__infI1,axiom,
! [A3: nat,X4: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ X4 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ X4 ) ) ).
% le_infI1
thf(fact_708_le__infI2,axiom,
! [B3: product_unit,X4: product_unit,A3: product_unit] :
( ( ord_le3221252021190050221t_unit @ B3 @ X4 )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A3 @ B3 ) @ X4 ) ) ).
% le_infI2
thf(fact_709_le__infI2,axiom,
! [B3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B3 @ X4 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A3 @ B3 ) @ X4 ) ) ).
% le_infI2
thf(fact_710_le__infI2,axiom,
! [B3: nat,X4: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ X4 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ X4 ) ) ).
% le_infI2
thf(fact_711_inf_OorderE,axiom,
! [A3: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ B3 )
=> ( A3
= ( inf_inf_Product_unit @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_712_inf_OorderE,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ( A3
= ( inf_in3450601097109690352c_fm_i @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_713_inf_OorderE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( A3
= ( inf_inf_nat @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_714_inf_OorderI,axiom,
! [A3: product_unit,B3: product_unit] :
( ( A3
= ( inf_inf_Product_unit @ A3 @ B3 ) )
=> ( ord_le3221252021190050221t_unit @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_715_inf_OorderI,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( A3
= ( inf_in3450601097109690352c_fm_i @ A3 @ B3 ) )
=> ( ord_le3843937902494030498c_fm_i @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_716_inf_OorderI,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B3 ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_717_inf__unique,axiom,
! [F: product_unit > product_unit > product_unit,X4: product_unit,Y3: product_unit] :
( ! [X: product_unit,Y5: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X @ Y5 ) @ X )
=> ( ! [X: product_unit,Y5: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X @ Y5 ) @ Y5 )
=> ( ! [X: product_unit,Y5: product_unit,Z5: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y5 )
=> ( ( ord_le3221252021190050221t_unit @ X @ Z5 )
=> ( ord_le3221252021190050221t_unit @ X @ ( F @ Y5 @ Z5 ) ) ) )
=> ( ( inf_inf_Product_unit @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_718_inf__unique,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( F @ X @ Y5 ) @ X )
=> ( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( F @ X @ Y5 ) @ Y5 )
=> ( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i,Z5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y5 )
=> ( ( ord_le3843937902494030498c_fm_i @ X @ Z5 )
=> ( ord_le3843937902494030498c_fm_i @ X @ ( F @ Y5 @ Z5 ) ) ) )
=> ( ( inf_in3450601097109690352c_fm_i @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_719_inf__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y3: nat] :
( ! [X: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X @ Y5 ) @ X )
=> ( ! [X: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X @ Y5 ) @ Y5 )
=> ( ! [X: nat,Y5: nat,Z5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
=> ( ( ord_less_eq_nat @ X @ Z5 )
=> ( ord_less_eq_nat @ X @ ( F @ Y5 @ Z5 ) ) ) )
=> ( ( inf_inf_nat @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_720_le__iff__inf,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X3: product_unit,Y4: product_unit] :
( ( inf_inf_Product_unit @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_721_le__iff__inf,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_722_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( inf_inf_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_723_inf_Oabsorb1,axiom,
! [A3: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ B3 )
=> ( ( inf_inf_Product_unit @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_724_inf_Oabsorb1,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ( ( inf_in3450601097109690352c_fm_i @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_725_inf_Oabsorb1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_726_inf_Oabsorb2,axiom,
! [B3: product_unit,A3: product_unit] :
( ( ord_le3221252021190050221t_unit @ B3 @ A3 )
=> ( ( inf_inf_Product_unit @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_727_inf_Oabsorb2,axiom,
! [B3: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B3 @ A3 )
=> ( ( inf_in3450601097109690352c_fm_i @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_728_inf_Oabsorb2,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_729_inf__absorb1,axiom,
! [X4: product_unit,Y3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X4 @ Y3 )
=> ( ( inf_inf_Product_unit @ X4 @ Y3 )
= X4 ) ) ).
% inf_absorb1
thf(fact_730_inf__absorb1,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
=> ( ( inf_in3450601097109690352c_fm_i @ X4 @ Y3 )
= X4 ) ) ).
% inf_absorb1
thf(fact_731_inf__absorb1,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( inf_inf_nat @ X4 @ Y3 )
= X4 ) ) ).
% inf_absorb1
thf(fact_732_inf__absorb2,axiom,
! [Y3: product_unit,X4: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y3 @ X4 )
=> ( ( inf_inf_Product_unit @ X4 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_733_inf__absorb2,axiom,
! [Y3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y3 @ X4 )
=> ( ( inf_in3450601097109690352c_fm_i @ X4 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_734_inf__absorb2,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( inf_inf_nat @ X4 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_735_inf_OboundedE,axiom,
! [A3: product_unit,B3: product_unit,C2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ ( inf_inf_Product_unit @ B3 @ C2 ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ A3 @ B3 )
=> ~ ( ord_le3221252021190050221t_unit @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_736_inf_OboundedE,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ ( inf_in3450601097109690352c_fm_i @ B3 @ C2 ) )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ~ ( ord_le3843937902494030498c_fm_i @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_737_inf_OboundedE,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A3 @ B3 )
=> ~ ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_738_inf_OboundedI,axiom,
! [A3: product_unit,B3: product_unit,C2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ B3 )
=> ( ( ord_le3221252021190050221t_unit @ A3 @ C2 )
=> ( ord_le3221252021190050221t_unit @ A3 @ ( inf_inf_Product_unit @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_739_inf_OboundedI,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ( ( ord_le3843937902494030498c_fm_i @ A3 @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ A3 @ ( inf_in3450601097109690352c_fm_i @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_740_inf_OboundedI,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_741_inf__greatest,axiom,
! [X4: product_unit,Y3: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X4 @ Y3 )
=> ( ( ord_le3221252021190050221t_unit @ X4 @ Z2 )
=> ( ord_le3221252021190050221t_unit @ X4 @ ( inf_inf_Product_unit @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_742_inf__greatest,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ Z2 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_743_inf__greatest,axiom,
! [X4: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ X4 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_744_inf_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A5: product_unit,B5: product_unit] :
( A5
= ( inf_inf_Product_unit @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_745_inf_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( A5
= ( inf_in3450601097109690352c_fm_i @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_746_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( A5
= ( inf_inf_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_747_inf_Ocobounded1,axiom,
! [A3: product_unit,B3: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_748_inf_Ocobounded1,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_749_inf_Ocobounded1,axiom,
! [A3: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_750_inf_Ocobounded2,axiom,
! [A3: product_unit,B3: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_751_inf_Ocobounded2,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_752_inf_Ocobounded2,axiom,
! [A3: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_753_inf_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A5: product_unit,B5: product_unit] :
( ( inf_inf_Product_unit @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_754_inf_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_755_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_756_inf_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B5: product_unit,A5: product_unit] :
( ( inf_inf_Product_unit @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_757_inf_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B5: set_Epistemic_fm_i,A5: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_758_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_759_inf_OcoboundedI1,axiom,
! [A3: product_unit,C2: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ C2 )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_760_inf_OcoboundedI1,axiom,
! [A3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_761_inf_OcoboundedI1,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_762_inf_OcoboundedI2,axiom,
! [B3: product_unit,C2: product_unit,A3: product_unit] :
( ( ord_le3221252021190050221t_unit @ B3 @ C2 )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_763_inf_OcoboundedI2,axiom,
! [B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B3 @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_764_inf_OcoboundedI2,axiom,
! [B3: nat,C2: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_765_mcs__conjunction__mult,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,S: list_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ S ) @ V )
& ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ S ) ) )
=> ( member6642669606046002379c_fm_i @ ( stalnaker_conjunct_i @ S ) @ V ) ) ) ) ).
% mcs_conjunction_mult
thf(fact_766_fm_Oexhaust,axiom,
! [Y3: epistemic_fm_i] :
( ( Y3 != epistemic_FF_i )
=> ( ! [X22: list_char] :
( Y3
!= ( epistemic_Pro_i @ X22 ) )
=> ( ! [X31: epistemic_fm_i,X32: epistemic_fm_i] :
( Y3
!= ( epistemic_Dis_i @ X31 @ X32 ) )
=> ( ! [X412: epistemic_fm_i,X422: epistemic_fm_i] :
( Y3
!= ( epistemic_Con_i @ X412 @ X422 ) )
=> ( ! [X512: epistemic_fm_i,X522: epistemic_fm_i] :
( Y3
!= ( epistemic_Imp_i @ X512 @ X522 ) )
=> ~ ! [X612: i,X622: epistemic_fm_i] :
( Y3
!= ( epistemic_K_i @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_767_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_768_list_Oset__intros_I2_J,axiom,
! [Y3: epistemic_fm_i,X222: list_Epistemic_fm_i,X21: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y3 @ ( set_Epistemic_fm_i2 @ X222 ) )
=> ( member6642669606046002379c_fm_i @ Y3 @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_769_list_Oset__intros_I1_J,axiom,
! [X21: epistemic_fm_i,X222: list_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X21 @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_770_list_Oset__cases,axiom,
! [E: epistemic_fm_i,A3: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ E @ ( set_Epistemic_fm_i2 @ A3 ) )
=> ( ! [Z22: list_Epistemic_fm_i] :
( A3
!= ( cons_Epistemic_fm_i @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_i,Z22: list_Epistemic_fm_i] :
( ( A3
= ( cons_Epistemic_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( set_Epistemic_fm_i2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_771_set__ConsD,axiom,
! [Y3: epistemic_fm_i,X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y3 @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X4 @ Xs ) ) )
=> ( ( Y3 = X4 )
| ( member6642669606046002379c_fm_i @ Y3 @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_772_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_773_fm_Odistinct_I23_J,axiom,
! [X312: epistemic_fm_i,X322: epistemic_fm_i,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Dis_i @ X312 @ X322 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(23)
thf(fact_774_fm_Odistinct_I11_J,axiom,
! [X2: list_char,X312: epistemic_fm_i,X322: epistemic_fm_i] :
( ( epistemic_Pro_i @ X2 )
!= ( epistemic_Dis_i @ X312 @ X322 ) ) ).
% fm.distinct(11)
thf(fact_775_set__subset__Cons,axiom,
! [Xs: list_Epistemic_fm_i,X4: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X4 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_776_inconsistent__finite,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ S )
=> ? [S4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S4 @ S )
& ( finite3304564979551393739c_fm_i @ S4 )
& ~ ( episte2285483198712856234tent_i @ A @ S4 ) ) ) ).
% inconsistent_finite
thf(fact_777_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_778_finite__Collect__subsets,axiom,
! [A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_779_finite__Collect__disjI,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( P2 @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) )
& ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_780_finite__Collect__disjI,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( P2 @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) )
& ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_781_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P2 @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_782_finite__Collect__conjI,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) )
| ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ Q ) ) )
=> ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_783_finite__Collect__conjI,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) )
| ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ Q ) ) )
=> ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_784_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_785_finite__has__maximal,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ? [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_786_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_787_finite__has__minimal,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ? [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_788_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_789_not__finite__existsD,axiom,
! [P2: set_Epistemic_fm_i > $o] :
( ~ ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) )
=> ? [X_1: set_Epistemic_fm_i] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_790_not__finite__existsD,axiom,
! [P2: epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) )
=> ? [X_1: epistemic_fm_i] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_791_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_792_pigeonhole__infinite__rel,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_nat,R3: set_Epistemic_fm_i > nat > $o] :
( ~ ( finite7933139204641697195c_fm_i @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R3 @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [A5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A5 @ A )
& ( R3 @ A5 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_793_pigeonhole__infinite__rel,axiom,
! [A: set_Epistemic_fm_i,B: set_nat,R3: epistemic_fm_i > nat > $o] :
( ~ ( finite3304564979551393739c_fm_i @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R3 @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [A5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A5 @ A )
& ( R3 @ A5 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_794_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R3: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R3 @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R3 @ A5 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_795_finite__has__maximal2,axiom,
! [A: set_se3485332733965609186c_fm_i,A3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A3 @ A )
=> ? [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( ord_le3843937902494030498c_fm_i @ A3 @ X )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_796_finite__has__maximal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ A3 @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_797_finite__has__minimal2,axiom,
! [A: set_se3485332733965609186c_fm_i,A3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A3 @ A )
=> ? [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( ord_le3843937902494030498c_fm_i @ X @ A3 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_798_finite__has__minimal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ X @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_799_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_800_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_801_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_802_finite__subset,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% finite_subset
thf(fact_803_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_804_infinite__super,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S @ T2 )
=> ( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ T2 ) ) ) ).
