TPTP Problem File: SLH0829^1.p
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- 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_00297_009805__6612300_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1631 ( 708 unt; 348 typ; 0 def)
% Number of atoms : 3770 (1319 equ; 0 cnn)
% Maximal formula atoms : 21 ( 2 avg)
% Number of connectives : 12676 ( 337 ~; 62 |; 254 &;10691 @)
% ( 0 <=>;1332 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 6 avg)
% Number of types : 38 ( 37 usr)
% Number of type conns : 1425 (1425 >; 0 *; 0 +; 0 <<)
% Number of symbols : 314 ( 311 usr; 20 con; 0-5 aty)
% Number of variables : 3903 ( 368 ^;3440 !; 95 ?;3903 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:12:17.715
%------------------------------------------------------------------------------
% Could-be-implicit typings (37)
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episte1560738328020401952t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__b_Mt__Set__Oset_Itf__b_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_Itf__b_J_Mt__Product____Type__Ounit_J_J,type,
episte9024493574998997535t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_Mt__Set__Oset_Itf__b_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_Itf__b_J_Mt__Product____Type__Ounit_J_J,type,
episte8559422309061589728t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_I_Eo_Mt__Set__Oset_Itf__b_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_Itf__b_J_Mt__Product____Type__Ounit_J_J,type,
episte6987468273373969862t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_Mt__Set__Oset_I_Eo_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_I_Eo_J_Mt__Product____Type__Ounit_J_J,type,
episte4442589092411611552t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_I_Eo_Mt__Set__Oset_I_Eo_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_I_Eo_J_Mt__Product____Type__Ounit_J_J,type,
episte399968703042273926t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__b_Mtf__b_Mt__Epistemic____Logic__Okripke__Okripke____ext_Itf__b_Mt__Product____Type__Ounit_J_J,type,
episte4519467831615171103t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__b_Mtf__a_Mt__Epistemic____Logic__Okripke__Okripke____ext_Itf__a_Mt__Product____Type__Ounit_J_J,type,
episte7477074710132283359t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_Mtf__b_Mt__Epistemic____Logic__Okripke__Okripke____ext_Itf__b_Mt__Product____Type__Ounit_J_J,type,
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thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_Mtf__a_Mt__Epistemic____Logic__Okripke__Okripke____ext_Itf__a_Mt__Product____Type__Ounit_J_J,type,
episte6182337868402532512t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_I_Eo_Mtf__b_Mt__Epistemic____Logic__Okripke__Okripke____ext_Itf__b_Mt__Product____Type__Ounit_J_J,type,
episte2543566300658443718t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_I_Eo_Mtf__a_Mt__Epistemic____Logic__Okripke__Okripke____ext_Itf__a_Mt__Product____Type__Ounit_J_J,type,
episte5501173179175555974t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__b_M_Eo_Mt__Epistemic____Logic__Okripke__Okripke____ext_I_Eo_Mt__Product____Type__Ounit_J_J,type,
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thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_M_Eo_Mt__Epistemic____Logic__Okripke__Okripke____ext_I_Eo_Mt__Product____Type__Ounit_J_J,type,
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thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_I_Eo_M_Eo_Mt__Epistemic____Logic__Okripke__Okripke____ext_I_Eo_Mt__Product____Type__Ounit_J_J,type,
episte8859593475650739846t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
episte1193835314949844379t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
set_Pr8948481099588399239c_fm_a: $tType ).
thf(ty_n_t__Epistemic____Logic__Okripke__Okripke____ext_Itf__b_Mt__Product____Type__Ounit_J,type,
episte2291185870209430742t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
set_se5208064806568342746c_fm_a: $tType ).
thf(ty_n_t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
list_Epistemic_fm_a: $tType ).
thf(ty_n_t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
set_Epistemic_fm_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_Itf__b_J_J,type,
list_set_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_set_b: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_I_Eo_J_J,type,
list_set_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Epistemic____Logic__Ofm_Itf__b_J,type,
epistemic_fm_b: $tType ).
thf(ty_n_t__Epistemic____Logic__Ofm_Itf__a_J,type,
epistemic_fm_a: $tType ).
thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__Epistemic____Logic__Ofm_I_Eo_J,type,
epistemic_fm_o: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__List__Olist_Itf__b_J,type,
list_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (311)
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
bNF_Ca1305897159876240246c_fm_a: set_Epistemic_fm_a > set_Pr8948481099588399239c_fm_a ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_I_Eo_J,type,
boolea379910186789422830_set_o: ( set_o > set_o > set_o ) > ( set_o > set_o > set_o ) > ( set_o > set_o ) > set_o > set_o > $o ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_Itf__b_J,type,
boolea6678413353003181397_set_b: ( set_b > set_b > set_b ) > ( set_b > set_b > set_b ) > ( set_b > set_b ) > set_b > set_b > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_Eo_J,type,
comple3063163877087187839_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__b_J,type,
comple6135023382983342438_set_b: set_set_b > set_b ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__b_J,type,
comple2307003614231284044_set_b: set_set_b > set_b ).
thf(sy_c_Epistemic__Logic_OAK_001_Eo,type,
epistemic_AK_o: ( epistemic_fm_o > $o ) > epistemic_fm_o > $o ).
thf(sy_c_Epistemic__Logic_OAK_001tf__a,type,
epistemic_AK_a: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAK_001tf__b,type,
epistemic_AK_b: ( epistemic_fm_b > $o ) > epistemic_fm_b > $o ).
thf(sy_c_Epistemic__Logic_OAx4_001_Eo,type,
epistemic_Ax4_o: epistemic_fm_o > $o ).
thf(sy_c_Epistemic__Logic_OAx4_001tf__a,type,
epistemic_Ax4_a: epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAx4_001tf__b,type,
epistemic_Ax4_b: epistemic_fm_b > $o ).
thf(sy_c_Epistemic__Logic_OAx5_001_Eo,type,
epistemic_Ax5_o: epistemic_fm_o > $o ).
thf(sy_c_Epistemic__Logic_OAx5_001tf__a,type,
epistemic_Ax5_a: epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAx5_001tf__b,type,
epistemic_Ax5_b: epistemic_fm_b > $o ).
thf(sy_c_Epistemic__Logic_OAxB_001_Eo,type,
epistemic_AxB_o: epistemic_fm_o > $o ).
thf(sy_c_Epistemic__Logic_OAxB_001tf__a,type,
epistemic_AxB_a: epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAxB_001tf__b,type,
epistemic_AxB_b: epistemic_fm_b > $o ).
thf(sy_c_Epistemic__Logic_OAxT_001_Eo,type,
epistemic_AxT_o: epistemic_fm_o > $o ).
thf(sy_c_Epistemic__Logic_OAxT_001tf__a,type,
epistemic_AxT_a: epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAxT_001tf__b,type,
epistemic_AxT_b: epistemic_fm_b > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__a_001_Eo_001t__Epistemic____Logic__Okripke__Okripke____ext_I_Eo_Mt__Product____Type__Ounit_J,type,
episte2052337558702062413t_unit: episte3259645218793129376t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__a_001t__Set__Oset_I_Eo_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_I_Eo_J_Mt__Product____Type__Ounit_J,type,
episte905247779971886925t_unit: episte4442589092411611552t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
episte2449151000174023629t_unit: episte1560738328020401952t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__a_001t__Set__Oset_Itf__b_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_Itf__b_J_Mt__Product____Type__Ounit_J,type,
episte3789892348602042125t_unit: episte8559422309061589728t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__a_001tf__a_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__a_Mt__Product____Type__Ounit_J,type,
episte2339904321507024205t_unit: episte6182337868402532512t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__a_001tf__b_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__b_Mt__Product____Type__Ounit_J,type,
episte6418945320598494989t_unit: episte3224730989885420256t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oconsistent_001tf__a,type,
episte2285483198712856226tent_a: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Oeval_001_Eo,type,
epistemic_eval_o: ( list_char > $o ) > ( epistemic_fm_o > $o ) > epistemic_fm_o > $o ).
thf(sy_c_Epistemic__Logic_Oeval_001tf__a,type,
epistemic_eval_a: ( list_char > $o ) > ( epistemic_fm_a > $o ) > epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Oeval_001tf__b,type,
epistemic_eval_b: ( list_char > $o ) > ( epistemic_fm_b > $o ) > epistemic_fm_b > $o ).
thf(sy_c_Epistemic__Logic_Ofm_OCon_001_Eo,type,
epistemic_Con_o: epistemic_fm_o > epistemic_fm_o > epistemic_fm_o ).
thf(sy_c_Epistemic__Logic_Ofm_OCon_001tf__a,type,
epistemic_Con_a: epistemic_fm_a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OCon_001tf__b,type,
epistemic_Con_b: epistemic_fm_b > epistemic_fm_b > epistemic_fm_b ).
thf(sy_c_Epistemic__Logic_Ofm_ODis_001_Eo,type,
epistemic_Dis_o: epistemic_fm_o > epistemic_fm_o > epistemic_fm_o ).
thf(sy_c_Epistemic__Logic_Ofm_ODis_001tf__a,type,
epistemic_Dis_a: epistemic_fm_a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_ODis_001tf__b,type,
epistemic_Dis_b: epistemic_fm_b > epistemic_fm_b > epistemic_fm_b ).
thf(sy_c_Epistemic__Logic_Ofm_OFF_001_Eo,type,
epistemic_FF_o: epistemic_fm_o ).
thf(sy_c_Epistemic__Logic_Ofm_OFF_001tf__a,type,
epistemic_FF_a: epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OFF_001tf__b,type,
epistemic_FF_b: epistemic_fm_b ).
thf(sy_c_Epistemic__Logic_Ofm_OImp_001_Eo,type,
epistemic_Imp_o: epistemic_fm_o > epistemic_fm_o > epistemic_fm_o ).
thf(sy_c_Epistemic__Logic_Ofm_OImp_001tf__a,type,
epistemic_Imp_a: epistemic_fm_a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OImp_001tf__b,type,
epistemic_Imp_b: epistemic_fm_b > epistemic_fm_b > epistemic_fm_b ).
thf(sy_c_Epistemic__Logic_Ofm_OK_001_Eo,type,
epistemic_K_o: $o > epistemic_fm_o > epistemic_fm_o ).
thf(sy_c_Epistemic__Logic_Ofm_OK_001tf__a,type,
epistemic_K_a: a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OK_001tf__b,type,
epistemic_K_b: b > epistemic_fm_b > epistemic_fm_b ).
thf(sy_c_Epistemic__Logic_Ofm_OPro_001_Eo,type,
epistemic_Pro_o: list_char > epistemic_fm_o ).
thf(sy_c_Epistemic__Logic_Ofm_OPro_001tf__a,type,
epistemic_Pro_a: list_char > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OPro_001tf__b,type,
epistemic_Pro_b: list_char > epistemic_fm_b ).
thf(sy_c_Epistemic__Logic_Ofm_Opred__fm_001tf__a,type,
epistemic_pred_fm_a: ( a > $o ) > epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Opred__fm_001tf__b,type,
epistemic_pred_fm_b: ( b > $o ) > epistemic_fm_b > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001tf__a_001tf__a,type,
epistemic_rel_fm_a_a: ( a > a > $o ) > epistemic_fm_a > epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001tf__b_001tf__b,type,
epistemic_rel_fm_b_b: ( b > b > $o ) > epistemic_fm_b > epistemic_fm_b > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001_Eo,type,
epistemic_set_fm_o: epistemic_fm_o > set_o ).
thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001tf__a,type,
epistemic_set_fm_a: epistemic_fm_a > set_a ).
thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001tf__b,type,
epistemic_set_fm_b: epistemic_fm_b > set_b ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060K_062_001_Eo_001_Eo_001t__Epistemic____Logic__Okripke__Okripke____ext_I_Eo_Mt__Product____Type__Ounit_J,type,
episte5693609662886803661t_unit: episte8859593475650739846t_unit > $o > $o > set_o ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060K_062_001_Eo_001t__Set__Oset_Itf__b_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_Itf__b_J_Mt__Product____Type__Ounit_J,type,
episte6509188576646022285t_unit: episte6987468273373969862t_unit > $o > set_b > set_set_b ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060K_062_001_Eo_001tf__b_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__b_Mt__Product____Type__Ounit_J,type,
episte3109837587766369421t_unit: episte2543566300658443718t_unit > $o > b > set_b ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060K_062_001tf__a_001_Eo_001t__Epistemic____Logic__Okripke__Okripke____ext_I_Eo_Mt__Product____Type__Ounit_J,type,
episte6681579530201160167t_unit: episte3259645218793129376t_unit > a > $o > set_o ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060K_062_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
episte6250069432388174439t_unit: episte1560738328020401952t_unit > a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060K_062_001tf__a_001t__Set__Oset_Itf__b_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_Itf__b_J_Mt__Product____Type__Ounit_J,type,
episte4990963547661007783t_unit: episte8559422309061589728t_unit > a > set_b > set_set_b ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060K_062_001tf__a_001tf__b_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__b_Mt__Product____Type__Ounit_J,type,
episte1071096959727133607t_unit: episte3224730989885420256t_unit > a > b > set_b ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060K_062_001tf__b_001_Eo_001t__Epistemic____Logic__Okripke__Okripke____ext_I_Eo_Mt__Product____Type__Ounit_J,type,
episte4738352111056093350t_unit: episte6655095512717024223t_unit > b > $o > set_o ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060K_062_001tf__b_001t__Set__Oset_Itf__b_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_Itf__b_J_Mt__Product____Type__Ounit_J,type,
episte7829654723686370150t_unit: episte9024493574998997535t_unit > b > set_b > set_set_b ).
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thf(sy_c_Fun_Oinj__on_001tf__b_001_Eo,type,
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thf(sy_c_Fun_Oinj__on_001tf__b_001tf__b,type,
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thf(sy_c_HOL_OThe_001_Eo,type,
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thf(sy_c_HOL_OThe_001tf__b,type,
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thf(sy_c_HOL_OUniq_001_Eo,type,
uniq_o: ( $o > $o ) > $o ).
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thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__b_M_Eo_J,type,
ord_less_eq_b_o: ( b > $o ) > ( b > $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__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
ord_le3275665582123262618c_fm_a: set_Epistemic_fm_a > set_Epistemic_fm_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
ord_le3795704787696855135_set_b: set_set_b > set_set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
top_top_o_o: $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
top_to7796028867103199306c_fm_a: set_Epistemic_fm_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
top_top_set_set_o: set_set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__b_J,type,
top_top_set_b: set_b ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
collec4904205152690461189c_fm_a: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_Eo_J,type,
collect_set_o: ( set_o > $o ) > set_set_o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
collec2519470961442302949c_fm_a: ( set_Epistemic_fm_a > $o ) > set_se5208064806568342746c_fm_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__b_J,type,
collect_set_b: ( set_b > $o ) > set_set_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_OPow_001_Eo,type,
pow_o: set_o > set_set_o ).
thf(sy_c_Set_OPow_001tf__b,type,
pow_b: set_b > set_set_b ).
thf(sy_c_Set_Obind_001_Eo_001_Eo,type,
bind_o_o: set_o > ( $o > set_o ) > set_o ).
thf(sy_c_Set_Ofilter_001tf__b,type,
filter_b: ( b > $o ) > set_b > set_b ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_Itf__b_J,type,
image_o_set_b: ( $o > set_b ) > set_o > set_set_b ).
thf(sy_c_Set_Oimage_001_Eo_001tf__b,type,
image_o_b: ( $o > b ) > set_o > set_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
image_set_b_set_b: ( set_b > set_b ) > set_set_b > set_set_b ).
thf(sy_c_Set_Oimage_001tf__b_001_Eo,type,
image_b_o: ( b > $o ) > set_b > set_o ).
thf(sy_c_Set_Oimage_001tf__b_001t__Set__Oset_Itf__b_J,type,
image_b_set_b: ( b > set_b ) > set_b > set_set_b ).
thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
insert7817948963269374698c_fm_a: epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
insert_set_o: set_o > set_set_o > set_set_o ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__b_J,type,
insert_set_b: set_b > set_set_b > set_set_b ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set_Ois__empty_001_Eo,type,
is_empty_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001_Eo,type,
is_singleton_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001tf__b,type,
is_singleton_b: set_b > $o ).
thf(sy_c_Set_Opairwise_001_Eo,type,
pairwise_o: ( $o > $o > $o ) > set_o > $o ).
thf(sy_c_Set_Opairwise_001tf__b,type,
pairwise_b: ( b > b > $o ) > set_b > $o ).
thf(sy_c_Set_Oremove_001_Eo,type,
remove_o: $o > set_o > set_o ).
thf(sy_c_Set_Oremove_001tf__b,type,
remove_b: b > set_b > set_b ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Ovimage_001_Eo_001_Eo,type,
vimage_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001tf__b,type,
vimage_o_b: ( $o > b ) > set_b > set_o ).
thf(sy_c_Set_Ovimage_001tf__b_001_Eo,type,
vimage_b_o: ( b > $o ) > set_o > set_b ).
thf(sy_c_Set_Ovimage_001tf__b_001tf__b,type,
vimage_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Stalnaker__Logic_OAx__2_001_Eo,type,
stalnaker_Ax_2_o: epistemic_fm_o > $o ).
thf(sy_c_Stalnaker__Logic_OAx__2_001tf__a,type,
stalnaker_Ax_2_a: epistemic_fm_a > $o ).
thf(sy_c_Stalnaker__Logic_OAx__2_001tf__b,type,
stalnaker_Ax_2_b: epistemic_fm_b > $o ).
thf(sy_c_Stalnaker__Logic_Oconjunct_001tf__a,type,
stalnaker_conjunct_a: list_Epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Stalnaker__Logic_Oweakly__directed_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
stalna9100096441501500634c_fm_a: episte1560738328020401952t_unit > $o ).
thf(sy_c_Stalnaker__Logic_Oweakly__directed_001tf__a_001tf__b,type,
stalna7560619637984394463ed_a_b: episte3224730989885420256t_unit > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
member6642669571620171971c_fm_a: epistemic_fm_a > set_Epistemic_fm_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
member536094252920883875c_fm_a: set_Epistemic_fm_a > set_se5208064806568342746c_fm_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_M,type,
m: episte3224730989885420256t_unit ).
thf(sy_v_i,type,
i: a ).
thf(sy_v_p,type,
p: epistemic_fm_a ).
thf(sy_v_w,type,
w: b ).
% Relevant facts (1279)
thf(fact_0_assms_I1_J,axiom,
stalna7560619637984394463ed_a_b @ m ).
% assms(1)
thf(fact_1_assms_I2_J,axiom,
member_b @ w @ ( episte1782384855165018035t_unit @ m ) ).
% assms(2)
thf(fact_2__092_060open_062_092_060exists_062v_092_060in_062_092_060W_062_AM_A_092_060inter_062_A_092_060K_062_AM_Ai_Aw_O_AM_M_Av_A_092_060Turnstile_062_AK_Ai_Ap_092_060close_062,axiom,
? [X: b] :
( ( member_b @ X @ ( inf_inf_set_b @ ( episte1782384855165018035t_unit @ m ) @ ( episte1071096959727133607t_unit @ m @ i @ w ) ) )
& ( episte295617885132580261cs_a_b @ m @ X @ ( epistemic_K_a @ i @ p ) ) ) ).
% \<open>\<exists>v\<in>\<W> M \<inter> \<K> M i w. M, v \<Turnstile> K i p\<close>
thf(fact_3_weakly__directed__def,axiom,
( stalna9100096441501500634c_fm_a
= ( ^ [M: episte1560738328020401952t_unit] :
! [I: a,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M ) )
=> ! [Y: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y @ ( episte8072386903178013299t_unit @ M ) )
=> ! [Z: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( episte8072386903178013299t_unit @ M ) )
=> ( ( ( member536094252920883875c_fm_a @ Z @ ( episte6250069432388174439t_unit @ M @ I @ X2 ) )
& ( member536094252920883875c_fm_a @ Y @ ( episte6250069432388174439t_unit @ M @ I @ X2 ) ) )
=> ? [Aa: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Aa @ ( episte8072386903178013299t_unit @ M ) )
& ( member536094252920883875c_fm_a @ Aa @ ( episte6250069432388174439t_unit @ M @ I @ Z ) )
& ( member536094252920883875c_fm_a @ Aa @ ( episte6250069432388174439t_unit @ M @ I @ Y ) ) ) ) ) ) ) ) ) ).