% infinite_super
thf(fact_805_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_806_rev__finite__subset,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% rev_finite_subset
thf(fact_807_fm_Orel__induct,axiom,
! [R3: i > i > $o,X4: epistemic_fm_i,Y3: epistemic_fm_i,Q: epistemic_fm_i > epistemic_fm_i > $o] :
( ( epistemic_rel_fm_i_i @ R3 @ X4 @ Y3 )
=> ( ( Q @ epistemic_FF_i @ epistemic_FF_i )
=> ( ! [A22: list_char,B22: list_char] :
( ( A22 = B22 )
=> ( Q @ ( epistemic_Pro_i @ A22 ) @ ( epistemic_Pro_i @ B22 ) ) )
=> ( ! [A31: epistemic_fm_i,A32: epistemic_fm_i,B31: epistemic_fm_i,B32: epistemic_fm_i] :
( ( Q @ A31 @ B31 )
=> ( ( Q @ A32 @ B32 )
=> ( Q @ ( epistemic_Dis_i @ A31 @ A32 ) @ ( epistemic_Dis_i @ B31 @ B32 ) ) ) )
=> ( ! [A41: epistemic_fm_i,A42: epistemic_fm_i,B41: epistemic_fm_i,B42: epistemic_fm_i] :
( ( Q @ A41 @ B41 )
=> ( ( Q @ A42 @ B42 )
=> ( Q @ ( epistemic_Con_i @ A41 @ A42 ) @ ( epistemic_Con_i @ B41 @ B42 ) ) ) )
=> ( ! [A51: epistemic_fm_i,A52: epistemic_fm_i,B51: epistemic_fm_i,B52: epistemic_fm_i] :
( ( Q @ A51 @ B51 )
=> ( ( Q @ A52 @ B52 )
=> ( Q @ ( epistemic_Imp_i @ A51 @ A52 ) @ ( epistemic_Imp_i @ B51 @ B52 ) ) ) )
=> ( ! [A61: i,A62: epistemic_fm_i,B61: i,B62: epistemic_fm_i] :
( ( R3 @ A61 @ B61 )
=> ( ( Q @ A62 @ B62 )
=> ( Q @ ( epistemic_K_i @ A61 @ A62 ) @ ( epistemic_K_i @ B61 @ B62 ) ) ) )
=> ( Q @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_808_fm_Orel__cases,axiom,
! [R3: i > i > $o,A3: epistemic_fm_i,B3: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R3 @ A3 @ B3 )
=> ( ( ( A3 = epistemic_FF_i )
=> ( B3 != epistemic_FF_i ) )
=> ( ! [X: list_char] :
( ( A3
= ( epistemic_Pro_i @ X ) )
=> ! [Y5: list_char] :
( ( B3
= ( epistemic_Pro_i @ Y5 ) )
=> ( X != Y5 ) ) )
=> ( ! [X1: epistemic_fm_i,X2a: epistemic_fm_i] :
( ( A3
= ( epistemic_Dis_i @ X1 @ X2a ) )
=> ! [Y1: epistemic_fm_i,Y2a: epistemic_fm_i] :
( ( B3
= ( epistemic_Dis_i @ Y1 @ Y2a ) )
=> ( ( epistemic_rel_fm_i_i @ R3 @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_i_i @ R3 @ X2a @ Y2a ) ) ) )
=> ( ! [X1a: epistemic_fm_i,X2b: epistemic_fm_i] :
( ( A3
= ( epistemic_Con_i @ X1a @ X2b ) )
=> ! [Y1a: epistemic_fm_i,Y2b: epistemic_fm_i] :
( ( B3
= ( epistemic_Con_i @ Y1a @ Y2b ) )
=> ( ( epistemic_rel_fm_i_i @ R3 @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_i_i @ R3 @ X2b @ Y2b ) ) ) )
=> ( ! [X1b: epistemic_fm_i,X2c: epistemic_fm_i] :
( ( A3
= ( epistemic_Imp_i @ X1b @ X2c ) )
=> ! [Y1b: epistemic_fm_i,Y2c: epistemic_fm_i] :
( ( B3
= ( epistemic_Imp_i @ Y1b @ Y2c ) )
=> ( ( epistemic_rel_fm_i_i @ R3 @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_i_i @ R3 @ X2c @ Y2c ) ) ) )
=> ~ ! [X1c: i,X2d: epistemic_fm_i] :
( ( A3
= ( epistemic_K_i @ X1c @ X2d ) )
=> ! [Y1c: i,Y2d: epistemic_fm_i] :
( ( B3
= ( epistemic_K_i @ Y1c @ Y2d ) )
=> ( ( R3 @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_i_i @ R3 @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_809_not__in__set__insert,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( insert177310195918387262c_fm_i @ X4 @ Xs )
= ( cons_Epistemic_fm_i @ X4 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_810_inter__Set__filter,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= ( filter6053540608743173075c_fm_i
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ A )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_811_inter__Set__filter,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= ( filter_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_812_fm_Oset__cases,axiom,
! [E: epistemic_fm_i,A3: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ A3 ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i] :
( ? [Z22: episte1308613106011496894c_fm_i] :
( A3
= ( episte6088726798904853267c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i,Z22: episte1308613106011496894c_fm_i] :
( ( A3
= ( episte6088726798904853267c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z22 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i] :
( ? [Z22: episte1308613106011496894c_fm_i] :
( A3
= ( episte3685526521632971807c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i,Z22: episte1308613106011496894c_fm_i] :
( ( A3
= ( episte3685526521632971807c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z22 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i] :
( ? [Z22: episte1308613106011496894c_fm_i] :
( A3
= ( episte260752253203357973c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i,Z22: episte1308613106011496894c_fm_i] :
( ( A3
= ( episte260752253203357973c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z22 ) ) )
=> ( ! [Z22: episte1308613106011496894c_fm_i] :
( A3
!= ( episte5657488666450005526c_fm_i @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_i,Z22: episte1308613106011496894c_fm_i] :
( ( A3
= ( episte5657488666450005526c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_813_fm_Oset__cases,axiom,
! [E: i,A3: epistemic_fm_i] :
( ( member_i @ E @ ( epistemic_set_fm_i @ A3 ) )
=> ( ! [Z1: epistemic_fm_i] :
( ? [Z22: epistemic_fm_i] :
( A3
= ( epistemic_Dis_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_i,Z22: epistemic_fm_i] :
( ( A3
= ( epistemic_Dis_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_i] :
( ? [Z22: epistemic_fm_i] :
( A3
= ( epistemic_Con_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_i,Z22: epistemic_fm_i] :
( ( A3
= ( epistemic_Con_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_i] :
( ? [Z22: epistemic_fm_i] :
( A3
= ( epistemic_Imp_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_i,Z22: epistemic_fm_i] :
( ( A3
= ( epistemic_Imp_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z22 ) ) )
=> ( ! [Z22: epistemic_fm_i] :
( A3
!= ( epistemic_K_i @ E @ Z22 ) )
=> ~ ! [Z1: i,Z22: epistemic_fm_i] :
( ( A3
= ( epistemic_K_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_814_member__filter,axiom,
! [X4: epistemic_fm_i,P2: epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ ( filter6053540608743173075c_fm_i @ P2 @ A ) )
= ( ( member6642669606046002379c_fm_i @ X4 @ A )
& ( P2 @ X4 ) ) ) ).
% member_filter
thf(fact_815_in__set__insert,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( insert177310195918387262c_fm_i @ X4 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_816_fm_Orel__intros_I6_J,axiom,
! [R3: i > i > $o,X61: i,Y61: i,X62: epistemic_fm_i,Y62: epistemic_fm_i] :
( ( R3 @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_i_i @ R3 @ X62 @ Y62 )
=> ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_K_i @ X61 @ X62 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_817_fm_Orel__inject_I6_J,axiom,
! [R3: i > i > $o,X61: i,X62: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_K_i @ X61 @ X62 ) @ ( epistemic_K_i @ Y61 @ Y62 ) )
= ( ( R3 @ X61 @ Y61 )
& ( epistemic_rel_fm_i_i @ R3 @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_818_fm_Orel__cong,axiom,
! [X4: episte1308613106011496894c_fm_i,Ya: episte1308613106011496894c_fm_i,Y3: episte1308613106011496894c_fm_i,Xa3: episte1308613106011496894c_fm_i,R3: epistemic_fm_i > epistemic_fm_i > $o,Ra: epistemic_fm_i > epistemic_fm_i > $o] :
( ( X4 = Ya )
=> ( ( Y3 = Xa3 )
=> ( ! [Z5: epistemic_fm_i,Yb: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z5 @ ( episte9089240992906287960c_fm_i @ Ya ) )
=> ( ( member6642669606046002379c_fm_i @ Yb @ ( episte9089240992906287960c_fm_i @ Xa3 ) )
=> ( ( R3 @ Z5 @ Yb )
= ( Ra @ Z5 @ Yb ) ) ) )
=> ( ( episte3445413862081925600c_fm_i @ R3 @ X4 @ Y3 )
= ( episte3445413862081925600c_fm_i @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_819_fm_Orel__mono__strong,axiom,
! [R3: epistemic_fm_i > epistemic_fm_i > $o,X4: episte1308613106011496894c_fm_i,Y3: episte1308613106011496894c_fm_i,Ra: epistemic_fm_i > epistemic_fm_i > $o] :
( ( episte3445413862081925600c_fm_i @ R3 @ X4 @ Y3 )
=> ( ! [Z5: epistemic_fm_i,Yb: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z5 @ ( episte9089240992906287960c_fm_i @ X4 ) )
=> ( ( member6642669606046002379c_fm_i @ Yb @ ( episte9089240992906287960c_fm_i @ Y3 ) )
=> ( ( R3 @ Z5 @ Yb )
=> ( Ra @ Z5 @ Yb ) ) ) )
=> ( episte3445413862081925600c_fm_i @ Ra @ X4 @ Y3 ) ) ) ).
% fm.rel_mono_strong
thf(fact_820_fm_Orel__refl__strong,axiom,
! [X4: episte1308613106011496894c_fm_i,Ra: epistemic_fm_i > epistemic_fm_i > $o] :
( ! [Z5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z5 @ ( episte9089240992906287960c_fm_i @ X4 ) )
=> ( Ra @ Z5 @ Z5 ) )
=> ( episte3445413862081925600c_fm_i @ Ra @ X4 @ X4 ) ) ).
% fm.rel_refl_strong
thf(fact_821_fm_Orel__inject_I2_J,axiom,
! [R3: i > i > $o,X2: list_char,Y2: list_char] :
( ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Pro_i @ X2 ) @ ( epistemic_Pro_i @ Y2 ) )
= ( X2 = Y2 ) ) ).
% fm.rel_inject(2)
thf(fact_822_fm_Orel__intros_I2_J,axiom,
! [X2: list_char,Y2: list_char,R3: i > i > $o] :
( ( X2 = Y2 )
=> ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Pro_i @ X2 ) @ ( epistemic_Pro_i @ Y2 ) ) ) ).
% fm.rel_intros(2)
thf(fact_823_Set_Ofilter__def,axiom,
( filter8450032351001387443c_fm_i
= ( ^ [P6: set_Epistemic_fm_i > $o,A2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [A5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A5 @ A2 )
& ( P6 @ A5 ) ) ) ) ) ).
% Set.filter_def
thf(fact_824_Set_Ofilter__def,axiom,
( filter6053540608743173075c_fm_i
= ( ^ [P6: epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [A5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A5 @ A2 )
& ( P6 @ A5 ) ) ) ) ) ).
% Set.filter_def
thf(fact_825_Set_Ofilter__def,axiom,
( filter_nat
= ( ^ [P6: nat > $o,A2: set_nat] :
( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( P6 @ A5 ) ) ) ) ) ).
% Set.filter_def
thf(fact_826_fm_Oset__intros_I6_J,axiom,
! [Yf: epistemic_fm_i,X52: episte1308613106011496894c_fm_i,X51: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Yf @ ( episte9089240992906287960c_fm_i @ X52 ) )
=> ( member6642669606046002379c_fm_i @ Yf @ ( episte9089240992906287960c_fm_i @ ( episte260752253203357973c_fm_i @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_827_fm_Oset__intros_I5_J,axiom,
! [Ye: epistemic_fm_i,X51: episte1308613106011496894c_fm_i,X52: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Ye @ ( episte9089240992906287960c_fm_i @ X51 ) )
=> ( member6642669606046002379c_fm_i @ Ye @ ( episte9089240992906287960c_fm_i @ ( episte260752253203357973c_fm_i @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_828_fm_Oset__intros_I8_J,axiom,
! [Yg: epistemic_fm_i,X62: episte1308613106011496894c_fm_i,X61: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Yg @ ( episte9089240992906287960c_fm_i @ X62 ) )
=> ( member6642669606046002379c_fm_i @ Yg @ ( episte9089240992906287960c_fm_i @ ( episte5657488666450005526c_fm_i @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_829_fm_Oset__intros_I8_J,axiom,
! [Yg: i,X62: epistemic_fm_i,X61: i] :
( ( member_i @ Yg @ ( epistemic_set_fm_i @ X62 ) )
=> ( member_i @ Yg @ ( epistemic_set_fm_i @ ( epistemic_K_i @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_830_fm_Oset__intros_I7_J,axiom,
! [X61: epistemic_fm_i,X62: episte1308613106011496894c_fm_i] : ( member6642669606046002379c_fm_i @ X61 @ ( episte9089240992906287960c_fm_i @ ( episte5657488666450005526c_fm_i @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_831_fm_Oset__intros_I7_J,axiom,
! [X61: i,X62: epistemic_fm_i] : ( member_i @ X61 @ ( epistemic_set_fm_i @ ( epistemic_K_i @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_832_fm_Oset__intros_I4_J,axiom,
! [Yd: epistemic_fm_i,X42: episte1308613106011496894c_fm_i,X41: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Yd @ ( episte9089240992906287960c_fm_i @ X42 ) )
=> ( member6642669606046002379c_fm_i @ Yd @ ( episte9089240992906287960c_fm_i @ ( episte3685526521632971807c_fm_i @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_833_fm_Oset__intros_I3_J,axiom,
! [Yc: epistemic_fm_i,X41: episte1308613106011496894c_fm_i,X42: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Yc @ ( episte9089240992906287960c_fm_i @ X41 ) )
=> ( member6642669606046002379c_fm_i @ Yc @ ( episte9089240992906287960c_fm_i @ ( episte3685526521632971807c_fm_i @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_834_fm_Oset__intros_I2_J,axiom,
! [Yb2: epistemic_fm_i,X322: episte1308613106011496894c_fm_i,X312: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Yb2 @ ( episte9089240992906287960c_fm_i @ X322 ) )
=> ( member6642669606046002379c_fm_i @ Yb2 @ ( episte9089240992906287960c_fm_i @ ( episte6088726798904853267c_fm_i @ X312 @ X322 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_835_fm_Oset__intros_I1_J,axiom,
! [Ya: epistemic_fm_i,X312: episte1308613106011496894c_fm_i,X322: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Ya @ ( episte9089240992906287960c_fm_i @ X312 ) )
=> ( member6642669606046002379c_fm_i @ Ya @ ( episte9089240992906287960c_fm_i @ ( episte6088726798904853267c_fm_i @ X312 @ X322 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_836_fm_Orel__distinct_I18_J,axiom,
! [R3: i > i > $o,Y61: i,Y62: epistemic_fm_i,X2: list_char] :
~ ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_K_i @ Y61 @ Y62 ) @ ( epistemic_Pro_i @ X2 ) ) ).
% fm.rel_distinct(18)
thf(fact_837_fm_Orel__distinct_I17_J,axiom,
! [R3: i > i > $o,X2: list_char,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Pro_i @ X2 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(17)
thf(fact_838_fm_Osimps_I91_J,axiom,
! [X2: list_char] :
( ( epistemic_set_fm_i @ ( epistemic_Pro_i @ X2 ) )
= bot_bot_set_i ) ).
% fm.simps(91)
thf(fact_839_List_Oinsert__def,axiom,
( insert177310195918387262c_fm_i
= ( ^ [X3: epistemic_fm_i,Xs3: list_Epistemic_fm_i] : ( if_lis3446954105117697518c_fm_i @ ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs3 ) ) @ Xs3 @ ( cons_Epistemic_fm_i @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_840_inconsistent__subset,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P3: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P3 @ bot_bo4194595901900360558c_fm_i ) @ V ) )
=> ~ ! [V5: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ V5 ) @ V )
=> ~ ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P3 @ V5 ) @ epistemic_FF_i ) ) ) ) ) ).
% inconsistent_subset
thf(fact_841_K__conjunction__in__mult,axiom,
! [A: epistemic_fm_i > $o,I2: i,G4: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( stalnaker_conjunct_i @ G4 ) ) @ ( stalnaker_conjunct_i @ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ I2 ) @ G4 ) ) ) ) ).