% weakly_directed_def
thf(fact_4_weakly__directed__def,axiom,
( stalna7560619637984394463ed_a_b
= ( ^ [M: episte3224730989885420256t_unit] :
! [I: a,X2: b] :
( ( member_b @ X2 @ ( episte1782384855165018035t_unit @ M ) )
=> ! [Y: b] :
( ( member_b @ Y @ ( episte1782384855165018035t_unit @ M ) )
=> ! [Z: b] :
( ( member_b @ Z @ ( episte1782384855165018035t_unit @ M ) )
=> ( ( ( member_b @ Z @ ( episte1071096959727133607t_unit @ M @ I @ X2 ) )
& ( member_b @ Y @ ( episte1071096959727133607t_unit @ M @ I @ X2 ) ) )
=> ? [Aa: b] :
( ( member_b @ Aa @ ( episte1782384855165018035t_unit @ M ) )
& ( member_b @ Aa @ ( episte1071096959727133607t_unit @ M @ I @ Z ) )
& ( member_b @ Aa @ ( episte1071096959727133607t_unit @ M @ I @ Y ) ) ) ) ) ) ) ) ) ).
% weakly_directed_def
thf(fact_5__092_060open_062M_M_Aw_A_092_060Turnstile_062_AL_Ai_A_IK_Ai_Ap_J_092_060close_062,axiom,
episte295617885132580261cs_a_b @ m @ w @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ p ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) ).
% \<open>M, w \<Turnstile> L i (K i p)\<close>
thf(fact_6_IntI,axiom,
! [C: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ A )
=> ( ( member536094252920883875c_fm_a @ C @ B )
=> ( member536094252920883875c_fm_a @ C @ ( inf_in1884693029477671368c_fm_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_7_IntI,axiom,
! [C: set_o,A: set_set_o,B: set_set_o] :
( ( member_set_o @ C @ A )
=> ( ( member_set_o @ C @ B )
=> ( member_set_o @ C @ ( inf_inf_set_set_o @ A @ B ) ) ) ) ).
% IntI
thf(fact_8_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_9_IntI,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ A )
=> ( ( member_o @ C @ B )
=> ( member_o @ C @ ( inf_inf_set_o @ A @ B ) ) ) ) ).
% IntI
thf(fact_10_IntI,axiom,
! [C: set_b,A: set_set_b,B: set_set_b] :
( ( member_set_b @ C @ A )
=> ( ( member_set_b @ C @ B )
=> ( member_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_11_IntI,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ A )
=> ( ( member_b @ C @ B )
=> ( member_b @ C @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_12_Int__iff,axiom,
! [C: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( inf_in1884693029477671368c_fm_a @ A @ B ) )
= ( ( member536094252920883875c_fm_a @ C @ A )
& ( member536094252920883875c_fm_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_13_Int__iff,axiom,
! [C: set_o,A: set_set_o,B: set_set_o] :
( ( member_set_o @ C @ ( inf_inf_set_set_o @ A @ B ) )
= ( ( member_set_o @ C @ A )
& ( member_set_o @ C @ B ) ) ) ).
% Int_iff
thf(fact_14_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_15_Int__iff,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
= ( ( member_o @ C @ A )
& ( member_o @ C @ B ) ) ) ).
% Int_iff
thf(fact_16_Int__iff,axiom,
! [C: set_b,A: set_set_b,B: set_set_b] :
( ( member_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) )
= ( ( member_set_b @ C @ A )
& ( member_set_b @ C @ B ) ) ) ).
% Int_iff
thf(fact_17_Int__iff,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
= ( ( member_b @ C @ A )
& ( member_b @ C @ B ) ) ) ).
% Int_iff
thf(fact_18_inf_Oidem,axiom,
! [A2: b > $o] :
( ( inf_inf_b_o @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_19_inf_Oidem,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_20_inf_Oidem,axiom,
! [A2: set_set_b] :
( ( inf_inf_set_set_b @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_21_inf_Oidem,axiom,
! [A2: $o] :
( ( inf_inf_o @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_22_inf_Oidem,axiom,
! [A2: set_b] :
( ( inf_inf_set_b @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_23_inf__idem,axiom,
! [X3: b > $o] :
( ( inf_inf_b_o @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_24_inf__idem,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_25_inf__idem,axiom,
! [X3: set_set_b] :
( ( inf_inf_set_set_b @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_26_inf__idem,axiom,
! [X3: $o] :
( ( inf_inf_o @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_27_inf__idem,axiom,
! [X3: set_b] :
( ( inf_inf_set_b @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_28_inf_Oleft__idem,axiom,
! [A2: b > $o,B2: b > $o] :
( ( inf_inf_b_o @ A2 @ ( inf_inf_b_o @ A2 @ B2 ) )
= ( inf_inf_b_o @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_29_inf_Oleft__idem,axiom,
! [A2: set_o,B2: set_o] :
( ( inf_inf_set_o @ A2 @ ( inf_inf_set_o @ A2 @ B2 ) )
= ( inf_inf_set_o @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_30_inf_Oleft__idem,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( inf_inf_set_set_b @ A2 @ ( inf_inf_set_set_b @ A2 @ B2 ) )
= ( inf_inf_set_set_b @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_31_inf_Oleft__idem,axiom,
! [A2: $o,B2: $o] :
( ( inf_inf_o @ A2 @ ( inf_inf_o @ A2 @ B2 ) )
= ( inf_inf_o @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_32_inf_Oleft__idem,axiom,
! [A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ A2 @ B2 ) )
= ( inf_inf_set_b @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_33_inf__left__idem,axiom,
! [X3: b > $o,Y2: b > $o] :
( ( inf_inf_b_o @ X3 @ ( inf_inf_b_o @ X3 @ Y2 ) )
= ( inf_inf_b_o @ X3 @ Y2 ) ) ).
% inf_left_idem
thf(fact_34_inf__left__idem,axiom,
! [X3: set_o,Y2: set_o] :
( ( inf_inf_set_o @ X3 @ ( inf_inf_set_o @ X3 @ Y2 ) )
= ( inf_inf_set_o @ X3 @ Y2 ) ) ).
% inf_left_idem
thf(fact_35_inf__left__idem,axiom,
! [X3: set_set_b,Y2: set_set_b] :
( ( inf_inf_set_set_b @ X3 @ ( inf_inf_set_set_b @ X3 @ Y2 ) )
= ( inf_inf_set_set_b @ X3 @ Y2 ) ) ).
% inf_left_idem
thf(fact_36_inf__left__idem,axiom,
! [X3: $o,Y2: $o] :
( ( inf_inf_o @ X3 @ ( inf_inf_o @ X3 @ Y2 ) )
= ( inf_inf_o @ X3 @ Y2 ) ) ).
% inf_left_idem
thf(fact_37_inf__left__idem,axiom,
! [X3: set_b,Y2: set_b] :
( ( inf_inf_set_b @ X3 @ ( inf_inf_set_b @ X3 @ Y2 ) )
= ( inf_inf_set_b @ X3 @ Y2 ) ) ).
% inf_left_idem
thf(fact_38_inf_Oright__idem,axiom,
! [A2: b > $o,B2: b > $o] :
( ( inf_inf_b_o @ ( inf_inf_b_o @ A2 @ B2 ) @ B2 )
= ( inf_inf_b_o @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_39_inf_Oright__idem,axiom,
! [A2: set_o,B2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_o @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_40_inf_Oright__idem,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( inf_inf_set_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_set_b @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_41_inf_Oright__idem,axiom,
! [A2: $o,B2: $o] :
( ( inf_inf_o @ ( inf_inf_o @ A2 @ B2 ) @ B2 )
= ( inf_inf_o @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_42_inf_Oright__idem,axiom,
! [A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_b @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_43_inf__right__idem,axiom,
! [X3: b > $o,Y2: b > $o] :
( ( inf_inf_b_o @ ( inf_inf_b_o @ X3 @ Y2 ) @ Y2 )
= ( inf_inf_b_o @ X3 @ Y2 ) ) ).
% inf_right_idem
thf(fact_44_inf__right__idem,axiom,
! [X3: set_o,Y2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ X3 @ Y2 ) @ Y2 )
= ( inf_inf_set_o @ X3 @ Y2 ) ) ).
% inf_right_idem
thf(fact_45_inf__right__idem,axiom,
! [X3: set_set_b,Y2: set_set_b] :
( ( inf_inf_set_set_b @ ( inf_inf_set_set_b @ X3 @ Y2 ) @ Y2 )
= ( inf_inf_set_set_b @ X3 @ Y2 ) ) ).
% inf_right_idem
thf(fact_46_inf__right__idem,axiom,
! [X3: $o,Y2: $o] :
( ( inf_inf_o @ ( inf_inf_o @ X3 @ Y2 ) @ Y2 )
= ( inf_inf_o @ X3 @ Y2 ) ) ).
% inf_right_idem
thf(fact_47_inf__right__idem,axiom,
! [X3: set_b,Y2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) @ Y2 )
= ( inf_inf_set_b @ X3 @ Y2 ) ) ).
% inf_right_idem
thf(fact_48_inf__apply,axiom,
( inf_inf_b_o
= ( ^ [F: b > $o,G: b > $o,X2: b] : ( inf_inf_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% inf_apply
thf(fact_49_semantics_Osimps_I6_J,axiom,
! [M2: episte4519467831615171103t_unit,W: b,I2: b,P: epistemic_fm_b] :
( ( episte6731534340014680036cs_b_b @ M2 @ W @ ( epistemic_K_b @ I2 @ P ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ ( inf_inf_set_b @ ( episte7096680817873233778t_unit @ M2 ) @ ( episte6385392922435349350t_unit @ M2 @ I2 @ W ) ) )
=> ( episte6731534340014680036cs_b_b @ M2 @ X2 @ P ) ) ) ) ).
% semantics.simps(6)
thf(fact_50_semantics_Osimps_I6_J,axiom,
! [M2: episte2543566300658443718t_unit,W: b,I2: $o,P: epistemic_fm_o] :
( ( episte1021116911515925003cs_o_b @ M2 @ W @ ( epistemic_K_o @ I2 @ P ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ ( inf_inf_set_b @ ( episte9142089827432095385t_unit @ M2 ) @ ( episte3109837587766369421t_unit @ M2 @ I2 @ W ) ) )
=> ( episte1021116911515925003cs_o_b @ M2 @ X2 @ P ) ) ) ) ).
% semantics.simps(6)
thf(fact_51_semantics_Osimps_I6_J,axiom,
! [M2: episte3259645218793129376t_unit,W: $o,I2: a,P: epistemic_fm_a] :
( ( episte1100597067981672766cs_a_o @ M2 @ W @ ( epistemic_K_a @ I2 @ P ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ ( inf_inf_set_o @ ( episte876160520765907443t_unit @ M2 ) @ ( episte6681579530201160167t_unit @ M2 @ I2 @ W ) ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ X2 @ P ) ) ) ) ).
% semantics.simps(6)
thf(fact_52_semantics_Osimps_I6_J,axiom,
! [M2: episte6655095512717024223t_unit,W: $o,I2: b,P: epistemic_fm_b] :
( ( episte7424746705372054269cs_b_o @ M2 @ W @ ( epistemic_K_b @ I2 @ P ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ ( inf_inf_set_o @ ( episte8156305138475616434t_unit @ M2 ) @ ( episte4738352111056093350t_unit @ M2 @ I2 @ W ) ) )
=> ( episte7424746705372054269cs_b_o @ M2 @ X2 @ P ) ) ) ) ).
% semantics.simps(6)
thf(fact_53_semantics_Osimps_I6_J,axiom,
! [M2: episte8859593475650739846t_unit,W: $o,I2: $o,P: epistemic_fm_o] :
( ( episte4695001281782364324cs_o_o @ M2 @ W @ ( epistemic_K_o @ I2 @ P ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ ( inf_inf_set_o @ ( episte1038022579713443545t_unit @ M2 ) @ ( episte5693609662886803661t_unit @ M2 @ I2 @ W ) ) )
=> ( episte4695001281782364324cs_o_o @ M2 @ X2 @ P ) ) ) ) ).
% semantics.simps(6)
thf(fact_54_semantics_Osimps_I6_J,axiom,
! [M2: episte8559422309061589728t_unit,W: set_b,I2: a,P: epistemic_fm_a] :
( ( episte586216548244424325_set_b @ M2 @ W @ ( epistemic_K_a @ I2 @ P ) )
= ( ! [X2: set_b] :
( ( member_set_b @ X2 @ ( inf_inf_set_set_b @ ( episte2726209618768418739t_unit @ M2 ) @ ( episte4990963547661007783t_unit @ M2 @ I2 @ W ) ) )
=> ( episte586216548244424325_set_b @ M2 @ X2 @ P ) ) ) ) ).
% semantics.simps(6)
thf(fact_55_semantics_Osimps_I6_J,axiom,
! [M2: episte9024493574998997535t_unit,W: set_b,I2: b,P: epistemic_fm_b] :
( ( episte1930681585053912260_set_b @ M2 @ W @ ( epistemic_K_b @ I2 @ P ) )
= ( ! [X2: set_b] :
( ( member_set_b @ X2 @ ( inf_inf_set_set_b @ ( episte5564900794793781106t_unit @ M2 ) @ ( episte7829654723686370150t_unit @ M2 @ I2 @ W ) ) )
=> ( episte1930681585053912260_set_b @ M2 @ X2 @ P ) ) ) ) ).
% semantics.simps(6)
thf(fact_56_semantics_Osimps_I6_J,axiom,
! [M2: episte6987468273373969862t_unit,W: set_b,I2: $o,P: epistemic_fm_o] :
( ( episte2297750978567635691_set_b @ M2 @ W @ ( epistemic_K_o @ I2 @ P ) )
= ( ! [X2: set_b] :
( ( member_set_b @ X2 @ ( inf_inf_set_set_b @ ( episte6857713297613678233t_unit @ M2 ) @ ( episte6509188576646022285t_unit @ M2 @ I2 @ W ) ) )
=> ( episte2297750978567635691_set_b @ M2 @ X2 @ P ) ) ) ) ).
% semantics.simps(6)
thf(fact_57_semantics_Osimps_I6_J,axiom,
! [M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I2: a,P: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M2 @ W @ ( epistemic_K_a @ I2 @ P ) )
= ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( inf_in1884693029477671368c_fm_a @ ( episte8072386903178013299t_unit @ M2 ) @ ( episte6250069432388174439t_unit @ M2 @ I2 @ W ) ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) ) ).
% semantics.simps(6)
thf(fact_58_semantics_Osimps_I6_J,axiom,
! [M2: episte3224730989885420256t_unit,W: b,I2: a,P: epistemic_fm_a] :
( ( episte295617885132580261cs_a_b @ M2 @ W @ ( epistemic_K_a @ I2 @ P ) )
= ( ! [X2: b] :
( ( member_b @ X2 @ ( inf_inf_set_b @ ( episte1782384855165018035t_unit @ M2 ) @ ( episte1071096959727133607t_unit @ M2 @ I2 @ W ) ) )
=> ( episte295617885132580261cs_a_b @ M2 @ X2 @ P ) ) ) ) ).
% semantics.simps(6)
thf(fact_59_fm_Oinject_I4_J,axiom,
! [X51: epistemic_fm_b,X52: epistemic_fm_b,Y51: epistemic_fm_b,Y52: epistemic_fm_b] :
( ( ( epistemic_Imp_b @ X51 @ X52 )
= ( epistemic_Imp_b @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_60_fm_Oinject_I4_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( ( epistemic_Imp_a @ X51 @ X52 )
= ( epistemic_Imp_a @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_61_fm_Oinject_I5_J,axiom,
! [X61: b,X62: epistemic_fm_b,Y61: b,Y62: epistemic_fm_b] :
( ( ( epistemic_K_b @ X61 @ X62 )
= ( epistemic_K_b @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_62_fm_Oinject_I5_J,axiom,
! [X61: $o,X62: epistemic_fm_o,Y61: $o,Y62: epistemic_fm_o] :
( ( ( epistemic_K_o @ X61 @ X62 )
= ( epistemic_K_o @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_63_fm_Oinject_I5_J,axiom,
! [X61: a,X62: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
( ( ( epistemic_K_a @ X61 @ X62 )
= ( epistemic_K_a @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_64_Imp__intro,axiom,
! [M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( ( episte7081087998767065248c_fm_a @ M2 @ W @ P )
=> ( episte7081087998767065248c_fm_a @ M2 @ W @ Q ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W @ ( epistemic_Imp_a @ P @ Q ) ) ) ).
% Imp_intro
thf(fact_65_Imp__intro,axiom,
! [M2: episte3224730989885420256t_unit,W: b,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( ( episte295617885132580261cs_a_b @ M2 @ W @ P )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ Q ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ ( epistemic_Imp_a @ P @ Q ) ) ) ).
% Imp_intro
thf(fact_66_fm_Odistinct_I29_J,axiom,
! [X51: epistemic_fm_o,X52: epistemic_fm_o,X61: $o,X62: epistemic_fm_o] :
( ( epistemic_Imp_o @ X51 @ X52 )
!= ( epistemic_K_o @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_67_fm_Odistinct_I29_J,axiom,
! [X51: epistemic_fm_b,X52: epistemic_fm_b,X61: b,X62: epistemic_fm_b] :
( ( epistemic_Imp_b @ X51 @ X52 )
!= ( epistemic_K_b @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_68_fm_Odistinct_I29_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Imp_a @ X51 @ X52 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_69_fm_Odistinct_I9_J,axiom,
! [X61: b,X62: epistemic_fm_b] :
( epistemic_FF_b
!= ( epistemic_K_b @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_70_fm_Odistinct_I9_J,axiom,
! [X61: $o,X62: epistemic_fm_o] :
( epistemic_FF_o
!= ( epistemic_K_o @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_71_fm_Odistinct_I9_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_72_fm_Odistinct_I7_J,axiom,
! [X51: epistemic_fm_o,X52: epistemic_fm_o] :
( epistemic_FF_o
!= ( epistemic_Imp_o @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_73_fm_Odistinct_I7_J,axiom,
! [X51: epistemic_fm_b,X52: epistemic_fm_b] :
( epistemic_FF_b
!= ( epistemic_Imp_b @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_74_fm_Odistinct_I7_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_75_semantics_Osimps_I5_J,axiom,
! [M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M2 @ W @ ( epistemic_Imp_a @ P @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M2 @ W @ P )
=> ( episte7081087998767065248c_fm_a @ M2 @ W @ Q ) ) ) ).
% semantics.simps(5)
thf(fact_76_semantics_Osimps_I5_J,axiom,
! [M2: episte3224730989885420256t_unit,W: b,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte295617885132580261cs_a_b @ M2 @ W @ ( epistemic_Imp_a @ P @ Q ) )
= ( ( episte295617885132580261cs_a_b @ M2 @ W @ P )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ Q ) ) ) ).
% semantics.simps(5)
thf(fact_77_semantics_Osimps_I1_J,axiom,
! [M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
~ ( episte7081087998767065248c_fm_a @ M2 @ W @ epistemic_FF_a ) ).
% semantics.simps(1)
thf(fact_78_semantics_Osimps_I1_J,axiom,
! [M2: episte3224730989885420256t_unit,W: b] :
~ ( episte295617885132580261cs_a_b @ M2 @ W @ epistemic_FF_a ) ).