% K_conjunction_in_mult
thf(fact_842_K__conjunction__out__mult,axiom,
! [A: epistemic_fm_i > $o,I2: i,G4: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ I2 ) @ G4 ) ) @ ( epistemic_K_i @ I2 @ ( stalnaker_conjunct_i @ G4 ) ) ) ) ).
% K_conjunction_out_mult
thf(fact_843_insert__iff,axiom,
! [A3: epistemic_fm_i,B3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ ( insert7817948997695205106c_fm_i @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member6642669606046002379c_fm_i @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_844_insertCI,axiom,
! [A3: epistemic_fm_i,B: set_Epistemic_fm_i,B3: epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ A3 @ B )
=> ( A3 = B3 ) )
=> ( member6642669606046002379c_fm_i @ A3 @ ( insert7817948997695205106c_fm_i @ B3 @ B ) ) ) ).
% insertCI
thf(fact_845_Un__iff,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
= ( ( member6642669606046002379c_fm_i @ C2 @ A )
| ( member6642669606046002379c_fm_i @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_846_UnCI,axiom,
! [C2: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ C2 @ B )
=> ( member6642669606046002379c_fm_i @ C2 @ A ) )
=> ( member6642669606046002379c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% UnCI
thf(fact_847_le__sup__iff,axiom,
! [X4: product_unit,Y3: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X4 @ Y3 ) @ Z2 )
= ( ( ord_le3221252021190050221t_unit @ X4 @ Z2 )
& ( ord_le3221252021190050221t_unit @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_848_le__sup__iff,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y3 ) @ Z2 )
= ( ( ord_le3843937902494030498c_fm_i @ X4 @ Z2 )
& ( ord_le3843937902494030498c_fm_i @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_849_le__sup__iff,axiom,
! [X4: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ Y3 ) @ Z2 )
= ( ( ord_less_eq_nat @ X4 @ Z2 )
& ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_850_sup_Obounded__iff,axiom,
! [B3: product_unit,C2: product_unit,A3: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B3 @ C2 ) @ A3 )
= ( ( ord_le3221252021190050221t_unit @ B3 @ A3 )
& ( ord_le3221252021190050221t_unit @ C2 @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_851_sup_Obounded__iff,axiom,
! [B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B3 @ C2 ) @ A3 )
= ( ( ord_le3843937902494030498c_fm_i @ B3 @ A3 )
& ( ord_le3843937902494030498c_fm_i @ C2 @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_852_sup_Obounded__iff,axiom,
! [B3: nat,C2: nat,A3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C2 ) @ A3 )
= ( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_853_sup__bot__left,axiom,
! [X4: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_854_sup__bot__right,axiom,
! [X4: product_unit] :
( ( sup_sup_Product_unit @ X4 @ bot_bot_Product_unit )
= X4 ) ).
% sup_bot_right
thf(fact_855_bot__eq__sup__iff,axiom,
! [X4: product_unit,Y3: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ X4 @ Y3 ) )
= ( ( X4 = bot_bot_Product_unit )
& ( Y3 = bot_bot_Product_unit ) ) ) ).
% bot_eq_sup_iff
thf(fact_856_sup__eq__bot__iff,axiom,
! [X4: product_unit,Y3: product_unit] :
( ( ( sup_sup_Product_unit @ X4 @ Y3 )
= bot_bot_Product_unit )
= ( ( X4 = bot_bot_Product_unit )
& ( Y3 = bot_bot_Product_unit ) ) ) ).
% sup_eq_bot_iff
thf(fact_857_sup__bot_Oeq__neutr__iff,axiom,
! [A3: product_unit,B3: product_unit] :
( ( ( sup_sup_Product_unit @ A3 @ B3 )
= bot_bot_Product_unit )
= ( ( A3 = bot_bot_Product_unit )
& ( B3 = bot_bot_Product_unit ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_858_sup__bot_Oleft__neutral,axiom,
! [A3: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_859_sup__bot_Oneutr__eq__iff,axiom,
! [A3: product_unit,B3: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ A3 @ B3 ) )
= ( ( A3 = bot_bot_Product_unit )
& ( B3 = bot_bot_Product_unit ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_860_sup__bot_Oright__neutral,axiom,
! [A3: product_unit] :
( ( sup_sup_Product_unit @ A3 @ bot_bot_Product_unit )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_861_singletonI,axiom,
! [A3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A3 @ ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singletonI
thf(fact_862_insert__subset,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 @ A ) @ B )
= ( ( member6642669606046002379c_fm_i @ X4 @ B )
& ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ) ).
% insert_subset
thf(fact_863_Un__subset__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C )
= ( ( ord_le3843937902494030498c_fm_i @ A @ C )
& ( ord_le3843937902494030498c_fm_i @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_864_Int__insert__right__if1,axiom,
! [A3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A3 @ B ) )
= ( insert7817948997695205106c_fm_i @ A3 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_865_Int__insert__right__if0,axiom,
! [A3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A3 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A3 @ B ) )
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_866_Int__insert__left__if1,axiom,
! [A3: epistemic_fm_i,C: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ C )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A3 @ B ) @ C )
= ( insert7817948997695205106c_fm_i @ A3 @ ( inf_in3450601097109690352c_fm_i @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_867_Int__insert__left__if0,axiom,
! [A3: epistemic_fm_i,C: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A3 @ C )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A3 @ B ) @ C )
= ( inf_in3450601097109690352c_fm_i @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_868_singleton__conv2,axiom,
! [A3: set_Epistemic_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z )
@ A3 ) )
= ( insert7698009978809854162c_fm_i @ A3 @ bot_bo145720340923748686c_fm_i ) ) ).
% singleton_conv2
thf(fact_869_singleton__conv2,axiom,
! [A3: epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ( ^ [Y: epistemic_fm_i,Z: epistemic_fm_i] : ( Y = Z )
@ A3 ) )
= ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singleton_conv2
thf(fact_870_singleton__conv2,axiom,
! [A3: nat] :
( ( collect_nat
@ ( ^ [Y: nat,Z: nat] : ( Y = Z )
@ A3 ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_871_singleton__conv,axiom,
! [A3: set_Epistemic_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] : ( X3 = A3 ) )
= ( insert7698009978809854162c_fm_i @ A3 @ bot_bo145720340923748686c_fm_i ) ) ).
% singleton_conv
thf(fact_872_singleton__conv,axiom,
! [A3: epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] : ( X3 = A3 ) )
= ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singleton_conv
thf(fact_873_singleton__conv,axiom,
! [A3: nat] :
( ( collect_nat
@ ^ [X3: nat] : ( X3 = A3 ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_874_singleton__insert__inj__eq,axiom,
! [B3: epistemic_fm_i,A3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ B3 @ bot_bo4194595901900360558c_fm_i )
= ( insert7817948997695205106c_fm_i @ A3 @ A ) )
= ( ( A3 = B3 )
& ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_875_singleton__insert__inj__eq_H,axiom,
! [A3: epistemic_fm_i,A: set_Epistemic_fm_i,B3: epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ A3 @ A )
= ( insert7817948997695205106c_fm_i @ B3 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( A3 = B3 )
& ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_876_insert__disjoint_I1_J,axiom,
! [A3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A3 @ A ) @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ~ ( member6642669606046002379c_fm_i @ A3 @ B )
& ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% insert_disjoint(1)
thf(fact_877_insert__disjoint_I2_J,axiom,
! [A3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A3 @ A ) @ B ) )
= ( ~ ( member6642669606046002379c_fm_i @ A3 @ B )
& ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_878_disjoint__insert_I1_J,axiom,
! [B: set_Epistemic_fm_i,A3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A3 @ A ) )
= bot_bo4194595901900360558c_fm_i )
= ( ~ ( member6642669606046002379c_fm_i @ A3 @ B )
& ( ( inf_in3450601097109690352c_fm_i @ B @ A )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% disjoint_insert(1)
thf(fact_879_disjoint__insert_I2_J,axiom,
! [A: set_Epistemic_fm_i,B3: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B3 @ B ) ) )
= ( ~ ( member6642669606046002379c_fm_i @ B3 @ A )
& ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_880_fm_Osimps_I95_J,axiom,
! [X61: i,X62: epistemic_fm_i] :
( ( epistemic_set_fm_i @ ( epistemic_K_i @ X61 @ X62 ) )
= ( insert_i @ X61 @ ( epistemic_set_fm_i @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_881_boolean__algebra_Odisj__zero__right,axiom,
! [X4: product_unit] :
( ( sup_sup_Product_unit @ X4 @ bot_bot_Product_unit )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_882_Collect__disj__eq,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( P2 @ X3 )
| ( Q @ X3 ) ) )
= ( sup_su2582925890723967158c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_883_Collect__disj__eq,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( P2 @ X3 )
| ( Q @ X3 ) ) )
= ( sup_su1936195050962291414c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_884_Collect__disj__eq,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P2 @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_885_Un__def,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A2 )
| ( member1104366573291651755c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_886_Un__def,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A2 )
| ( member6642669606046002379c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_887_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
| ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_888_insert__Collect,axiom,
! [A3: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( insert7698009978809854162c_fm_i @ A3 @ ( collec3087743281813070829c_fm_i @ P2 ) )
= ( collec3087743281813070829c_fm_i
@ ^ [U2: set_Epistemic_fm_i] :
( ( U2 != A3 )
=> ( P2 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_889_insert__Collect,axiom,
! [A3: epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( insert7817948997695205106c_fm_i @ A3 @ ( collec4904205187116291597c_fm_i @ P2 ) )
= ( collec4904205187116291597c_fm_i
@ ^ [U2: epistemic_fm_i] :
( ( U2 != A3 )
=> ( P2 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_890_insert__Collect,axiom,
! [A3: nat,P2: nat > $o] :
( ( insert_nat @ A3 @ ( collect_nat @ P2 ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A3 )
=> ( P2 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_891_insert__compr,axiom,
( insert7698009978809854162c_fm_i
= ( ^ [A5: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( X3 = A5 )
| ( member1104366573291651755c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_892_insert__compr,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A5: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( X3 = A5 )
| ( member6642669606046002379c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_893_insert__compr,axiom,
( insert_nat
= ( ^ [A5: nat,B2: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( X3 = A5 )
| ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_894_mk__disjoint__insert,axiom,
! [A3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ A )
=> ? [B6: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ A3 @ B6 ) )
& ~ ( member6642669606046002379c_fm_i @ A3 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_895_insert__eq__iff,axiom,
! [A3: epistemic_fm_i,A: set_Epistemic_fm_i,B3: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A3 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ B3 @ B )
=> ( ( ( insert7817948997695205106c_fm_i @ A3 @ A )
= ( insert7817948997695205106c_fm_i @ B3 @ B ) )
= ( ( ( A3 = B3 )
=> ( A = B ) )
& ( ( A3 != B3 )
=> ? [C3: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ B3 @ C3 ) )
& ~ ( member6642669606046002379c_fm_i @ B3 @ C3 )
& ( B
= ( insert7817948997695205106c_fm_i @ A3 @ C3 ) )
& ~ ( member6642669606046002379c_fm_i @ A3 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_896_insert__absorb,axiom,
! [A3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ A )
=> ( ( insert7817948997695205106c_fm_i @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_897_insert__ident,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ X4 @ B )
=> ( ( ( insert7817948997695205106c_fm_i @ X4 @ A )
= ( insert7817948997695205106c_fm_i @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_898_Set_Oset__insert,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ~ ! [B6: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ X4 @ B6 ) )
=> ( member6642669606046002379c_fm_i @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_899_insert__def,axiom,
( insert7698009978809854162c_fm_i
= ( ^ [A5: set_Epistemic_fm_i] :
( sup_su2582925890723967158c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] : ( X3 = A5 ) ) ) ) ) ).
% insert_def
thf(fact_900_insert__def,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A5: epistemic_fm_i] :
( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] : ( X3 = A5 ) ) ) ) ) ).
% insert_def
thf(fact_901_insert__def,axiom,
( insert_nat
= ( ^ [A5: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X3: nat] : ( X3 = A5 ) ) ) ) ) ).
% insert_def
thf(fact_902_insertI2,axiom,
! [A3: epistemic_fm_i,B: set_Epistemic_fm_i,B3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ B )
=> ( member6642669606046002379c_fm_i @ A3 @ ( insert7817948997695205106c_fm_i @ B3 @ B ) ) ) ).
% insertI2
thf(fact_903_insertI1,axiom,
! [A3: epistemic_fm_i,B: set_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A3 @ ( insert7817948997695205106c_fm_i @ A3 @ B ) ) ).
% insertI1
thf(fact_904_insertE,axiom,
! [A3: epistemic_fm_i,B3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ ( insert7817948997695205106c_fm_i @ B3 @ A ) )
=> ( ( A3 != B3 )
=> ( member6642669606046002379c_fm_i @ A3 @ A ) ) ) ).
% insertE
thf(fact_905_UnI2,axiom,
! [C2: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ B )
=> ( member6642669606046002379c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% UnI2
thf(fact_906_UnI1,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ A )
=> ( member6642669606046002379c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% UnI1
thf(fact_907_UnE,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
=> ( ~ ( member6642669606046002379c_fm_i @ C2 @ A )
=> ( member6642669606046002379c_fm_i @ C2 @ B ) ) ) ).
% UnE
thf(fact_908_sup_OcoboundedI2,axiom,
! [C2: product_unit,B3: product_unit,A3: product_unit] :
( ( ord_le3221252021190050221t_unit @ C2 @ B3 )
=> ( ord_le3221252021190050221t_unit @ C2 @ ( sup_sup_Product_unit @ A3 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_909_sup_OcoboundedI2,axiom,
! [C2: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ B3 )
=> ( ord_le3843937902494030498c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_910_sup_OcoboundedI2,axiom,
! [C2: nat,B3: nat,A3: nat] :
( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_911_sup_OcoboundedI1,axiom,
! [C2: product_unit,A3: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ C2 @ A3 )
=> ( ord_le3221252021190050221t_unit @ C2 @ ( sup_sup_Product_unit @ A3 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_912_sup_OcoboundedI1,axiom,
! [C2: set_Epistemic_fm_i,A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ A3 )
=> ( ord_le3843937902494030498c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_913_sup_OcoboundedI1,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_914_sup_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A5: product_unit,B5: product_unit] :
( ( sup_sup_Product_unit @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_915_sup_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_916_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_917_sup_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B5: product_unit,A5: product_unit] :
( ( sup_sup_Product_unit @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_918_sup_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B5: set_Epistemic_fm_i,A5: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_919_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_920_sup_Ocobounded2,axiom,
! [B3: product_unit,A3: product_unit] : ( ord_le3221252021190050221t_unit @ B3 @ ( sup_sup_Product_unit @ A3 @ B3 ) ) ).