% semantics.simps(1)
thf(fact_79_inf__fun__def,axiom,
( inf_inf_b_o
= ( ^ [F: b > $o,G: b > $o,X2: b] : ( inf_inf_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% inf_fun_def
thf(fact_80_inf__left__commute,axiom,
! [X3: b > $o,Y2: b > $o,Z2: b > $o] :
( ( inf_inf_b_o @ X3 @ ( inf_inf_b_o @ Y2 @ Z2 ) )
= ( inf_inf_b_o @ Y2 @ ( inf_inf_b_o @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_81_inf__left__commute,axiom,
! [X3: set_o,Y2: set_o,Z2: set_o] :
( ( inf_inf_set_o @ X3 @ ( inf_inf_set_o @ Y2 @ Z2 ) )
= ( inf_inf_set_o @ Y2 @ ( inf_inf_set_o @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_82_inf__left__commute,axiom,
! [X3: set_set_b,Y2: set_set_b,Z2: set_set_b] :
( ( inf_inf_set_set_b @ X3 @ ( inf_inf_set_set_b @ Y2 @ Z2 ) )
= ( inf_inf_set_set_b @ Y2 @ ( inf_inf_set_set_b @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_83_inf__left__commute,axiom,
! [X3: $o,Y2: $o,Z2: $o] :
( ( inf_inf_o @ X3 @ ( inf_inf_o @ Y2 @ Z2 ) )
= ( inf_inf_o @ Y2 @ ( inf_inf_o @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_84_inf__left__commute,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X3 @ ( inf_inf_set_b @ Y2 @ Z2 ) )
= ( inf_inf_set_b @ Y2 @ ( inf_inf_set_b @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_85_inf_Oleft__commute,axiom,
! [B2: b > $o,A2: b > $o,C: b > $o] :
( ( inf_inf_b_o @ B2 @ ( inf_inf_b_o @ A2 @ C ) )
= ( inf_inf_b_o @ A2 @ ( inf_inf_b_o @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_86_inf_Oleft__commute,axiom,
! [B2: set_o,A2: set_o,C: set_o] :
( ( inf_inf_set_o @ B2 @ ( inf_inf_set_o @ A2 @ C ) )
= ( inf_inf_set_o @ A2 @ ( inf_inf_set_o @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_87_inf_Oleft__commute,axiom,
! [B2: set_set_b,A2: set_set_b,C: set_set_b] :
( ( inf_inf_set_set_b @ B2 @ ( inf_inf_set_set_b @ A2 @ C ) )
= ( inf_inf_set_set_b @ A2 @ ( inf_inf_set_set_b @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_88_inf_Oleft__commute,axiom,
! [B2: $o,A2: $o,C: $o] :
( ( inf_inf_o @ B2 @ ( inf_inf_o @ A2 @ C ) )
= ( inf_inf_o @ A2 @ ( inf_inf_o @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_89_inf_Oleft__commute,axiom,
! [B2: set_b,A2: set_b,C: set_b] :
( ( inf_inf_set_b @ B2 @ ( inf_inf_set_b @ A2 @ C ) )
= ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_90_inf__commute,axiom,
( inf_inf_b_o
= ( ^ [X2: b > $o,Y: b > $o] : ( inf_inf_b_o @ Y @ X2 ) ) ) ).
% inf_commute
thf(fact_91_inf__commute,axiom,
( inf_inf_set_o
= ( ^ [X2: set_o,Y: set_o] : ( inf_inf_set_o @ Y @ X2 ) ) ) ).
% inf_commute
thf(fact_92_inf__commute,axiom,
( inf_inf_set_set_b
= ( ^ [X2: set_set_b,Y: set_set_b] : ( inf_inf_set_set_b @ Y @ X2 ) ) ) ).
% inf_commute
thf(fact_93_inf__commute,axiom,
( inf_inf_o
= ( ^ [X2: $o,Y: $o] : ( inf_inf_o @ Y @ X2 ) ) ) ).
% inf_commute
thf(fact_94_inf__commute,axiom,
( inf_inf_set_b
= ( ^ [X2: set_b,Y: set_b] : ( inf_inf_set_b @ Y @ X2 ) ) ) ).
% inf_commute
thf(fact_95_inf_Ocommute,axiom,
( inf_inf_b_o
= ( ^ [A3: b > $o,B3: b > $o] : ( inf_inf_b_o @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_96_inf_Ocommute,axiom,
( inf_inf_set_o
= ( ^ [A3: set_o,B3: set_o] : ( inf_inf_set_o @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_97_inf_Ocommute,axiom,
( inf_inf_set_set_b
= ( ^ [A3: set_set_b,B3: set_set_b] : ( inf_inf_set_set_b @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_98_inf_Ocommute,axiom,
( inf_inf_o
= ( ^ [A3: $o,B3: $o] : ( inf_inf_o @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_99_inf_Ocommute,axiom,
( inf_inf_set_b
= ( ^ [A3: set_b,B3: set_b] : ( inf_inf_set_b @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_100_inf__assoc,axiom,
! [X3: b > $o,Y2: b > $o,Z2: b > $o] :
( ( inf_inf_b_o @ ( inf_inf_b_o @ X3 @ Y2 ) @ Z2 )
= ( inf_inf_b_o @ X3 @ ( inf_inf_b_o @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_101_inf__assoc,axiom,
! [X3: set_o,Y2: set_o,Z2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ X3 @ Y2 ) @ Z2 )
= ( inf_inf_set_o @ X3 @ ( inf_inf_set_o @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_102_inf__assoc,axiom,
! [X3: set_set_b,Y2: set_set_b,Z2: set_set_b] :
( ( inf_inf_set_set_b @ ( inf_inf_set_set_b @ X3 @ Y2 ) @ Z2 )
= ( inf_inf_set_set_b @ X3 @ ( inf_inf_set_set_b @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_103_inf__assoc,axiom,
! [X3: $o,Y2: $o,Z2: $o] :
( ( inf_inf_o @ ( inf_inf_o @ X3 @ Y2 ) @ Z2 )
= ( inf_inf_o @ X3 @ ( inf_inf_o @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_104_inf__assoc,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) @ Z2 )
= ( inf_inf_set_b @ X3 @ ( inf_inf_set_b @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_105_inf_Oassoc,axiom,
! [A2: b > $o,B2: b > $o,C: b > $o] :
( ( inf_inf_b_o @ ( inf_inf_b_o @ A2 @ B2 ) @ C )
= ( inf_inf_b_o @ A2 @ ( inf_inf_b_o @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_106_inf_Oassoc,axiom,
! [A2: set_o,B2: set_o,C: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ C )
= ( inf_inf_set_o @ A2 @ ( inf_inf_set_o @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_107_inf_Oassoc,axiom,
! [A2: set_set_b,B2: set_set_b,C: set_set_b] :
( ( inf_inf_set_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ C )
= ( inf_inf_set_set_b @ A2 @ ( inf_inf_set_set_b @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_108_inf_Oassoc,axiom,
! [A2: $o,B2: $o,C: $o] :
( ( inf_inf_o @ ( inf_inf_o @ A2 @ B2 ) @ C )
= ( inf_inf_o @ A2 @ ( inf_inf_o @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_109_inf_Oassoc,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C )
= ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_110_inf__sup__aci_I1_J,axiom,
( inf_inf_b_o
= ( ^ [X2: b > $o,Y: b > $o] : ( inf_inf_b_o @ Y @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_111_inf__sup__aci_I1_J,axiom,
( inf_inf_set_o
= ( ^ [X2: set_o,Y: set_o] : ( inf_inf_set_o @ Y @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_112_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_b
= ( ^ [X2: set_set_b,Y: set_set_b] : ( inf_inf_set_set_b @ Y @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_113_inf__sup__aci_I1_J,axiom,
( inf_inf_o
= ( ^ [X2: $o,Y: $o] : ( inf_inf_o @ Y @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_114_inf__sup__aci_I1_J,axiom,
( inf_inf_set_b
= ( ^ [X2: set_b,Y: set_b] : ( inf_inf_set_b @ Y @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_115_inf__sup__aci_I2_J,axiom,
! [X3: b > $o,Y2: b > $o,Z2: b > $o] :
( ( inf_inf_b_o @ ( inf_inf_b_o @ X3 @ Y2 ) @ Z2 )
= ( inf_inf_b_o @ X3 @ ( inf_inf_b_o @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_116_inf__sup__aci_I2_J,axiom,
! [X3: set_o,Y2: set_o,Z2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ X3 @ Y2 ) @ Z2 )
= ( inf_inf_set_o @ X3 @ ( inf_inf_set_o @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_117_inf__sup__aci_I2_J,axiom,
! [X3: set_set_b,Y2: set_set_b,Z2: set_set_b] :
( ( inf_inf_set_set_b @ ( inf_inf_set_set_b @ X3 @ Y2 ) @ Z2 )
= ( inf_inf_set_set_b @ X3 @ ( inf_inf_set_set_b @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_118_inf__sup__aci_I2_J,axiom,
! [X3: $o,Y2: $o,Z2: $o] :
( ( inf_inf_o @ ( inf_inf_o @ X3 @ Y2 ) @ Z2 )
= ( inf_inf_o @ X3 @ ( inf_inf_o @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_119_inf__sup__aci_I2_J,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) @ Z2 )
= ( inf_inf_set_b @ X3 @ ( inf_inf_set_b @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_120_inf__sup__aci_I3_J,axiom,
! [X3: b > $o,Y2: b > $o,Z2: b > $o] :
( ( inf_inf_b_o @ X3 @ ( inf_inf_b_o @ Y2 @ Z2 ) )
= ( inf_inf_b_o @ Y2 @ ( inf_inf_b_o @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_121_inf__sup__aci_I3_J,axiom,
! [X3: set_o,Y2: set_o,Z2: set_o] :
( ( inf_inf_set_o @ X3 @ ( inf_inf_set_o @ Y2 @ Z2 ) )
= ( inf_inf_set_o @ Y2 @ ( inf_inf_set_o @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_122_inf__sup__aci_I3_J,axiom,
! [X3: set_set_b,Y2: set_set_b,Z2: set_set_b] :
( ( inf_inf_set_set_b @ X3 @ ( inf_inf_set_set_b @ Y2 @ Z2 ) )
= ( inf_inf_set_set_b @ Y2 @ ( inf_inf_set_set_b @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_123_inf__sup__aci_I3_J,axiom,
! [X3: $o,Y2: $o,Z2: $o] :
( ( inf_inf_o @ X3 @ ( inf_inf_o @ Y2 @ Z2 ) )
= ( inf_inf_o @ Y2 @ ( inf_inf_o @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_124_inf__sup__aci_I3_J,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X3 @ ( inf_inf_set_b @ Y2 @ Z2 ) )
= ( inf_inf_set_b @ Y2 @ ( inf_inf_set_b @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_125_inf__sup__aci_I4_J,axiom,
! [X3: b > $o,Y2: b > $o] :
( ( inf_inf_b_o @ X3 @ ( inf_inf_b_o @ X3 @ Y2 ) )
= ( inf_inf_b_o @ X3 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_126_inf__sup__aci_I4_J,axiom,
! [X3: set_o,Y2: set_o] :
( ( inf_inf_set_o @ X3 @ ( inf_inf_set_o @ X3 @ Y2 ) )
= ( inf_inf_set_o @ X3 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_127_inf__sup__aci_I4_J,axiom,
! [X3: set_set_b,Y2: set_set_b] :
( ( inf_inf_set_set_b @ X3 @ ( inf_inf_set_set_b @ X3 @ Y2 ) )
= ( inf_inf_set_set_b @ X3 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_128_inf__sup__aci_I4_J,axiom,
! [X3: $o,Y2: $o] :
( ( inf_inf_o @ X3 @ ( inf_inf_o @ X3 @ Y2 ) )
= ( inf_inf_o @ X3 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_129_inf__sup__aci_I4_J,axiom,
! [X3: set_b,Y2: set_b] :
( ( inf_inf_set_b @ X3 @ ( inf_inf_set_b @ X3 @ Y2 ) )
= ( inf_inf_set_b @ X3 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_130_Int__left__commute,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( inf_inf_set_o @ A @ ( inf_inf_set_o @ B @ C2 ) )
= ( inf_inf_set_o @ B @ ( inf_inf_set_o @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_131_Int__left__commute,axiom,
! [A: set_set_b,B: set_set_b,C2: set_set_b] :
( ( inf_inf_set_set_b @ A @ ( inf_inf_set_set_b @ B @ C2 ) )
= ( inf_inf_set_set_b @ B @ ( inf_inf_set_set_b @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_132_Int__left__commute,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) )
= ( inf_inf_set_b @ B @ ( inf_inf_set_b @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_133_Int__left__absorb,axiom,
! [A: set_o,B: set_o] :
( ( inf_inf_set_o @ A @ ( inf_inf_set_o @ A @ B ) )
= ( inf_inf_set_o @ A @ B ) ) ).