% sup.cobounded2
thf(fact_921_sup_Ocobounded2,axiom,
! [B3: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B3 @ ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) ) ).
% sup.cobounded2
thf(fact_922_sup_Ocobounded2,axiom,
! [B3: nat,A3: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A3 @ B3 ) ) ).
% sup.cobounded2
thf(fact_923_sup_Ocobounded1,axiom,
! [A3: product_unit,B3: product_unit] : ( ord_le3221252021190050221t_unit @ A3 @ ( sup_sup_Product_unit @ A3 @ B3 ) ) ).
% sup.cobounded1
thf(fact_924_sup_Ocobounded1,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A3 @ ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) ) ).
% sup.cobounded1
thf(fact_925_sup_Ocobounded1,axiom,
! [A3: nat,B3: nat] : ( ord_less_eq_nat @ A3 @ ( sup_sup_nat @ A3 @ B3 ) ) ).
% sup.cobounded1
thf(fact_926_sup_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B5: product_unit,A5: product_unit] :
( A5
= ( sup_sup_Product_unit @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_927_sup_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B5: set_Epistemic_fm_i,A5: set_Epistemic_fm_i] :
( A5
= ( sup_su1936195050962291414c_fm_i @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_928_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_929_sup_OboundedI,axiom,
! [B3: product_unit,A3: product_unit,C2: product_unit] :
( ( ord_le3221252021190050221t_unit @ B3 @ A3 )
=> ( ( ord_le3221252021190050221t_unit @ C2 @ A3 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B3 @ C2 ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_930_sup_OboundedI,axiom,
! [B3: set_Epistemic_fm_i,A3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B3 @ A3 )
=> ( ( ord_le3843937902494030498c_fm_i @ C2 @ A3 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B3 @ C2 ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_931_sup_OboundedI,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C2 ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_932_sup_OboundedE,axiom,
! [B3: product_unit,C2: product_unit,A3: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B3 @ C2 ) @ A3 )
=> ~ ( ( ord_le3221252021190050221t_unit @ B3 @ A3 )
=> ~ ( ord_le3221252021190050221t_unit @ C2 @ A3 ) ) ) ).
% sup.boundedE
thf(fact_933_sup_OboundedE,axiom,
! [B3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B3 @ C2 ) @ A3 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ B3 @ A3 )
=> ~ ( ord_le3843937902494030498c_fm_i @ C2 @ A3 ) ) ) ).
% sup.boundedE
thf(fact_934_sup_OboundedE,axiom,
! [B3: nat,C2: nat,A3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq_nat @ B3 @ A3 )
=> ~ ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% sup.boundedE
thf(fact_935_sup__absorb2,axiom,
! [X4: product_unit,Y3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X4 @ Y3 )
=> ( ( sup_sup_Product_unit @ X4 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_936_sup__absorb2,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
=> ( ( sup_su1936195050962291414c_fm_i @ X4 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_937_sup__absorb2,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( sup_sup_nat @ X4 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_938_sup__absorb1,axiom,
! [Y3: product_unit,X4: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y3 @ X4 )
=> ( ( sup_sup_Product_unit @ X4 @ Y3 )
= X4 ) ) ).
% sup_absorb1
thf(fact_939_sup__absorb1,axiom,
! [Y3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y3 @ X4 )
=> ( ( sup_su1936195050962291414c_fm_i @ X4 @ Y3 )
= X4 ) ) ).
% sup_absorb1
thf(fact_940_sup__absorb1,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( sup_sup_nat @ X4 @ Y3 )
= X4 ) ) ).
% sup_absorb1
thf(fact_941_sup_Oabsorb2,axiom,
! [A3: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ B3 )
=> ( ( sup_sup_Product_unit @ A3 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_942_sup_Oabsorb2,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
=> ( ( sup_su1936195050962291414c_fm_i @ A3 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_943_sup_Oabsorb2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_944_sup_Oabsorb1,axiom,
! [B3: product_unit,A3: product_unit] :
( ( ord_le3221252021190050221t_unit @ B3 @ A3 )
=> ( ( sup_sup_Product_unit @ A3 @ B3 )
= A3 ) ) ).
% sup.absorb1
thf(fact_945_sup_Oabsorb1,axiom,
! [B3: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B3 @ A3 )
=> ( ( sup_su1936195050962291414c_fm_i @ A3 @ B3 )
= A3 ) ) ).
% sup.absorb1
thf(fact_946_sup_Oabsorb1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ).
% sup.absorb1
thf(fact_947_sup__unique,axiom,
! [F: product_unit > product_unit > product_unit,X4: product_unit,Y3: product_unit] :
( ! [X: product_unit,Y5: product_unit] : ( ord_le3221252021190050221t_unit @ X @ ( F @ X @ Y5 ) )
=> ( ! [X: product_unit,Y5: product_unit] : ( ord_le3221252021190050221t_unit @ Y5 @ ( F @ X @ Y5 ) )
=> ( ! [X: product_unit,Y5: product_unit,Z5: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y5 @ X )
=> ( ( ord_le3221252021190050221t_unit @ Z5 @ X )
=> ( ord_le3221252021190050221t_unit @ ( F @ Y5 @ Z5 ) @ X ) ) )
=> ( ( sup_sup_Product_unit @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_948_sup__unique,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X @ ( F @ X @ Y5 ) )
=> ( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y5 @ ( F @ X @ Y5 ) )
=> ( ! [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i,Z5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y5 @ X )
=> ( ( ord_le3843937902494030498c_fm_i @ Z5 @ X )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ Y5 @ Z5 ) @ X ) ) )
=> ( ( sup_su1936195050962291414c_fm_i @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_949_sup__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y3: nat] :
( ! [X: nat,Y5: nat] : ( ord_less_eq_nat @ X @ ( F @ X @ Y5 ) )
=> ( ! [X: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ ( F @ X @ Y5 ) )
=> ( ! [X: nat,Y5: nat,Z5: nat] :
( ( ord_less_eq_nat @ Y5 @ X )
=> ( ( ord_less_eq_nat @ Z5 @ X )
=> ( ord_less_eq_nat @ ( F @ Y5 @ Z5 ) @ X ) ) )
=> ( ( sup_sup_nat @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_950_sup_OorderI,axiom,
! [A3: product_unit,B3: product_unit] :
( ( A3
= ( sup_sup_Product_unit @ A3 @ B3 ) )
=> ( ord_le3221252021190050221t_unit @ B3 @ A3 ) ) ).
% sup.orderI
thf(fact_951_sup_OorderI,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( A3
= ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) )
=> ( ord_le3843937902494030498c_fm_i @ B3 @ A3 ) ) ).
% sup.orderI
thf(fact_952_sup_OorderI,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B3 ) )
=> ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% sup.orderI
thf(fact_953_sup_OorderE,axiom,
! [B3: product_unit,A3: product_unit] :
( ( ord_le3221252021190050221t_unit @ B3 @ A3 )
=> ( A3
= ( sup_sup_Product_unit @ A3 @ B3 ) ) ) ).
% sup.orderE
thf(fact_954_sup_OorderE,axiom,
! [B3: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B3 @ A3 )
=> ( A3
= ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) ) ) ).
% sup.orderE
thf(fact_955_sup_OorderE,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% sup.orderE
thf(fact_956_le__iff__sup,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X3: product_unit,Y4: product_unit] :
( ( sup_sup_Product_unit @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_957_le__iff__sup,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X3: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_958_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( sup_sup_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_959_sup__least,axiom,
! [Y3: product_unit,X4: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y3 @ X4 )
=> ( ( ord_le3221252021190050221t_unit @ Z2 @ X4 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ Y3 @ Z2 ) @ X4 ) ) ) ).
% sup_least
thf(fact_960_sup__least,axiom,
! [Y3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y3 @ X4 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z2 @ X4 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y3 @ Z2 ) @ X4 ) ) ) ).
% sup_least
thf(fact_961_sup__least,axiom,
! [Y3: nat,X4: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_nat @ Z2 @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z2 ) @ X4 ) ) ) ).
% sup_least
thf(fact_962_sup__mono,axiom,
! [A3: product_unit,C2: product_unit,B3: product_unit,D2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ C2 )
=> ( ( ord_le3221252021190050221t_unit @ B3 @ D2 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A3 @ B3 ) @ ( sup_sup_Product_unit @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_963_sup__mono,axiom,
! [A3: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B3 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_964_sup__mono,axiom,
! [A3: nat,C2: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B3 ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_965_sup_Omono,axiom,
! [C2: product_unit,A3: product_unit,D2: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ C2 @ A3 )
=> ( ( ord_le3221252021190050221t_unit @ D2 @ B3 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ C2 @ D2 ) @ ( sup_sup_Product_unit @ A3 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_966_sup_Omono,axiom,
! [C2: set_Epistemic_fm_i,A3: set_Epistemic_fm_i,D2: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ A3 )
=> ( ( ord_le3843937902494030498c_fm_i @ D2 @ B3 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ C2 @ D2 ) @ ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_967_sup_Omono,axiom,
! [C2: nat,A3: nat,D2: nat,B3: nat] :
( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ( ord_less_eq_nat @ D2 @ B3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_968_le__supI2,axiom,
! [X4: product_unit,B3: product_unit,A3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X4 @ B3 )
=> ( ord_le3221252021190050221t_unit @ X4 @ ( sup_sup_Product_unit @ A3 @ B3 ) ) ) ).
% le_supI2
thf(fact_969_le__supI2,axiom,
! [X4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ B3 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) ) ) ).
% le_supI2
thf(fact_970_le__supI2,axiom,
! [X4: nat,B3: nat,A3: nat] :
( ( ord_less_eq_nat @ X4 @ B3 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% le_supI2
thf(fact_971_le__supI1,axiom,
! [X4: product_unit,A3: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X4 @ A3 )
=> ( ord_le3221252021190050221t_unit @ X4 @ ( sup_sup_Product_unit @ A3 @ B3 ) ) ) ).
% le_supI1
thf(fact_972_le__supI1,axiom,
! [X4: set_Epistemic_fm_i,A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ A3 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) ) ) ).
% le_supI1
thf(fact_973_le__supI1,axiom,
! [X4: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ X4 @ A3 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% le_supI1
thf(fact_974_sup__ge2,axiom,
! [Y3: product_unit,X4: product_unit] : ( ord_le3221252021190050221t_unit @ Y3 @ ( sup_sup_Product_unit @ X4 @ Y3 ) ) ).
% sup_ge2
thf(fact_975_sup__ge2,axiom,
! [Y3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y3 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y3 ) ) ).
% sup_ge2
thf(fact_976_sup__ge2,axiom,
! [Y3: nat,X4: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X4 @ Y3 ) ) ).
% sup_ge2
thf(fact_977_sup__ge1,axiom,
! [X4: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ X4 @ ( sup_sup_Product_unit @ X4 @ Y3 ) ) ).
% sup_ge1
thf(fact_978_sup__ge1,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y3 ) ) ).
% sup_ge1
thf(fact_979_sup__ge1,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y3 ) ) ).
% sup_ge1
thf(fact_980_le__supI,axiom,
! [A3: product_unit,X4: product_unit,B3: product_unit] :
( ( ord_le3221252021190050221t_unit @ A3 @ X4 )
=> ( ( ord_le3221252021190050221t_unit @ B3 @ X4 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A3 @ B3 ) @ X4 ) ) ) ).
% le_supI
thf(fact_981_le__supI,axiom,
! [A3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ X4 )
=> ( ( ord_le3843937902494030498c_fm_i @ B3 @ X4 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) @ X4 ) ) ) ).
% le_supI
thf(fact_982_le__supI,axiom,
! [A3: nat,X4: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ X4 )
=> ( ( ord_less_eq_nat @ B3 @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B3 ) @ X4 ) ) ) ).
% le_supI
thf(fact_983_le__supE,axiom,
! [A3: product_unit,B3: product_unit,X4: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A3 @ B3 ) @ X4 )
=> ~ ( ( ord_le3221252021190050221t_unit @ A3 @ X4 )
=> ~ ( ord_le3221252021190050221t_unit @ B3 @ X4 ) ) ) ).
% le_supE
thf(fact_984_le__supE,axiom,
! [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A3 @ B3 ) @ X4 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A3 @ X4 )
=> ~ ( ord_le3843937902494030498c_fm_i @ B3 @ X4 ) ) ) ).
% le_supE
thf(fact_985_le__supE,axiom,
! [A3: nat,B3: nat,X4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B3 ) @ X4 )
=> ~ ( ( ord_less_eq_nat @ A3 @ X4 )
=> ~ ( ord_less_eq_nat @ B3 @ X4 ) ) ) ).
% le_supE
thf(fact_986_inf__sup__ord_I3_J,axiom,
! [X4: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ X4 @ ( sup_sup_Product_unit @ X4 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_987_inf__sup__ord_I3_J,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_988_inf__sup__ord_I3_J,axiom,
! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_989_inf__sup__ord_I4_J,axiom,
! [Y3: product_unit,X4: product_unit] : ( ord_le3221252021190050221t_unit @ Y3 @ ( sup_sup_Product_unit @ X4 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_990_inf__sup__ord_I4_J,axiom,
! [Y3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y3 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_991_inf__sup__ord_I4_J,axiom,
! [Y3: nat,X4: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X4 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_992_list_Omap__ident__strong,axiom,
! [T3: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ! [Z5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z5 @ ( set_Epistemic_fm_i2 @ T3 ) )
=> ( ( F @ Z5 )
= Z5 ) )
=> ( ( map_Ep2755178516647988292c_fm_i @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_993_map__idI,axiom,
! [Xs: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_994_subset__Un__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A2 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_995_subset__UnE,axiom,
! [C: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
=> ~ ! [A6: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A6 @ A )
=> ! [B7: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B7 @ B )
=> ( C
!= ( sup_su1936195050962291414c_fm_i @ A6 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_996_Un__absorb2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_997_Un__absorb1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_998_Un__upper2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_upper2
thf(fact_999_Un__upper1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_upper1
thf(fact_1000_Un__least,axiom,
! [A: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_1001_Un__mono,axiom,
! [A: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ ( sup_su1936195050962291414c_fm_i @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_1002_singleton__iff,axiom,
! [B3: epistemic_fm_i,A3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B3 @ ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_1003_singletonD,axiom,
! [B3: epistemic_fm_i,A3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B3 @ ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_1004_subset__insertI2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,B3: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_1005_subset__insertI,axiom,
! [B: set_Epistemic_fm_i,A3: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A3 @ B ) ) ).
% subset_insertI
thf(fact_1006_subset__insert,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ B ) )
= ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ) ).