% Int_left_absorb
thf(fact_134_Int__left__absorb,axiom,
! [A: set_set_b,B: set_set_b] :
( ( inf_inf_set_set_b @ A @ ( inf_inf_set_set_b @ A @ B ) )
= ( inf_inf_set_set_b @ A @ B ) ) ).
% Int_left_absorb
thf(fact_135_Int__left__absorb,axiom,
! [A: set_b,B: set_b] :
( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ A @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ).
% Int_left_absorb
thf(fact_136_Int__commute,axiom,
( inf_inf_set_o
= ( ^ [A4: set_o,B4: set_o] : ( inf_inf_set_o @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_137_Int__commute,axiom,
( inf_inf_set_set_b
= ( ^ [A4: set_set_b,B4: set_set_b] : ( inf_inf_set_set_b @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_138_Int__commute,axiom,
( inf_inf_set_b
= ( ^ [A4: set_b,B4: set_b] : ( inf_inf_set_b @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_139_Int__absorb,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ A )
= A ) ).
% Int_absorb
thf(fact_140_Int__absorb,axiom,
! [A: set_set_b] :
( ( inf_inf_set_set_b @ A @ A )
= A ) ).
% Int_absorb
thf(fact_141_Int__absorb,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ A )
= A ) ).
% Int_absorb
thf(fact_142_Int__assoc,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A @ B ) @ C2 )
= ( inf_inf_set_o @ A @ ( inf_inf_set_o @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_143_Int__assoc,axiom,
! [A: set_set_b,B: set_set_b,C2: set_set_b] :
( ( inf_inf_set_set_b @ ( inf_inf_set_set_b @ A @ B ) @ C2 )
= ( inf_inf_set_set_b @ A @ ( inf_inf_set_set_b @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_144_Int__assoc,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B ) @ C2 )
= ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_145_IntD2,axiom,
! [C: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( inf_in1884693029477671368c_fm_a @ A @ B ) )
=> ( member536094252920883875c_fm_a @ C @ B ) ) ).
% IntD2
thf(fact_146_IntD2,axiom,
! [C: set_o,A: set_set_o,B: set_set_o] :
( ( member_set_o @ C @ ( inf_inf_set_set_o @ A @ B ) )
=> ( member_set_o @ C @ B ) ) ).
% IntD2
thf(fact_147_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_148_IntD2,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
=> ( member_o @ C @ B ) ) ).
% IntD2
thf(fact_149_IntD2,axiom,
! [C: set_b,A: set_set_b,B: set_set_b] :
( ( member_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) )
=> ( member_set_b @ C @ B ) ) ).
% IntD2
thf(fact_150_IntD2,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
=> ( member_b @ C @ B ) ) ).
% IntD2
thf(fact_151_IntD1,axiom,
! [C: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( inf_in1884693029477671368c_fm_a @ A @ B ) )
=> ( member536094252920883875c_fm_a @ C @ A ) ) ).
% IntD1
thf(fact_152_IntD1,axiom,
! [C: set_o,A: set_set_o,B: set_set_o] :
( ( member_set_o @ C @ ( inf_inf_set_set_o @ A @ B ) )
=> ( member_set_o @ C @ A ) ) ).
% IntD1
thf(fact_153_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_154_IntD1,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
=> ( member_o @ C @ A ) ) ).
% IntD1
thf(fact_155_IntD1,axiom,
! [C: set_b,A: set_set_b,B: set_set_b] :
( ( member_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) )
=> ( member_set_b @ C @ A ) ) ).
% IntD1
thf(fact_156_IntD1,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
=> ( member_b @ C @ A ) ) ).
% IntD1
thf(fact_157_IntE,axiom,
! [C: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,B: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ C @ ( inf_in1884693029477671368c_fm_a @ A @ B ) )
=> ~ ( ( member536094252920883875c_fm_a @ C @ A )
=> ~ ( member536094252920883875c_fm_a @ C @ B ) ) ) ).
% IntE
thf(fact_158_IntE,axiom,
! [C: set_o,A: set_set_o,B: set_set_o] :
( ( member_set_o @ C @ ( inf_inf_set_set_o @ A @ B ) )
=> ~ ( ( member_set_o @ C @ A )
=> ~ ( member_set_o @ C @ B ) ) ) ).
% IntE
thf(fact_159_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_160_IntE,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
=> ~ ( ( member_o @ C @ A )
=> ~ ( member_o @ C @ B ) ) ) ).
% IntE
thf(fact_161_IntE,axiom,
! [C: set_b,A: set_set_b,B: set_set_b] :
( ( member_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) )
=> ~ ( ( member_set_b @ C @ A )
=> ~ ( member_set_b @ C @ B ) ) ) ).
% IntE
thf(fact_162_IntE,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
=> ~ ( ( member_b @ C @ A )
=> ~ ( member_b @ C @ B ) ) ) ).
% IntE
thf(fact_163_generalization,axiom,
! [P: epistemic_fm_a,W: b,M2: episte3224730989885420256t_unit,I2: a] :
( ! [M3: episte3224730989885420256t_unit,X: b] :
( ( member_b @ X @ ( episte1782384855165018035t_unit @ M3 ) )
=> ( episte295617885132580261cs_a_b @ M3 @ X @ P ) )
=> ( ( member_b @ W @ ( episte1782384855165018035t_unit @ M2 ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ ( epistemic_K_a @ I2 @ P ) ) ) ) ).
% generalization
thf(fact_164_generalization,axiom,
! [P: epistemic_fm_a,W: $o,M2: episte3259645218793129376t_unit,I2: a] :
( ! [M3: episte3259645218793129376t_unit,X: $o] :
( ( member_o @ X @ ( episte876160520765907443t_unit @ M3 ) )
=> ( episte1100597067981672766cs_a_o @ M3 @ X @ P ) )
=> ( ( member_o @ W @ ( episte876160520765907443t_unit @ M2 ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ W @ ( epistemic_K_a @ I2 @ P ) ) ) ) ).
% generalization
thf(fact_165_generalization,axiom,
! [P: epistemic_fm_a,W: a,M2: episte6182337868402532512t_unit,I2: a] :
( ! [M3: episte6182337868402532512t_unit,X: a] :
( ( member_a @ X @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ X @ P ) )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W @ ( epistemic_K_a @ I2 @ P ) ) ) ) ).
% generalization
thf(fact_166_generalization,axiom,
! [P: epistemic_fm_b,W: b,M2: episte4519467831615171103t_unit,I2: b] :
( ! [M3: episte4519467831615171103t_unit,X: b] :
( ( member_b @ X @ ( episte7096680817873233778t_unit @ M3 ) )
=> ( episte6731534340014680036cs_b_b @ M3 @ X @ P ) )
=> ( ( member_b @ W @ ( episte7096680817873233778t_unit @ M2 ) )
=> ( episte6731534340014680036cs_b_b @ M2 @ W @ ( epistemic_K_b @ I2 @ P ) ) ) ) ).
% generalization
thf(fact_167_generalization,axiom,
! [P: epistemic_fm_b,W: $o,M2: episte6655095512717024223t_unit,I2: b] :
( ! [M3: episte6655095512717024223t_unit,X: $o] :
( ( member_o @ X @ ( episte8156305138475616434t_unit @ M3 ) )
=> ( episte7424746705372054269cs_b_o @ M3 @ X @ P ) )
=> ( ( member_o @ W @ ( episte8156305138475616434t_unit @ M2 ) )
=> ( episte7424746705372054269cs_b_o @ M2 @ W @ ( epistemic_K_b @ I2 @ P ) ) ) ) ).
% generalization
thf(fact_168_generalization,axiom,
! [P: epistemic_fm_b,W: a,M2: episte7477074710132283359t_unit,I2: b] :
( ! [M3: episte7477074710132283359t_unit,X: a] :
( ( member_a @ X @ ( episte3017639818781762994t_unit @ M3 ) )
=> ( episte6731534340014680035cs_b_a @ M3 @ X @ P ) )
=> ( ( member_a @ W @ ( episte3017639818781762994t_unit @ M2 ) )
=> ( episte6731534340014680035cs_b_a @ M2 @ W @ ( epistemic_K_b @ I2 @ P ) ) ) ) ).
% generalization
thf(fact_169_generalization,axiom,
! [P: epistemic_fm_o,W: b,M2: episte2543566300658443718t_unit,I2: $o] :
( ! [M3: episte2543566300658443718t_unit,X: b] :
( ( member_b @ X @ ( episte9142089827432095385t_unit @ M3 ) )
=> ( episte1021116911515925003cs_o_b @ M3 @ X @ P ) )
=> ( ( member_b @ W @ ( episte9142089827432095385t_unit @ M2 ) )
=> ( episte1021116911515925003cs_o_b @ M2 @ W @ ( epistemic_K_o @ I2 @ P ) ) ) ) ).