% subset_insert
thf(fact_1007_insert__mono,axiom,
! [C: set_Epistemic_fm_i,D: set_Epistemic_fm_i,A3: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ A3 @ C ) @ ( insert7817948997695205106c_fm_i @ A3 @ D ) ) ) ).
% insert_mono
thf(fact_1008_insert__subsetI,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,X5: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X5 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1009_Int__insert__right,axiom,
! [A3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ A3 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A3 @ B ) )
= ( insert7817948997695205106c_fm_i @ A3 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ A3 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A3 @ B ) )
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1010_Int__insert__left,axiom,
! [A3: epistemic_fm_i,C: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ A3 @ C )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A3 @ B ) @ C )
= ( insert7817948997695205106c_fm_i @ A3 @ ( inf_in3450601097109690352c_fm_i @ B @ C ) ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ A3 @ C )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A3 @ B ) @ C )
= ( inf_in3450601097109690352c_fm_i @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_1011_Collect__conv__if2,axiom,
! [P2: set_Epistemic_fm_i > $o,A3: set_Epistemic_fm_i] :
( ( ( P2 @ A3 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( A3 = X3 )
& ( P2 @ X3 ) ) )
= ( insert7698009978809854162c_fm_i @ A3 @ bot_bo145720340923748686c_fm_i ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( A3 = X3 )
& ( P2 @ X3 ) ) )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_conv_if2
thf(fact_1012_Collect__conv__if2,axiom,
! [P2: epistemic_fm_i > $o,A3: epistemic_fm_i] :
( ( ( P2 @ A3 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( A3 = X3 )
& ( P2 @ X3 ) ) )
= ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( A3 = X3 )
& ( P2 @ X3 ) ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_conv_if2
thf(fact_1013_Collect__conv__if2,axiom,
! [P2: nat > $o,A3: nat] :
( ( ( P2 @ A3 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A3 = X3 )
& ( P2 @ X3 ) ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A3 = X3 )
& ( P2 @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_1014_Collect__conv__if,axiom,
! [P2: set_Epistemic_fm_i > $o,A3: set_Epistemic_fm_i] :
( ( ( P2 @ A3 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( X3 = A3 )
& ( P2 @ X3 ) ) )
= ( insert7698009978809854162c_fm_i @ A3 @ bot_bo145720340923748686c_fm_i ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( X3 = A3 )
& ( P2 @ X3 ) ) )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_conv_if
thf(fact_1015_Collect__conv__if,axiom,
! [P2: epistemic_fm_i > $o,A3: epistemic_fm_i] :
( ( ( P2 @ A3 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( X3 = A3 )
& ( P2 @ X3 ) ) )
= ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( X3 = A3 )
& ( P2 @ X3 ) ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_conv_if
thf(fact_1016_Collect__conv__if,axiom,
! [P2: nat > $o,A3: nat] :
( ( ( P2 @ A3 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A3 )
& ( P2 @ X3 ) ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A3 )
& ( P2 @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_1017_consistent__disjuncts,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P3: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Dis_i @ P3 @ Q2 ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P3 @ bot_bo4194595901900360558c_fm_i ) @ V ) )
| ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q2 @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_disjuncts
thf(fact_1018_maximal__def,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S )
= ( ! [P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P4 @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ P4 @ S ) ) ) ) ).
% maximal_def
thf(fact_1019_distrib__sup__le,axiom,
! [X4: product_unit,Y3: product_unit,Z2: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X4 @ ( inf_inf_Product_unit @ Y3 @ Z2 ) ) @ ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X4 @ Y3 ) @ ( sup_sup_Product_unit @ X4 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1020_distrib__sup__le,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ Y3 @ Z2 ) ) @ ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y3 ) @ ( sup_su1936195050962291414c_fm_i @ X4 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1021_distrib__sup__le,axiom,
! [X4: nat,Y3: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ ( inf_inf_nat @ Y3 @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X4 @ Y3 ) @ ( sup_sup_nat @ X4 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1022_distrib__inf__le,axiom,
! [X4: product_unit,Y3: product_unit,Z2: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X4 @ Y3 ) @ ( inf_inf_Product_unit @ X4 @ Z2 ) ) @ ( inf_inf_Product_unit @ X4 @ ( sup_sup_Product_unit @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1023_distrib__inf__le,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y3 ) @ ( inf_in3450601097109690352c_fm_i @ X4 @ Z2 ) ) @ ( inf_in3450601097109690352c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1024_distrib__inf__le,axiom,
! [X4: nat,Y3: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X4 @ Y3 ) @ ( inf_inf_nat @ X4 @ Z2 ) ) @ ( inf_inf_nat @ X4 @ ( sup_sup_nat @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1025_infinite__finite__induct,axiom,
! [P2: set_Epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ! [A7: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ A7 )
=> ( P2 @ A7 ) )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ~ ( member6642669606046002379c_fm_i @ X @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ X @ F4 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1026_infinite__finite__induct,axiom,
! [P2: set_nat > $o,A: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P2 @ A7 ) )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ X @ F4 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1027_finite__ne__induct,axiom,
! [F5: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F5 )
=> ( ( F5 != bot_bo4194595901900360558c_fm_i )
=> ( ! [X: epistemic_fm_i] : ( P2 @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) )
=> ( ! [X: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ( F4 != bot_bo4194595901900360558c_fm_i )
=> ( ~ ( member6642669606046002379c_fm_i @ X @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ X @ F4 ) ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1028_finite__ne__induct,axiom,
! [F5: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F5 )
=> ( ( F5 != bot_bot_set_nat )
=> ( ! [X: nat] : ( P2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ! [X: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ X @ F4 ) ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1029_finite__induct,axiom,
! [F5: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F5 )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ~ ( member6642669606046002379c_fm_i @ X @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ X @ F4 ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ).
% finite_induct
thf(fact_1030_finite__induct,axiom,
! [F5: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F5 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ X @ F4 ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ).
% finite_induct
thf(fact_1031_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A2: set_nat,B5: nat] :
( ( A5
= ( insert_nat @ B5 @ A2 ) )
& ( finite_finite_nat @ A2 ) ) ) ) ) ).
% finite.simps
thf(fact_1032_finite_Ocases,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A4: nat] :
( A3
= ( insert_nat @ A4 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_1033_subset__singleton__iff,axiom,
! [X5: set_Epistemic_fm_i,A3: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X5 @ ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( X5 = bot_bo4194595901900360558c_fm_i )
| ( X5
= ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% subset_singleton_iff
thf(fact_1034_subset__singletonD,axiom,
! [A: set_Epistemic_fm_i,X4: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) )
=> ( ( A = bot_bo4194595901900360558c_fm_i )
| ( A
= ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% subset_singletonD
thf(fact_1035_Un__Int__assoc__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ C )
= ( inf_in3450601097109690352c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C ) ) )
= ( ord_le3843937902494030498c_fm_i @ C @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1036_consistent__consequent,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P3: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( member6642669606046002379c_fm_i @ P3 @ V )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P3 @ Q2 ) )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q2 @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_consequent
thf(fact_1037_consistent__consequent_H,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P3: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( member6642669606046002379c_fm_i @ P3 @ V )
=> ( ! [G3: list_char > $o,H2: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H2 @ ( epistemic_Imp_i @ P3 @ Q2 ) )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q2 @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_consequent'
thf(fact_1038_finite__subset__induct_H,axiom,
! [F5: set_nat,A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F5 )
=> ( ( ord_less_eq_set_nat @ F5 @ A )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A4 @ A )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ~ ( member_nat @ A4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ A4 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1039_finite__subset__induct_H,axiom,
! [F5: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F5 )
=> ( ( ord_le3843937902494030498c_fm_i @ F5 @ A )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A4: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ( member6642669606046002379c_fm_i @ A4 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ F4 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ A4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ A4 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1040_finite__subset__induct,axiom,
! [F5: set_nat,A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F5 )
=> ( ( ord_less_eq_set_nat @ F5 @ A )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A4 @ A )
=> ( ~ ( member_nat @ A4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_nat @ A4 @ F4 ) ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1041_finite__subset__induct,axiom,
! [F5: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F5 )
=> ( ( ord_le3843937902494030498c_fm_i @ F5 @ A )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A4: epistemic_fm_i,F4: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F4 )
=> ( ( member6642669606046002379c_fm_i @ A4 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ A4 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ A4 @ F4 ) ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1042_K__distrib__K__imp,axiom,
! [A: epistemic_fm_i > $o,I2: i,G4: list_Epistemic_fm_i,Q2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_K_i @ I2 @ ( epistemic_imply_i @ G4 @ Q2 ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ I2 ) @ G4 ) @ ( epistemic_K_i @ I2 @ Q2 ) ) ) ) ).
% K_distrib_K_imp
thf(fact_1043_exists__finite__inconsistent,axiom,
! [A: epistemic_fm_i > $o,P3: epistemic_fm_i,V: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ V ) )
=> ~ ! [W2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ W2 ) @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ V ) )
=> ( ~ ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) @ W2 )
=> ( ( finite3304564979551393739c_fm_i @ W2 )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ W2 ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_1044_inconsistent__imply,axiom,
! [A: epistemic_fm_i > $o,P3: epistemic_fm_i,G4: list_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ ( set_Epistemic_fm_i2 @ G4 ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G4 @ P3 ) ) ) ).
% inconsistent_imply
thf(fact_1045_finite__ranking__induct,axiom,
! [S: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o,F: epistemic_fm_i > nat] :
( ( finite3304564979551393739c_fm_i @ S )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X: epistemic_fm_i,S5: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ S5 )
=> ( ! [Y6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y6 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X ) ) )
=> ( ( P2 @ S5 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ X @ S5 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1046_finite__ranking__induct,axiom,
! [S: set_nat,P2: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X: nat,S5: set_nat] :
( ( finite_finite_nat @ S5 )
=> ( ! [Y6: nat] :
( ( member_nat @ Y6 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X ) ) )
=> ( ( P2 @ S5 )
=> ( P2 @ ( insert_nat @ X @ S5 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1047_extendS__def,axiom,
! [A: epistemic_fm_i > $o,N: epistemic_fm_i,Prev: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S @ N @ Prev )
= ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) ) )
& ( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S @ N @ Prev )
= Prev ) ) ) ).
% extendS_def
thf(fact_1048_sup__unit__def,axiom,
( sup_sup_Product_unit
= ( ^ [Uu2: product_unit,Uv: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_1049_sup__Un__eq,axiom,
! [R3: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( sup_su2844414564337698183fm_i_o
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ R3 )
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ S ) )
= ( ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ R3 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1050_sup__set__def,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ( sup_su6723579903606884263fm_i_o
@ ^ [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ A2 )
@ ^ [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_1051_sup__set__def,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( sup_su2844414564337698183fm_i_o
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ A2 )
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_1052_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_1053_is__singletonI_H,axiom,
! [A: set_Epistemic_fm_i] :
( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ! [X: epistemic_fm_i,Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ( ( member6642669606046002379c_fm_i @ Y5 @ A )
=> ( X = Y5 ) ) )
=> ( is_sin4923117429174228502c_fm_i @ A ) ) ) ).
% is_singletonI'
thf(fact_1054_arg__min__least,axiom,
! [S: set_Epistemic_fm_i,Y3: epistemic_fm_i,F: epistemic_fm_i > nat] :
( ( finite3304564979551393739c_fm_i @ S )
=> ( ( S != bot_bo4194595901900360558c_fm_i )
=> ( ( member6642669606046002379c_fm_i @ Y3 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7487310264250698137_i_nat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1055_arg__min__least,axiom,
! [S: set_nat,Y3: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y3 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1056_Sup__fin_Oinsert,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A ) )
= ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1057_Sup__fin_Oinsert,axiom,
! [A: set_Product_unit,X4: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X4 @ A ) )
= ( sup_sup_Product_unit @ X4 @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1058_Inf__fin_Oinsert,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ A ) )
= ( inf_inf_nat @ X4 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1059_Inf__fin_Oinsert,axiom,
! [A: set_Product_unit,X4: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X4 @ A ) )
= ( inf_inf_Product_unit @ X4 @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1060_subset__singleton__iff__Uniq,axiom,
! [A: set_Epistemic_fm_i] :
( ( ? [A5: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A5 @ bot_bo4194595901900360558c_fm_i ) ) )
= ( uniq_Epistemic_fm_i
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1061_Inf__fin__le__Sup__fin,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1062_Inf__fin__le__Sup__fin,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1063_Inf__fin_OcoboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,A3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ A3 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1064_Inf__fin_OcoboundedI,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ A3 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1065_Sup__fin_OcoboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,A3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A3 @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1066_Sup__fin_OcoboundedI,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ( ord_less_eq_nat @ A3 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1067_Sup__fin_Obounded__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X4 )
= ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ X4 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1068_Sup__fin_Obounded__iff,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X4 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ X4 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1069_Inf__fin_Obounded__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ ( lattic8495942116038605215c_fm_i @ A ) )
= ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1070_Inf__fin_Obounded__iff,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X4 @ ( lattic5238388535129920115in_nat @ A ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X4 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1071_Sup__fin_OboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [A4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A4 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A4 @ X4 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X4 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1072_Sup__fin_OboundedI,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ A4 @ X4 ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X4 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1073_Sup__fin_OboundedE,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X4 )
=> ! [A8: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A8 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A8 @ X4 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1074_Sup__fin_OboundedE,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X4 )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A )
=> ( ord_less_eq_nat @ A8 @ X4 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1075_Inf__fin_OboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [A4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A4 @ A )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ A4 ) )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1076_Inf__fin_OboundedI,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ X4 @ A4 ) )
=> ( ord_less_eq_nat @ X4 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1077_Inf__fin_OboundedE,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ ( lattic8495942116038605215c_fm_i @ A ) )
=> ! [A8: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A8 @ A )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1078_Inf__fin_OboundedE,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X4 @ ( lattic5238388535129920115in_nat @ A ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A )
=> ( ord_less_eq_nat @ X4 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1079_Inf__fin_Osubset__imp,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ B ) @ ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1080_Inf__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1081_Sup__fin_Osubset__imp,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1082_Sup__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1083_Inf__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1084_Inf__fin_Osubset,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( B != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ B @ A )
=> ( ( inf_inf_Product_unit @ ( lattic1263872656861969706t_unit @ B ) @ ( lattic1263872656861969706t_unit @ A ) )
= ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1085_Sup__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1086_Sup__fin_Osubset,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( B != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ B @ A )
=> ( ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ B ) @ ( lattic5294303975357428420t_unit @ A ) )
= ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1087_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X4 @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ A ) )
= ( inf_inf_nat @ X4 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1088_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_Product_unit,X4: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X4 @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X4 @ A ) )
= ( inf_inf_Product_unit @ X4 @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1089_Inf__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X: nat,Y5: nat] : ( member_nat @ ( inf_inf_nat @ X @ Y5 ) @ ( insert_nat @ X @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_1090_Inf__fin_Oclosed,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit,Y5: product_unit] : ( member_Product_unit @ ( inf_inf_Product_unit @ X @ Y5 ) @ ( insert_Product_unit @ X @ ( insert_Product_unit @ Y5 @ bot_bo3957492148770167129t_unit ) ) )
=> ( member_Product_unit @ ( lattic1263872656861969706t_unit @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_1091_Sup__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X: nat,Y5: nat] : ( member_nat @ ( sup_sup_nat @ X @ Y5 ) @ ( insert_nat @ X @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1092_Sup__fin_Oclosed,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit,Y5: product_unit] : ( member_Product_unit @ ( sup_sup_Product_unit @ X @ Y5 ) @ ( insert_Product_unit @ X @ ( insert_Product_unit @ Y5 @ bot_bo3957492148770167129t_unit ) ) )
=> ( member_Product_unit @ ( lattic5294303975357428420t_unit @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1093_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X4 @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A ) )
= ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1094_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_Product_unit,X4: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X4 @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X4 @ A ) )
= ( sup_sup_Product_unit @ X4 @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1095_Inf__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1096_Inf__fin_Ounion,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( B != bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( sup_su793286257634532545t_unit @ A @ B ) )
= ( inf_inf_Product_unit @ ( lattic1263872656861969706t_unit @ A ) @ ( lattic1263872656861969706t_unit @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1097_Sup__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1098_Sup__fin_Ounion,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( B != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( sup_su793286257634532545t_unit @ A @ B ) )
= ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ A ) @ ( lattic5294303975357428420t_unit @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1099_Sup__fin_Oinsert__remove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A ) )
= X4 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A ) )
= ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1100_Sup__fin_Oinsert__remove,axiom,
! [A: set_Product_unit,X4: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X4 @ A ) )
= X4 ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X4 @ A ) )
= ( sup_sup_Product_unit @ X4 @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1101_Sup__fin_Oremove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X4 @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= X4 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1102_Sup__fin_Oremove,axiom,
! [A: set_Product_unit,X4: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X4 @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ A )
= X4 ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ A )
= ( sup_sup_Product_unit @ X4 @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1103_Inf__fin_Oremove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X4 @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= X4 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= ( inf_inf_nat @ X4 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1104_Inf__fin_Oremove,axiom,
! [A: set_Product_unit,X4: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X4 @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ A )
= X4 ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ A )
= ( inf_inf_Product_unit @ X4 @ ( lattic1263872656861969706t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1105_DiffI,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ C2 @ B )
=> ( member6642669606046002379c_fm_i @ C2 @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1106_Diff__iff,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
= ( ( member6642669606046002379c_fm_i @ C2 @ A )
& ~ ( member6642669606046002379c_fm_i @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_1107_insert__Diff1,axiom,
! [X4: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 @ A ) @ B )
= ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1108_Diff__insert0,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ B ) )
= ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1109_Diff__eq__empty__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1110_double__diff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ( minus_1543973939109951465c_fm_i @ B @ ( minus_1543973939109951465c_fm_i @ C @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1111_Diff__subset,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1112_Diff__mono,axiom,
! [A: set_Epistemic_fm_i,C: set_Epistemic_fm_i,D: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ D @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_1113_DiffE,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C2 @ A )
=> ( member6642669606046002379c_fm_i @ C2 @ B ) ) ) ).
% DiffE
thf(fact_1114_DiffD1,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
=> ( member6642669606046002379c_fm_i @ C2 @ A ) ) ).
% DiffD1
thf(fact_1115_DiffD2,axiom,
! [C2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C2 @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
=> ~ ( member6642669606046002379c_fm_i @ C2 @ B ) ) ).
% DiffD2
thf(fact_1116_set__diff__eq,axiom,
( minus_1073301138667672009c_fm_i
= ( ^ [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A2 )
& ~ ( member1104366573291651755c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1117_set__diff__eq,axiom,
( minus_1543973939109951465c_fm_i
= ( ^ [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A2 )
& ~ ( member6642669606046002379c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1118_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1119_insert__Diff__if,axiom,
! [X4: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ X4 @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 @ A ) @ B )
= ( minus_1543973939109951465c_fm_i @ A @ B ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ X4 @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 @ A ) @ B )
= ( insert7817948997695205106c_fm_i @ X4 @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1120_diff__shunt__var,axiom,
! [X4: product_unit,Y3: product_unit] :
( ( ( minus_3524152463667985524t_unit @ X4 @ Y3 )
= bot_bot_Product_unit )
= ( ord_le3221252021190050221t_unit @ X4 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1121_diff__shunt__var,axiom,
! [X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ X4 @ Y3 )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1122_Diff__insert__absorb,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 @ A ) @ ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1123_insert__Diff,axiom,
! [A3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ A )
=> ( ( insert7817948997695205106c_fm_i @ A3 @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1124_subset__Diff__insert,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X4: epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ X4 @ C ) ) )
= ( ( ord_le3843937902494030498c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ C ) )
& ~ ( member6642669606046002379c_fm_i @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1125_Diff__partition,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1126_Diff__subset__conv,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ C )
= ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_1127_infinite__remove,axiom,
! [S: set_nat,A3: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1128_infinite__coinduct,axiom,
! [X5: set_nat > $o,A: set_nat] :
( ( X5 @ A )
=> ( ! [A7: set_nat] :
( ( X5 @ A7 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A7 )
& ( ( X5 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_1129_finite__empty__induct,axiom,
! [A: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( P2 @ A )
=> ( ! [A4: epistemic_fm_i,A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( member6642669606046002379c_fm_i @ A4 @ A7 )
=> ( ( P2 @ A7 )
=> ( P2 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ A4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ( P2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% finite_empty_induct
thf(fact_1130_finite__empty__induct,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P2 @ A )
=> ( ! [A4: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( member_nat @ A4 @ A7 )
=> ( ( P2 @ A7 )
=> ( P2 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1131_Diff__single__insert,axiom,
! [A: set_Epistemic_fm_i,X4: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) @ B )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1132_subset__insert__iff,axiom,
! [A: set_Epistemic_fm_i,X4: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ B ) )
= ( ( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) @ B ) )
& ( ~ ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1133_finite__remove__induct,axiom,
! [B: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( P2 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1134_finite__remove__induct,axiom,
! [B: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( A7 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A7 @ B )
=> ( ! [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A7 )
=> ( P2 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1135_remove__induct,axiom,
! [P2: set_nat > $o,B: set_nat] :
( ( P2 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P2 @ B ) )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( P2 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_1136_remove__induct,axiom,
! [P2: set_Epistemic_fm_i > $o,B: set_Epistemic_fm_i] :
( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ( ~ ( finite3304564979551393739c_fm_i @ B )
=> ( P2 @ B ) )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( A7 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A7 @ B )
=> ( ! [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A7 )
=> ( P2 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_1137_Inf__fin_Oinsert__remove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ A ) )
= X4 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ A ) )
= ( inf_inf_nat @ X4 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1138_Inf__fin_Oinsert__remove,axiom,
! [A: set_Product_unit,X4: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X4 @ A ) )
= X4 ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X4 @ A ) )
= ( inf_inf_Product_unit @ X4 @ ( lattic1263872656861969706t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1139_removeAll__id,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remove817376155714978094c_fm_i @ X4 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1140_minus__set__def,axiom,
( minus_1073301138667672009c_fm_i
= ( ^ [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ( minus_5671197887077578836fm_i_o
@ ^ [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ A2 )
@ ^ [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_1141_minus__set__def,axiom,
( minus_1543973939109951465c_fm_i
= ( ^ [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( minus_6519238007984220724fm_i_o
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ A2 )
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_1142_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_1143_MCS_Omaximal__def,axiom,
! [Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ( ( maxima3264069618988350929c_fm_i @ Consistent @ S )
= ( ! [P4: epistemic_fm_i] :
( ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P4 @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ P4 @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_1144_member__remove,axiom,
! [X4: epistemic_fm_i,Y3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ ( remove5552923401211475591c_fm_i @ Y3 @ A ) )
= ( ( member6642669606046002379c_fm_i @ X4 @ A )
& ( X4 != Y3 ) ) ) ).
% member_remove
thf(fact_1145_MCS__axioms,axiom,
! [A: epistemic_fm_i > $o] : ( maxima1924290493700099598c_fm_i @ ( episte2285483198712856234tent_i @ A ) ) ).
% MCS_axioms
thf(fact_1146_MCS_Omaximal_Ocong,axiom,
maxima3264069618988350929c_fm_i = maxima3264069618988350929c_fm_i ).
% MCS.maximal.cong
thf(fact_1147_Set__filter__fold,axiom,
! [A: set_nat,P2: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( filter_nat @ P2 @ A )
= ( finite5529483035118572448et_nat
@ ^ [X3: nat,A9: set_nat] : ( if_set_nat @ ( P2 @ X3 ) @ ( insert_nat @ X3 @ A9 ) @ A9 )
@ bot_bot_set_nat
@ A ) ) ) ).
% Set_filter_fold
thf(fact_1148_MCS__Lim__Ord_Oconsistent__hereditary,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i,S2: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( Consistent @ S )
=> ( ( ord_le3843937902494030498c_fm_i @ S2 @ S )
=> ( Consistent @ S2 ) ) ) ) ).
% MCS_Lim_Ord.consistent_hereditary
thf(fact_1149_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R: set_Pr1261947904930325089at_nat,Consistent: set_nat > $o,S: set_nat] :
( ( maxima4501515429357695765rd_nat @ R @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S4: set_nat] :
( ( ord_less_eq_set_nat @ S4 @ S )
& ( finite_finite_nat @ S4 )
& ~ ( Consistent @ S4 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_1150_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S4 @ S )
& ( finite3304564979551393739c_fm_i @ S4 )
& ~ ( Consistent @ S4 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_1151_flat__lub__def,axiom,
( partia6511039473171299566c_fm_i
= ( ^ [B5: epistemic_fm_i,A2: set_Epistemic_fm_i] :
( if_Epistemic_fm_i @ ( ord_le3843937902494030498c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B5 @ bot_bo4194595901900360558c_fm_i ) ) @ B5
@ ( the_Epistemic_fm_i
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ ( minus_1543973939109951465c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B5 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_1152_image__eqI,axiom,
! [B3: epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,X4: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( B3
= ( F @ X4 ) )
=> ( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( member6642669606046002379c_fm_i @ B3 @ ( image_120052958407486725c_fm_i @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1153_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P2: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) ) )
=> ( P2 @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P2 @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1154_all__finite__subset__image,axiom,
! [F: epistemic_fm_i > nat,A: set_Epistemic_fm_i,P2: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_2291338433772226221_i_nat @ F @ A ) ) )
=> ( P2 @ B2 ) ) )
= ( ! [B2: set_Epistemic_fm_i] :
( ( ( finite3304564979551393739c_fm_i @ B2 )
& ( ord_le3843937902494030498c_fm_i @ B2 @ A ) )
=> ( P2 @ ( image_2291338433772226221_i_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1155_all__finite__subset__image,axiom,
! [F: nat > epistemic_fm_i,A: set_nat,P2: set_Epistemic_fm_i > $o] :
( ( ! [B2: set_Epistemic_fm_i] :
( ( ( finite3304564979551393739c_fm_i @ B2 )
& ( ord_le3843937902494030498c_fm_i @ B2 @ ( image_3894954817185171339c_fm_i @ F @ A ) ) )
=> ( P2 @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P2 @ ( image_3894954817185171339c_fm_i @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1156_all__finite__subset__image,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( ! [B2: set_Epistemic_fm_i] :
( ( ( finite3304564979551393739c_fm_i @ B2 )
& ( ord_le3843937902494030498c_fm_i @ B2 @ ( image_120052958407486725c_fm_i @ F @ A ) ) )
=> ( P2 @ B2 ) ) )
= ( ! [B2: set_Epistemic_fm_i] :
( ( ( finite3304564979551393739c_fm_i @ B2 )
& ( ord_le3843937902494030498c_fm_i @ B2 @ A ) )
=> ( P2 @ ( image_120052958407486725c_fm_i @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1157_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P2: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
& ( P2 @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P2 @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1158_ex__finite__subset__image,axiom,
! [F: epistemic_fm_i > nat,A: set_Epistemic_fm_i,P2: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_2291338433772226221_i_nat @ F @ A ) )
& ( P2 @ B2 ) ) )
= ( ? [B2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B2 )
& ( ord_le3843937902494030498c_fm_i @ B2 @ A )
& ( P2 @ ( image_2291338433772226221_i_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1159_ex__finite__subset__image,axiom,
! [F: nat > epistemic_fm_i,A: set_nat,P2: set_Epistemic_fm_i > $o] :
( ( ? [B2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B2 )
& ( ord_le3843937902494030498c_fm_i @ B2 @ ( image_3894954817185171339c_fm_i @ F @ A ) )
& ( P2 @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P2 @ ( image_3894954817185171339c_fm_i @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1160_ex__finite__subset__image,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( ? [B2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B2 )
& ( ord_le3843937902494030498c_fm_i @ B2 @ ( image_120052958407486725c_fm_i @ F @ A ) )
& ( P2 @ B2 ) ) )
= ( ? [B2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B2 )
& ( ord_le3843937902494030498c_fm_i @ B2 @ A )
& ( P2 @ ( image_120052958407486725c_fm_i @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1161_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1162_finite__subset__image,axiom,
! [B: set_nat,F: epistemic_fm_i > nat,A: set_Epistemic_fm_i] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_2291338433772226221_i_nat @ F @ A ) )
=> ? [C4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C4 @ A )
& ( finite3304564979551393739c_fm_i @ C4 )
& ( B
= ( image_2291338433772226221_i_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1163_finite__subset__image,axiom,
! [B: set_Epistemic_fm_i,F: nat > epistemic_fm_i,A: set_nat] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ ( image_3894954817185171339c_fm_i @ F @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_3894954817185171339c_fm_i @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1164_finite__subset__image,axiom,
! [B: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ ( image_120052958407486725c_fm_i @ F @ A ) )
=> ? [C4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C4 @ A )
& ( finite3304564979551393739c_fm_i @ C4 )
& ( B
= ( image_120052958407486725c_fm_i @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1165_finite__surj,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_1166_finite__surj,axiom,
! [A: set_nat,B: set_Epistemic_fm_i,F: nat > epistemic_fm_i] :
( ( finite_finite_nat @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ ( image_3894954817185171339c_fm_i @ F @ A ) )
=> ( finite3304564979551393739c_fm_i @ B ) ) ) ).
% finite_surj
thf(fact_1167_image__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( image_120052958407486725c_fm_i @ F @ A ) @ ( image_120052958407486725c_fm_i @ F @ B ) ) ) ).