% generalization
thf(fact_170_generalization,axiom,
! [P: epistemic_fm_o,W: $o,M2: episte8859593475650739846t_unit,I2: $o] :
( ! [M3: episte8859593475650739846t_unit,X: $o] :
( ( member_o @ X @ ( episte1038022579713443545t_unit @ M3 ) )
=> ( episte4695001281782364324cs_o_o @ M3 @ X @ P ) )
=> ( ( member_o @ W @ ( episte1038022579713443545t_unit @ M2 ) )
=> ( episte4695001281782364324cs_o_o @ M2 @ W @ ( epistemic_K_o @ I2 @ P ) ) ) ) ).
% generalization
thf(fact_171_generalization,axiom,
! [P: epistemic_fm_o,W: a,M2: episte5501173179175555974t_unit,I2: $o] :
( ! [M3: episte5501173179175555974t_unit,X: a] :
( ( member_a @ X @ ( episte5063048828340624601t_unit @ M3 ) )
=> ( episte1021116911515925002cs_o_a @ M3 @ X @ P ) )
=> ( ( member_a @ W @ ( episte5063048828340624601t_unit @ M2 ) )
=> ( episte1021116911515925002cs_o_a @ M2 @ W @ ( epistemic_K_o @ I2 @ P ) ) ) ) ).
% generalization
thf(fact_172_generalization,axiom,
! [P: epistemic_fm_a,W: set_o,M2: episte4442589092411611552t_unit,I2: a] :
( ! [M3: episte4442589092411611552t_unit,X: set_o] :
( ( member_set_o @ X @ ( episte7994720633578350067t_unit @ M3 ) )
=> ( episte5787648899026124830_set_o @ M3 @ X @ P ) )
=> ( ( member_set_o @ W @ ( episte7994720633578350067t_unit @ M2 ) )
=> ( episte5787648899026124830_set_o @ M2 @ W @ ( epistemic_K_a @ I2 @ P ) ) ) ) ).
% generalization
thf(fact_173_Ax__2_Ocases,axiom,
! [A2: epistemic_fm_o] :
( ( stalnaker_Ax_2_o @ A2 )
=> ~ ! [I3: $o,P2: epistemic_fm_o] :
( A2
!= ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I3 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I3 @ P2 ) @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I3 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I3 @ ( epistemic_Imp_o @ P2 @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ) ).
% Ax_2.cases
thf(fact_174_Ax__2_Ocases,axiom,
! [A2: epistemic_fm_b] :
( ( stalnaker_Ax_2_b @ A2 )
=> ~ ! [I3: b,P2: epistemic_fm_b] :
( A2
!= ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I3 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I3 @ P2 ) @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ ( epistemic_K_b @ I3 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I3 @ ( epistemic_Imp_b @ P2 @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ) ).
% Ax_2.cases
thf(fact_175_Ax__2_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( stalnaker_Ax_2_a @ A2 )
=> ~ ! [I3: a,P2: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P2 ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ).
% Ax_2.cases
thf(fact_176_Ax__2_Osimps,axiom,
( stalnaker_Ax_2_o
= ( ^ [A3: epistemic_fm_o] :
? [I: $o,P3: epistemic_fm_o] :
( A3
= ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I @ ( epistemic_Imp_o @ ( epistemic_K_o @ I @ P3 ) @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I @ ( epistemic_Imp_o @ ( epistemic_K_o @ I @ ( epistemic_Imp_o @ P3 @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ) ) ).
% Ax_2.simps
thf(fact_177_Ax__2_Osimps,axiom,
( stalnaker_Ax_2_b
= ( ^ [A3: epistemic_fm_b] :
? [I: b,P3: epistemic_fm_b] :
( A3
= ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I @ ( epistemic_Imp_b @ ( epistemic_K_b @ I @ P3 ) @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ ( epistemic_K_b @ I @ ( epistemic_Imp_b @ ( epistemic_K_b @ I @ ( epistemic_Imp_b @ P3 @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ) ) ).
% Ax_2.simps
thf(fact_178_Ax__2_Osimps,axiom,
( stalnaker_Ax_2_a
= ( ^ [A3: epistemic_fm_a] :
? [I: a,P3: epistemic_fm_a] :
( A3
= ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P3 ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% Ax_2.simps
thf(fact_179_mem__Collect__eq,axiom,
! [A2: set_o,P4: set_o > $o] :
( ( member_set_o @ A2 @ ( collect_set_o @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_180_mem__Collect__eq,axiom,
! [A2: set_b,P4: set_b > $o] :
( ( member_set_b @ A2 @ ( collect_set_b @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_181_mem__Collect__eq,axiom,
! [A2: a,P4: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_182_mem__Collect__eq,axiom,
! [A2: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ A2 @ ( collec2519470961442302949c_fm_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_183_mem__Collect__eq,axiom,
! [A2: epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ A2 @ ( collec4904205152690461189c_fm_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_184_mem__Collect__eq,axiom,
! [A2: $o,P4: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_185_mem__Collect__eq,axiom,
! [A2: b,P4: b > $o] :
( ( member_b @ A2 @ ( collect_b @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_186_Collect__mem__eq,axiom,
! [A: set_set_o] :
( ( collect_set_o
@ ^ [X2: set_o] : ( member_set_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_187_Collect__mem__eq,axiom,
! [A: set_set_b] :
( ( collect_set_b
@ ^ [X2: set_b] : ( member_set_b @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_188_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_189_Collect__mem__eq,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [X2: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_190_Collect__mem__eq,axiom,
! [A: set_Epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [X2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_191_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_192_Collect__mem__eq,axiom,
! [A: set_b] :
( ( collect_b
@ ^ [X2: b] : ( member_b @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_193_Collect__cong,axiom,
! [P4: b > $o,Q2: b > $o] :
( ! [X: b] :
( ( P4 @ X )
= ( Q2 @ X ) )
=> ( ( collect_b @ P4 )
= ( collect_b @ Q2 ) ) ) ).
% Collect_cong
thf(fact_194_Collect__cong,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o] :
( ! [X: set_Epistemic_fm_a] :
( ( P4 @ X )
= ( Q2 @ X ) )
=> ( ( collec2519470961442302949c_fm_a @ P4 )
= ( collec2519470961442302949c_fm_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_195_Collect__cong,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o] :
( ! [X: epistemic_fm_a] :
( ( P4 @ X )
= ( Q2 @ X ) )
=> ( ( collec4904205152690461189c_fm_a @ P4 )
= ( collec4904205152690461189c_fm_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_196_Collect__cong,axiom,
! [P4: $o > $o,Q2: $o > $o] :
( ! [X: $o] :
( ( P4 @ X )
= ( Q2 @ X ) )
=> ( ( collect_o @ P4 )
= ( collect_o @ Q2 ) ) ) ).
% Collect_cong
thf(fact_197_Ax__2_Ointros,axiom,
! [I2: $o,P: epistemic_fm_o] : ( stalnaker_Ax_2_o @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ P @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ).
% Ax_2.intros
thf(fact_198_Ax__2_Ointros,axiom,
! [I2: b,P: epistemic_fm_b] : ( stalnaker_Ax_2_b @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ P @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ).
% Ax_2.intros
thf(fact_199_Ax__2_Ointros,axiom,
! [I2: a,P: epistemic_fm_a] : ( stalnaker_Ax_2_a @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% Ax_2.intros
thf(fact_200_transitive__def,axiom,
( episte8364071018013720454t_unit
= ( ^ [M: episte1560738328020401952t_unit] :
! [I: a,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M ) )
=> ! [Y: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y @ ( episte8072386903178013299t_unit @ M ) )
=> ! [Z: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( episte8072386903178013299t_unit @ M ) )
=> ( ( ( member536094252920883875c_fm_a @ Z @ ( episte6250069432388174439t_unit @ M @ I @ Y ) )
& ( member536094252920883875c_fm_a @ X2 @ ( episte6250069432388174439t_unit @ M @ I @ Z ) ) )
=> ( member536094252920883875c_fm_a @ X2 @ ( episte6250069432388174439t_unit @ M @ I @ Y ) ) ) ) ) ) ) ) ).
% transitive_def
thf(fact_201_transitive__def,axiom,
( episte9018110198556821958t_unit
= ( ^ [M: episte3224730989885420256t_unit] :
! [I: a,X2: b] :
( ( member_b @ X2 @ ( episte1782384855165018035t_unit @ M ) )
=> ! [Y: b] :
( ( member_b @ Y @ ( episte1782384855165018035t_unit @ M ) )
=> ! [Z: b] :
( ( member_b @ Z @ ( episte1782384855165018035t_unit @ M ) )
=> ( ( ( member_b @ Z @ ( episte1071096959727133607t_unit @ M @ I @ Y ) )
& ( member_b @ X2 @ ( episte1071096959727133607t_unit @ M @ I @ Z ) ) )
=> ( member_b @ X2 @ ( episte1071096959727133607t_unit @ M @ I @ Y ) ) ) ) ) ) ) ) ).
% transitive_def
thf(fact_202_symmetric__def,axiom,
( episte5478016696552465318t_unit
= ( ^ [M: episte1560738328020401952t_unit] :
! [I: a,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M ) )
=> ! [Y: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y @ ( episte8072386903178013299t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ X2 @ ( episte6250069432388174439t_unit @ M @ I @ Y ) )
= ( member536094252920883875c_fm_a @ Y @ ( episte6250069432388174439t_unit @ M @ I @ X2 ) ) ) ) ) ) ) ).
% symmetric_def
thf(fact_203_symmetric__def,axiom,
( episte4899516349225340390t_unit
= ( ^ [M: episte3224730989885420256t_unit] :
! [I: a,X2: b] :
( ( member_b @ X2 @ ( episte1782384855165018035t_unit @ M ) )
=> ! [Y: b] :
( ( member_b @ Y @ ( episte1782384855165018035t_unit @ M ) )
=> ( ( member_b @ X2 @ ( episte1071096959727133607t_unit @ M @ I @ Y ) )
= ( member_b @ Y @ ( episte1071096959727133607t_unit @ M @ I @ X2 ) ) ) ) ) ) ) ).
% symmetric_def
thf(fact_204_reflexive__def,axiom,
( episte5648423998891577755t_unit
= ( ^ [M: episte1560738328020401952t_unit] :
! [I: a,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M ) )
=> ( member536094252920883875c_fm_a @ X2 @ ( episte6250069432388174439t_unit @ M @ I @ X2 ) ) ) ) ) ).