% image_mono
thf(fact_1168_image__subsetI,axiom,
! [A: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ( member6642669606046002379c_fm_i @ ( F @ X ) @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ ( image_120052958407486725c_fm_i @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1169_subset__imageE,axiom,
! [B: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ ( image_120052958407486725c_fm_i @ F @ A ) )
=> ~ ! [C4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C4 @ A )
=> ( B
!= ( image_120052958407486725c_fm_i @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1170_subset__image__iff,axiom,
! [B: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ ( image_120052958407486725c_fm_i @ F @ A ) )
= ( ? [AA: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ AA @ A )
& ( B
= ( image_120052958407486725c_fm_i @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1171_all__subset__image,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( ! [B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ ( image_120052958407486725c_fm_i @ F @ A ) )
=> ( P2 @ B2 ) ) )
= ( ! [B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( P2 @ ( image_120052958407486725c_fm_i @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_1172_imageI,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( member6642669606046002379c_fm_i @ ( F @ X4 ) @ ( image_120052958407486725c_fm_i @ F @ A ) ) ) ).
% imageI
thf(fact_1173_rev__image__eqI,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,B3: epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ( B3
= ( F @ X4 ) )
=> ( member6642669606046002379c_fm_i @ B3 @ ( image_120052958407486725c_fm_i @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1174_imageE,axiom,
! [B3: epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B3 @ ( image_120052958407486725c_fm_i @ F @ A ) )
=> ~ ! [X: epistemic_fm_i] :
( ( B3
= ( F @ X ) )
=> ~ ( member6642669606046002379c_fm_i @ X @ A ) ) ) ).
% imageE
thf(fact_1175_Compr__image__eq,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( image_2698589095351095557c_fm_i @ F @ A ) )
& ( P2 @ X3 ) ) )
= ( image_2698589095351095557c_fm_i @ F
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1176_Compr__image__eq,axiom,
! [F: epistemic_fm_i > set_Epistemic_fm_i,A: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( image_2913337242412145381c_fm_i @ F @ A ) )
& ( P2 @ X3 ) ) )
= ( image_2913337242412145381c_fm_i @ F
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1177_Compr__image__eq,axiom,
! [F: nat > set_Epistemic_fm_i,A: set_nat,P2: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( image_3332579458907315563c_fm_i @ F @ A ) )
& ( P2 @ X3 ) ) )
= ( image_3332579458907315563c_fm_i @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1178_Compr__image__eq,axiom,
! [F: set_Epistemic_fm_i > epistemic_fm_i,A: set_se3485332733965609186c_fm_i,P2: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( image_809639422248493861c_fm_i @ F @ A ) )
& ( P2 @ X3 ) ) )
= ( image_809639422248493861c_fm_i @ F
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1179_Compr__image__eq,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( image_120052958407486725c_fm_i @ F @ A ) )
& ( P2 @ X3 ) ) )
= ( image_120052958407486725c_fm_i @ F
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1180_Compr__image__eq,axiom,
! [F: nat > epistemic_fm_i,A: set_nat,P2: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( image_3894954817185171339c_fm_i @ F @ A ) )
& ( P2 @ X3 ) ) )
= ( image_3894954817185171339c_fm_i @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1181_Compr__image__eq,axiom,
! [F: set_Epistemic_fm_i > nat,A: set_se3485332733965609186c_fm_i,P2: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_9130930184777177229_i_nat @ F @ A ) )
& ( P2 @ X3 ) ) )
= ( image_9130930184777177229_i_nat @ F
@ ( collec3087743281813070829c_fm_i
@ ^ [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1182_Compr__image__eq,axiom,
! [F: epistemic_fm_i > nat,A: set_Epistemic_fm_i,P2: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_2291338433772226221_i_nat @ F @ A ) )
& ( P2 @ X3 ) ) )
= ( image_2291338433772226221_i_nat @ F
@ ( collec4904205187116291597c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1183_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P2: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
& ( P2 @ X3 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1184_image__Collect__subsetI,axiom,
! [P2: set_Epistemic_fm_i > $o,F: set_Epistemic_fm_i > epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X: set_Epistemic_fm_i] :
( ( P2 @ X )
=> ( member6642669606046002379c_fm_i @ ( F @ X ) @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ ( image_809639422248493861c_fm_i @ F @ ( collec3087743281813070829c_fm_i @ P2 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1185_image__Collect__subsetI,axiom,
! [P2: epistemic_fm_i > $o,F: epistemic_fm_i > epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X: epistemic_fm_i] :
( ( P2 @ X )
=> ( member6642669606046002379c_fm_i @ ( F @ X ) @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ ( image_120052958407486725c_fm_i @ F @ ( collec4904205187116291597c_fm_i @ P2 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1186_image__Collect__subsetI,axiom,
! [P2: nat > $o,F: nat > epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X: nat] :
( ( P2 @ X )
=> ( member6642669606046002379c_fm_i @ ( F @ X ) @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ ( image_3894954817185171339c_fm_i @ F @ ( collect_nat @ P2 ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1187_pigeonhole__infinite,axiom,
! [A: set_se3485332733965609186c_fm_i,F: set_Epistemic_fm_i > nat] :
( ~ ( finite7933139204641697195c_fm_i @ A )
=> ( ( finite_finite_nat @ ( image_9130930184777177229_i_nat @ F @ A ) )
=> ? [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ~ ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [A5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A5 @ A )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1188_pigeonhole__infinite,axiom,
! [A: set_Epistemic_fm_i,F: epistemic_fm_i > nat] :
( ~ ( finite3304564979551393739c_fm_i @ A )
=> ( ( finite_finite_nat @ ( image_2291338433772226221_i_nat @ F @ A ) )
=> ? [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ~ ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [A5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A5 @ A )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1189_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1190_in__image__insert__iff,axiom,
! [B: set_se3485332733965609186c_fm_i,X4: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ! [C4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ C4 @ B )
=> ~ ( member6642669606046002379c_fm_i @ X4 @ C4 ) )
=> ( ( member1104366573291651755c_fm_i @ A @ ( image_2698589095351095557c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 ) @ B ) )
= ( ( member6642669606046002379c_fm_i @ X4 @ A )
& ( member1104366573291651755c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1191_Inf__fin_Ohom__commute,axiom,
! [H: nat > nat,N2: set_nat] :
( ! [X: nat,Y5: nat] :
( ( H @ ( inf_inf_nat @ X @ Y5 ) )
= ( inf_inf_nat @ ( H @ X ) @ ( H @ Y5 ) ) )
=> ( ( finite_finite_nat @ N2 )
=> ( ( N2 != bot_bot_set_nat )
=> ( ( H @ ( lattic5238388535129920115in_nat @ N2 ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H @ N2 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1192_Inf__fin_Ohom__commute,axiom,
! [H: product_unit > product_unit,N2: set_Product_unit] :
( ! [X: product_unit,Y5: product_unit] :
( ( H @ ( inf_inf_Product_unit @ X @ Y5 ) )
= ( inf_inf_Product_unit @ ( H @ X ) @ ( H @ Y5 ) ) )
=> ( ( finite4290736615968046902t_unit @ N2 )
=> ( ( N2 != bot_bo3957492148770167129t_unit )
=> ( ( H @ ( lattic1263872656861969706t_unit @ N2 ) )
= ( lattic1263872656861969706t_unit @ ( image_405062704495631173t_unit @ H @ N2 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1193_Sup__fin_Ohom__commute,axiom,
! [H: nat > nat,N2: set_nat] :
( ! [X: nat,Y5: nat] :
( ( H @ ( sup_sup_nat @ X @ Y5 ) )
= ( sup_sup_nat @ ( H @ X ) @ ( H @ Y5 ) ) )
=> ( ( finite_finite_nat @ N2 )
=> ( ( N2 != bot_bot_set_nat )
=> ( ( H @ ( lattic1093996805478795353in_nat @ N2 ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N2 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1194_Sup__fin_Ohom__commute,axiom,
! [H: product_unit > product_unit,N2: set_Product_unit] :
( ! [X: product_unit,Y5: product_unit] :
( ( H @ ( sup_sup_Product_unit @ X @ Y5 ) )
= ( sup_sup_Product_unit @ ( H @ X ) @ ( H @ Y5 ) ) )
=> ( ( finite4290736615968046902t_unit @ N2 )
=> ( ( N2 != bot_bo3957492148770167129t_unit )
=> ( ( H @ ( lattic5294303975357428420t_unit @ N2 ) )
= ( lattic5294303975357428420t_unit @ ( image_405062704495631173t_unit @ H @ N2 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1195_Inf__fin_Oset__eq__fold,axiom,
! [X4: product_unit,Xs: list_Product_unit] :
( ( lattic1263872656861969706t_unit @ ( set_Product_unit2 @ ( cons_Product_unit @ X4 @ Xs ) ) )
= ( fold_P8565935628131843659t_unit @ inf_inf_Product_unit @ Xs @ X4 ) ) ).
% Inf_fin.set_eq_fold
thf(fact_1196_Sup__fin_Oset__eq__fold,axiom,
! [X4: product_unit,Xs: list_Product_unit] :
( ( lattic5294303975357428420t_unit @ ( set_Product_unit2 @ ( cons_Product_unit @ X4 @ Xs ) ) )
= ( fold_P8565935628131843659t_unit @ sup_sup_Product_unit @ Xs @ X4 ) ) ).
% Sup_fin.set_eq_fold
thf(fact_1197_remdups_Osimps_I2_J,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remdup2862625226703050045c_fm_i @ ( cons_Epistemic_fm_i @ X4 @ Xs ) )
= ( remdup2862625226703050045c_fm_i @ Xs ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remdup2862625226703050045c_fm_i @ ( cons_Epistemic_fm_i @ X4 @ Xs ) )
= ( cons_Epistemic_fm_i @ X4 @ ( remdup2862625226703050045c_fm_i @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_1198_PowI,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( member1104366573291651755c_fm_i @ A @ ( pow_Epistemic_fm_i @ B ) ) ) ).
% PowI
thf(fact_1199_Pow__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A @ ( pow_Epistemic_fm_i @ B ) )
= ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% Pow_iff
thf(fact_1200_Pow__def,axiom,
( pow_Epistemic_fm_i
= ( ^ [A2: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B2 @ A2 ) ) ) ) ).
% Pow_def
thf(fact_1201_PowD,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A @ ( pow_Epistemic_fm_i @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% PowD
thf(fact_1202_Pow__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ord_le5389487502678872194c_fm_i @ ( pow_Epistemic_fm_i @ A ) @ ( pow_Epistemic_fm_i @ B ) ) ) ).
% Pow_mono
thf(fact_1203_Pow__fold,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( pow_nat @ A )
= ( finite4178521680790401110et_nat
@ ^ [X3: nat,A2: set_set_nat] : ( sup_sup_set_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X3 ) @ A2 ) )
@ ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat )
@ A ) ) ) ).
% Pow_fold
thf(fact_1204_chains__extend,axiom,
! [C2: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i,Z2: set_Epistemic_fm_i] :
( ( member1461078328125707403c_fm_i @ C2 @ ( chains830531755891748338c_fm_i @ S ) )
=> ( ( member1104366573291651755c_fm_i @ Z2 @ S )
=> ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ X @ Z2 ) )
=> ( member1461078328125707403c_fm_i @ ( sup_su2582925890723967158c_fm_i @ ( insert7698009978809854162c_fm_i @ Z2 @ bot_bo145720340923748686c_fm_i ) @ C2 ) @ ( chains830531755891748338c_fm_i @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1205_chainsD,axiom,
! [C2: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( member1461078328125707403c_fm_i @ C2 @ ( chains830531755891748338c_fm_i @ S ) )
=> ( ( member1104366573291651755c_fm_i @ X4 @ C2 )
=> ( ( member1104366573291651755c_fm_i @ Y3 @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ Y3 )
| ( ord_le3843937902494030498c_fm_i @ Y3 @ X4 ) ) ) ) ) ).
% chainsD
thf(fact_1206_Zorn__Lemma2,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [X: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ ( chains830531755891748338c_fm_i @ A ) )
=> ? [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
& ! [Xb: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xb @ X )
=> ( ord_le3843937902494030498c_fm_i @ Xb @ Xa ) ) ) )
=> ? [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1207_empty__Shift,axiom,
! [Kl: set_li8269783359719514504c_fm_i,K4: epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ nil_Epistemic_fm_i @ Kl )
=> ( ( member6642669606046002379c_fm_i @ K4 @ ( bNF_Gr1093487169986386109c_fm_i @ Kl @ nil_Epistemic_fm_i ) )
=> ( member6475149752759350353c_fm_i @ nil_Epistemic_fm_i @ ( bNF_Gr8437504169225076033c_fm_i @ Kl @ K4 ) ) ) ) ).
% empty_Shift
thf(fact_1208_chain__subset__def,axiom,
( chain_7805378461283022331c_fm_i
= ( ^ [C3: set_se3485332733965609186c_fm_i] :
! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ C3 )
=> ! [Y4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y4 @ C3 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Y4 )
| ( ord_le3843937902494030498c_fm_i @ Y4 @ X3 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_1209_Fpow__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ord_le5389487502678872194c_fm_i @ ( finite7994277067554142856c_fm_i @ A ) @ ( finite7994277067554142856c_fm_i @ B ) ) ) ).
% Fpow_mono
thf(fact_1210_Fpow__def,axiom,
( finite_Fpow_nat
= ( ^ [A2: set_nat] :
( collect_set_nat
@ ^ [X7: set_nat] :
( ( ord_less_eq_set_nat @ X7 @ A2 )
& ( finite_finite_nat @ X7 ) ) ) ) ) ).
% Fpow_def
thf(fact_1211_Fpow__def,axiom,
( finite7994277067554142856c_fm_i
= ( ^ [A2: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X7: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X7 @ A2 )
& ( finite3304564979551393739c_fm_i @ X7 ) ) ) ) ) ).
% Fpow_def
thf(fact_1212_Fpow__Pow__finite,axiom,
( finite7994277067554142856c_fm_i
= ( ^ [A2: set_Epistemic_fm_i] : ( inf_in161960956874937808c_fm_i @ ( pow_Epistemic_fm_i @ A2 ) @ ( collec3087743281813070829c_fm_i @ finite3304564979551393739c_fm_i ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_1213_Fpow__Pow__finite,axiom,
( finite_Fpow_nat
= ( ^ [A2: set_nat] : ( inf_inf_set_set_nat @ ( pow_nat @ A2 ) @ ( collect_set_nat @ finite_finite_nat ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_1214_in__listsI,axiom,
! [Xs: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X @ A ) )
=> ( member6475149752759350353c_fm_i @ Xs @ ( lists_Epistemic_fm_i @ A ) ) ) ).
% in_listsI
thf(fact_1215_in__lists__conv__set,axiom,
! [Xs: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Xs @ ( lists_Epistemic_fm_i @ A ) )
= ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X3 @ A ) ) ) ) ).
% in_lists_conv_set
thf(fact_1216_in__listsD,axiom,
! [Xs: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Xs @ ( lists_Epistemic_fm_i @ A ) )
=> ! [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X6 @ A ) ) ) ).
% in_listsD
thf(fact_1217_lists__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ord_le1209247765174941992c_fm_i @ ( lists_Epistemic_fm_i @ A ) @ ( lists_Epistemic_fm_i @ B ) ) ) ).
% lists_mono
thf(fact_1218_lists__def2,axiom,
( lists_Epistemic_fm_i
= ( ^ [A2: set_Epistemic_fm_i] :
( collec5759350117362652307c_fm_i
@ ^ [L: list_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ L ) @ A2 ) ) ) ) ).
% lists_def2
thf(fact_1219_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( linord2614967742042102400et_nat @ A )
= nil_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1220_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A ) )
= A ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_1221_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ ( insert_nat @ X4 @ A ) )
= ( linord8961336180081300637at_nat
@ ^ [X3: nat] : X3
@ X4
@ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_1222_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X4 @ A )
=> ( ( linord2614967742042102400et_nat @ ( insert_nat @ X4 @ A ) )
= ( linord8961336180081300637at_nat
@ ^ [X3: nat] : X3
@ X4
@ ( linord2614967742042102400et_nat @ A ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_1223_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X4 @ A )
=> ( ( linord2614967742042102400et_nat @ A )
= ( linord8961336180081300637at_nat
@ ^ [X3: nat] : X3
@ X4
@ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_1224_sorted__list__of__set__nonempty,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( linord2614967742042102400et_nat @ A )
= ( cons_nat @ ( lattic8721135487736765967in_nat @ A ) @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ ( lattic8721135487736765967in_nat @ A ) @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_1225_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
= ( remove1_nat @ X4 @ ( linord2614967742042102400et_nat @ A ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_1226_in__set__remove1,axiom,
! [A3: epistemic_fm_i,B3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( A3 != B3 )
=> ( ( member6642669606046002379c_fm_i @ A3 @ ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ B3 @ Xs ) ) )
= ( member6642669606046002379c_fm_i @ A3 @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1227_Min_Obounded__iff,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X4 @ ( lattic8721135487736765967in_nat @ A ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X4 @ X3 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_1228_Min_OcoboundedI,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ A3 ) ) ) ).
% Min.coboundedI
thf(fact_1229_Min__eqI,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ A )
=> ( ord_less_eq_nat @ X4 @ Y5 ) )
=> ( ( member_nat @ X4 @ A )
=> ( ( lattic8721135487736765967in_nat @ A )
= X4 ) ) ) ) ).
% Min_eqI
thf(fact_1230_Min__le,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X4 @ A )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X4 ) ) ) ).
% Min_le
thf(fact_1231_set__remove1__subset,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ X4 @ Xs ) ) @ ( set_Epistemic_fm_i2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1232_remove1__idem,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remove8001893566138066622c_fm_i @ X4 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1233_notin__set__remove1,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i,Y3: epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ Y3 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1234_Min__in,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A ) @ A ) ) ) ).
% Min_in
thf(fact_1235_Min__eq__iff,axiom,
! [A: set_nat,M5: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A )
= M5 )
= ( ( member_nat @ M5 @ A )
& ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ M5 @ X3 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1236_Min__le__iff,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X4 )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ X4 ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_1237_eq__Min__iff,axiom,
! [A: set_nat,M5: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M5
= ( lattic8721135487736765967in_nat @ A ) )
= ( ( member_nat @ M5 @ A )
& ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ M5 @ X3 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1238_Min_OboundedE,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X4 @ ( lattic8721135487736765967in_nat @ A ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A )
=> ( ord_less_eq_nat @ X4 @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1239_Min_OboundedI,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ X4 @ A4 ) )
=> ( ord_less_eq_nat @ X4 @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_1240_Min__insert2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ A )
=> ( ord_less_eq_nat @ A3 @ B4 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ A3 @ A ) )
= A3 ) ) ) ).
% Min_insert2
thf(fact_1241_Min__antimono,axiom,
! [M: set_nat,N2: set_nat] :
( ( ord_less_eq_set_nat @ M @ N2 )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N2 ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_1242_Min_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_1243_dual__Max,axiom,
( ( lattices_Max_nat
@ ^ [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X3 ) )
= lattic8721135487736765967in_nat ) ).
% dual_Max
thf(fact_1244_distinct_Osimps_I2_J,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( distin5379319190628067867c_fm_i @ ( cons_Epistemic_fm_i @ X4 @ Xs ) )
= ( ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( distin5379319190628067867c_fm_i @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1245_finite__distinct__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= A )
& ( distinct_nat @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_1246_someI2__bex,axiom,
! [A: set_Epistemic_fm_i,P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ? [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A )
& ( P2 @ X6 ) )
=> ( ! [X: epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ X @ A )
& ( P2 @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoic6950144925765728886c_fm_i
@ ^ [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_1247_some__in__eq,axiom,
! [A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i
@ ( fChoic6950144925765728886c_fm_i
@ ^ [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ A ) )
@ A )
= ( A != bot_bo4194595901900360558c_fm_i ) ) ).
% some_in_eq
thf(fact_1248_finite__lists__distinct__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= N )
& ( distinct_nat @ Xs3 )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_1249_finite__lists__distinct__length__eq,axiom,
! [A: set_Epistemic_fm_i,N: nat] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3106398504192302929c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [Xs3: list_Epistemic_fm_i] :
( ( ( size_s3357407047445144980c_fm_i @ Xs3 )
= N )
& ( distin5379319190628067867c_fm_i @ Xs3 )
& ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs3 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_1250_finite__lists__length__le,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1251_finite__lists__length__le,axiom,
! [A: set_Epistemic_fm_i,N: nat] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3106398504192302929c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [Xs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs3 ) @ A )
& ( ord_less_eq_nat @ ( size_s3357407047445144980c_fm_i @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1252_finite__lists__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A )
& ( ( size_size_list_nat @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1253_finite__lists__length__eq,axiom,
! [A: set_Epistemic_fm_i,N: nat] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3106398504192302929c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [Xs3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs3 ) @ A )
& ( ( size_s3357407047445144980c_fm_i @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1254_set__n__lists,axiom,
! [N: nat,Xs: list_Epistemic_fm_i] :
( ( set_li9010496130497932989c_fm_i @ ( n_list3185984609264368688c_fm_i @ N @ Xs ) )
= ( collec5759350117362652307c_fm_i
@ ^ [Ys: list_Epistemic_fm_i] :
( ( ( size_s3357407047445144980c_fm_i @ Ys )
= N )
& ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ys ) @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1255_finite__Collect__le__nat,axiom,
! [K4: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K4 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1256_nlists__def,axiom,
( cardin549358664437198940c_fm_i
= ( ^ [A2: set_Epistemic_fm_i,N3: nat] :
( collec5759350117362652307c_fm_i
@ ^ [L: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ L ) @ A2 )
& ( ( size_s3357407047445144980c_fm_i @ L )
= N3 ) ) ) ) ) ).
% nlists_def
thf(fact_1257_subset__subseqs,axiom,
! [X5: set_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X5 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member1104366573291651755c_fm_i @ X5 @ ( image_4310783338035737183c_fm_i @ set_Epistemic_fm_i2 @ ( set_li9010496130497932989c_fm_i @ ( subseq859285874047815415c_fm_i @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1258_finite__less__ub,axiom,
! [F: nat > nat,U3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U3 ) ) ) ) ).
% finite_less_ub
thf(fact_1259_Sup__bot__conv_I1_J,axiom,
! [A: set_Product_unit] :
( ( ( comple4687483117567863418t_unit @ A )
= bot_bot_Product_unit )
= ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( X3 = bot_bot_Product_unit ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1260_Sup__bot__conv_I2_J,axiom,
! [A: set_Product_unit] :
( ( bot_bot_Product_unit
= ( comple4687483117567863418t_unit @ A ) )
= ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( X3 = bot_bot_Product_unit ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1261_UN__I,axiom,
! [A3: epistemic_fm_i,A: set_Epistemic_fm_i,B3: epistemic_fm_i,B: epistemic_fm_i > set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ A )
=> ( ( member6642669606046002379c_fm_i @ B3 @ ( B @ A3 ) )
=> ( member6642669606046002379c_fm_i @ B3 @ ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1262_Sup__empty,axiom,
( ( comple4687483117567863418t_unit @ bot_bo3957492148770167129t_unit )
= bot_bot_Product_unit ) ).
% Sup_empty
thf(fact_1263_finite__UN__I,axiom,
! [A: set_Epistemic_fm_i,B: epistemic_fm_i > set_nat] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ! [A4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A4 @ A )
=> ( finite_finite_nat @ ( B @ A4 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_5232765980333981283et_nat @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_1264_finite__UN__I,axiom,
! [A: set_nat,B: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( finite_finite_nat @ ( B @ A4 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_1265_SUP__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: epistemic_fm_i > set_Epistemic_fm_i,G: epistemic_fm_i > set_Epistemic_fm_i] :
( ! [I3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ I3 @ A )
=> ? [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ B )
& ( ord_le3843937902494030498c_fm_i @ ( F @ I3 ) @ ( G @ X6 ) ) ) )
=> ( ! [J: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ J @ B )
=> ? [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A )
& ( ord_le3843937902494030498c_fm_i @ ( G @ J ) @ ( F @ X6 ) ) ) )
=> ( ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ F @ A ) )
= ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1266_SUP__eqI,axiom,
! [A: set_Epistemic_fm_i,F: epistemic_fm_i > set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ! [I3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ I3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ I3 ) @ X4 ) )
=> ( ! [Y5: set_Epistemic_fm_i] :
( ! [I4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ I4 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ I4 ) @ Y5 ) )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ Y5 ) )
=> ( ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ F @ A ) )
= X4 ) ) ) ).
% SUP_eqI
thf(fact_1267_SUP__least,axiom,
! [A: set_Epistemic_fm_i,F: epistemic_fm_i > set_Epistemic_fm_i,U3: set_Epistemic_fm_i] :
( ! [I3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ I3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ I3 ) @ U3 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ F @ A ) ) @ U3 ) ) ).
% SUP_least
thf(fact_1268_SUP__upper,axiom,
! [I2: epistemic_fm_i,A: set_Epistemic_fm_i,F: epistemic_fm_i > set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ I2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ I2 ) @ ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1269_SUP__upper2,axiom,
! [I2: epistemic_fm_i,A: set_Epistemic_fm_i,U3: set_Epistemic_fm_i,F: epistemic_fm_i > set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ I2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ U3 @ ( F @ I2 ) )
=> ( ord_le3843937902494030498c_fm_i @ U3 @ ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1270_UN__upper,axiom,
! [A3: epistemic_fm_i,A: set_Epistemic_fm_i,B: epistemic_fm_i > set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( B @ A3 ) @ ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1271_UN__least,axiom,
! [A: set_Epistemic_fm_i,B: epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( B @ X ) @ C ) )
=> ( ord_le3843937902494030498c_fm_i @ ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ B @ A ) ) @ C ) ) ).
% UN_least
thf(fact_1272_UN__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: epistemic_fm_i > set_Epistemic_fm_i,G: epistemic_fm_i > set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ F @ A ) ) @ ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1273_SUP__eq__iff,axiom,
! [I5: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,F: epistemic_fm_i > set_Epistemic_fm_i] :
( ( I5 != bot_bo4194595901900360558c_fm_i )
=> ( ! [I3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ I3 @ I5 )
=> ( ord_le3843937902494030498c_fm_i @ C2 @ ( F @ I3 ) ) )
=> ( ( ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ F @ I5 ) )
= C2 )
= ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ I5 )
=> ( ( F @ X3 )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1274_Sup__inf,axiom,
! [B: set_Product_unit,A3: product_unit] :
( ( inf_inf_Product_unit @ ( comple4687483117567863418t_unit @ B ) @ A3 )
= ( comple4687483117567863418t_unit
@ ( image_405062704495631173t_unit
@ ^ [B5: product_unit] : ( inf_inf_Product_unit @ B5 @ A3 )
@ B ) ) ) ).
% Sup_inf
thf(fact_1275_SUP__absorb,axiom,
! [K4: epistemic_fm_i,I5: set_Epistemic_fm_i,A: epistemic_fm_i > product_unit] :
( ( member6642669606046002379c_fm_i @ K4 @ I5 )
=> ( ( sup_sup_Product_unit @ ( A @ K4 ) @ ( comple4687483117567863418t_unit @ ( image_7958789640865678704t_unit @ A @ I5 ) ) )
= ( comple4687483117567863418t_unit @ ( image_7958789640865678704t_unit @ A @ I5 ) ) ) ) ).
% SUP_absorb
thf(fact_1276_UN__E,axiom,
! [B3: epistemic_fm_i,B: epistemic_fm_i > set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B3 @ ( comple8437045806746640111c_fm_i @ ( image_2913337242412145381c_fm_i @ B @ A ) ) )
=> ~ ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ~ ( member6642669606046002379c_fm_i @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
% Helper facts (8)
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( if_set_nat @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X4: set_nat,Y3: set_nat] :
( ( if_set_nat @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001t__Epistemic____Logic__Ofm_Itf__i_J_T,axiom,
! [X4: epistemic_fm_i,Y3: epistemic_fm_i] :
( ( if_Epistemic_fm_i @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Epistemic____Logic__Ofm_Itf__i_J_T,axiom,
! [X4: epistemic_fm_i,Y3: epistemic_fm_i] :
( ( if_Epistemic_fm_i @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_fChoice_1_1_fChoice_001t__Epistemic____Logic__Ofm_Itf__i_J_T,axiom,
! [P2: epistemic_fm_i > $o] :
( ( P2 @ ( fChoic6950144925765728886c_fm_i @ P2 ) )
= ( ? [X7: epistemic_fm_i] : ( P2 @ X7 ) ) ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X4: list_Epistemic_fm_i,Y3: list_Epistemic_fm_i] :
( ( if_lis3446954105117697518c_fm_i @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X4: list_Epistemic_fm_i,Y3: list_Epistemic_fm_i] :
( ( if_lis3446954105117697518c_fm_i @ $true @ X4 @ Y3 )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( stalna4517306826702924506c_fm_i
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W5: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ a @ W5 )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ W5 ) ) )
@ ^ [I: i,V2: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P4 ) @ V2 ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V2: set_Epistemic_fm_i,X3: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X3 ) @ V2 )
@ product_Unity ) ) ) ).
%------------------------------------------------------------------------------