% reflexive_def
thf(fact_205_reflexive__def,axiom,
( episte3426825378771607259t_unit
= ( ^ [M: episte3224730989885420256t_unit] :
! [I: a,X2: b] :
( ( member_b @ X2 @ ( episte1782384855165018035t_unit @ M ) )
=> ( member_b @ X2 @ ( episte1071096959727133607t_unit @ M @ I @ X2 ) ) ) ) ) ).
% reflexive_def
thf(fact_206_Euclidean__def,axiom,
( episte2449151000174023629t_unit
= ( ^ [M: episte1560738328020401952t_unit] :
! [I: a,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M ) )
=> ! [Y: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y @ ( episte8072386903178013299t_unit @ M ) )
=> ! [Z: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( episte8072386903178013299t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ Y @ ( episte6250069432388174439t_unit @ M @ I @ X2 ) )
=> ( ( member536094252920883875c_fm_a @ Z @ ( episte6250069432388174439t_unit @ M @ I @ X2 ) )
=> ( member536094252920883875c_fm_a @ Z @ ( episte6250069432388174439t_unit @ M @ I @ Y ) ) ) ) ) ) ) ) ) ).
% Euclidean_def
thf(fact_207_Euclidean__def,axiom,
( episte6418945320598494989t_unit
= ( ^ [M: episte3224730989885420256t_unit] :
! [I: a,X2: b] :
( ( member_b @ X2 @ ( episte1782384855165018035t_unit @ M ) )
=> ! [Y: b] :
( ( member_b @ Y @ ( episte1782384855165018035t_unit @ M ) )
=> ! [Z: b] :
( ( member_b @ Z @ ( episte1782384855165018035t_unit @ M ) )
=> ( ( member_b @ Y @ ( episte1071096959727133607t_unit @ M @ I @ X2 ) )
=> ( ( member_b @ Z @ ( episte1071096959727133607t_unit @ M @ I @ X2 ) )
=> ( member_b @ Z @ ( episte1071096959727133607t_unit @ M @ I @ Y ) ) ) ) ) ) ) ) ) ).
% Euclidean_def
thf(fact_208_truth,axiom,
! [M2: episte3224730989885420256t_unit,W: b,I2: a,P: epistemic_fm_a] :
( ( episte3426825378771607259t_unit @ M2 )
=> ( ( member_b @ W @ ( episte1782384855165018035t_unit @ M2 ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_209_truth,axiom,
! [M2: episte2543566300658443718t_unit,W: b,I2: $o,P: epistemic_fm_o] :
( ( episte4955991936609055169t_unit @ M2 )
=> ( ( member_b @ W @ ( episte9142089827432095385t_unit @ M2 ) )
=> ( episte1021116911515925003cs_o_b @ M2 @ W @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_210_truth,axiom,
! [M2: episte8859593475650739846t_unit,W: $o,I2: $o,P: epistemic_fm_o] :
( ( episte6484863922062934529t_unit @ M2 )
=> ( ( member_o @ W @ ( episte1038022579713443545t_unit @ M2 ) )
=> ( episte4695001281782364324cs_o_o @ M2 @ W @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_211_truth,axiom,
! [M2: episte5501173179175555974t_unit,W: a,I2: $o,P: epistemic_fm_o] :
( ( episte876950937517584385t_unit @ M2 )
=> ( ( member_a @ W @ ( episte5063048828340624601t_unit @ M2 ) )
=> ( episte1021116911515925002cs_o_a @ M2 @ W @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_212_truth,axiom,
! [M2: episte3259645218793129376t_unit,W: $o,I2: a,P: epistemic_fm_a] :
( ( episte2671743178186770715t_unit @ M2 )
=> ( ( member_o @ W @ ( episte876160520765907443t_unit @ M2 ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_213_truth,axiom,
! [M2: episte6182337868402532512t_unit,W: a,I2: a,P: epistemic_fm_a] :
( ( episte8571156416534912283t_unit @ M2 )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_214_truth,axiom,
! [M2: episte4519467831615171103t_unit,W: b,I2: b,P: epistemic_fm_b] :
( ( episte8741121341479823002t_unit @ M2 )
=> ( ( member_b @ W @ ( episte7096680817873233778t_unit @ M2 ) )
=> ( episte6731534340014680036cs_b_b @ M2 @ W @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_215_truth,axiom,
! [M2: episte6655095512717024223t_unit,W: $o,I2: b,P: epistemic_fm_b] :
( ( episte728515759041703898t_unit @ M2 )
=> ( ( member_o @ W @ ( episte8156305138475616434t_unit @ M2 ) )
=> ( episte7424746705372054269cs_b_o @ M2 @ W @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_216_truth,axiom,
! [M2: episte7477074710132283359t_unit,W: a,I2: b,P: epistemic_fm_b] :
( ( episte4662080342388352218t_unit @ M2 )
=> ( ( member_a @ W @ ( episte3017639818781762994t_unit @ M2 ) )
=> ( episte6731534340014680035cs_b_a @ M2 @ W @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_217_truth,axiom,
! [M2: episte399968703042273926t_unit,W: set_o,I2: $o,P: epistemic_fm_o] :
( ( episte603136233781302785t_unit @ M2 )
=> ( ( member_set_o @ W @ ( episte7065225365757467353t_unit @ M2 ) )
=> ( episte1745421503759077252_set_o @ M2 @ W @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ P ) ) ) ) ).
% truth
thf(fact_218_pos__introspection,axiom,
! [M2: episte3224730989885420256t_unit,W: b,I2: a,P: epistemic_fm_a] :
( ( episte9018110198556821958t_unit @ M2 )
=> ( ( member_b @ W @ ( episte1782384855165018035t_unit @ M2 ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ ( epistemic_K_a @ I2 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_219_pos__introspection,axiom,
! [M2: episte2543566300658443718t_unit,W: b,I2: $o,P: epistemic_fm_o] :
( ( episte7950819053531357740t_unit @ M2 )
=> ( ( member_b @ W @ ( episte9142089827432095385t_unit @ M2 ) )
=> ( episte1021116911515925003cs_o_b @ M2 @ W @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ ( epistemic_K_o @ I2 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_220_pos__introspection,axiom,
! [M2: episte8859593475650739846t_unit,W: $o,I2: $o,P: epistemic_fm_o] :
( ( episte4142360316505390444t_unit @ M2 )
=> ( ( member_o @ W @ ( episte1038022579713443545t_unit @ M2 ) )
=> ( episte4695001281782364324cs_o_o @ M2 @ W @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ ( epistemic_K_o @ I2 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_221_pos__introspection,axiom,
! [M2: episte5501173179175555974t_unit,W: a,I2: $o,P: epistemic_fm_o] :
( ( episte3871778054439886956t_unit @ M2 )
=> ( ( member_a @ W @ ( episte5063048828340624601t_unit @ M2 ) )
=> ( episte1021116911515925002cs_o_a @ M2 @ W @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ ( epistemic_K_o @ I2 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_222_pos__introspection,axiom,
! [M2: episte3259645218793129376t_unit,W: $o,I2: a,P: epistemic_fm_a] :
( ( episte1265558571371986694t_unit @ M2 )
=> ( ( member_o @ W @ ( episte876160520765907443t_unit @ M2 ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ ( epistemic_K_a @ I2 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_223_pos__introspection,axiom,
! [M2: episte6182337868402532512t_unit,W: a,I2: a,P: epistemic_fm_a] :
( ( episte4939069199465351174t_unit @ M2 )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ ( epistemic_K_a @ I2 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_224_pos__introspection,axiom,
! [M2: episte4519467831615171103t_unit,W: b,I2: b,P: epistemic_fm_b] :
( ( episte5109034124410261893t_unit @ M2 )
=> ( ( member_b @ W @ ( episte7096680817873233778t_unit @ M2 ) )
=> ( episte6731534340014680036cs_b_b @ M2 @ W @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_K_b @ I2 @ ( epistemic_K_b @ I2 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_225_pos__introspection,axiom,
! [M2: episte6655095512717024223t_unit,W: $o,I2: b,P: epistemic_fm_b] :
( ( episte8545703189081695685t_unit @ M2 )
=> ( ( member_o @ W @ ( episte8156305138475616434t_unit @ M2 ) )
=> ( episte7424746705372054269cs_b_o @ M2 @ W @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_K_b @ I2 @ ( epistemic_K_b @ I2 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_226_pos__introspection,axiom,
! [M2: episte7477074710132283359t_unit,W: a,I2: b,P: epistemic_fm_b] :
( ( episte1029993125318791109t_unit @ M2 )
=> ( ( member_a @ W @ ( episte3017639818781762994t_unit @ M2 ) )
=> ( episte6731534340014680035cs_b_a @ M2 @ W @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_K_b @ I2 @ ( epistemic_K_b @ I2 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_227_pos__introspection,axiom,
! [M2: episte399968703042273926t_unit,W: set_o,I2: $o,P: epistemic_fm_o] :
( ( episte6499144448312146284t_unit @ M2 )
=> ( ( member_set_o @ W @ ( episte7065225365757467353t_unit @ M2 ) )
=> ( episte1745421503759077252_set_o @ M2 @ W @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ ( epistemic_K_o @ I2 @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_228_frame_Oequality,axiom,
! [R: episte1560738328020401952t_unit,R2: episte1560738328020401952t_unit] :
( ( ( episte8072386903178013299t_unit @ R )
= ( episte8072386903178013299t_unit @ R2 ) )
=> ( ( ( episte6250069432388174439t_unit @ R )
= ( episte6250069432388174439t_unit @ R2 ) )
=> ( ( ( episte3309513806868946049t_unit @ R )
= ( episte3309513806868946049t_unit @ R2 ) )
=> ( R = R2 ) ) ) ) ).
% frame.equality
thf(fact_229_frame_Oequality,axiom,
! [R: episte3224730989885420256t_unit,R2: episte3224730989885420256t_unit] :
( ( ( episte1782384855165018035t_unit @ R )
= ( episte1782384855165018035t_unit @ R2 ) )
=> ( ( ( episte1071096959727133607t_unit @ R )
= ( episte1071096959727133607t_unit @ R2 ) )
=> ( ( ( episte6669210725420356545t_unit @ R )
= ( episte6669210725420356545t_unit @ R2 ) )
=> ( R = R2 ) ) ) ) ).
% frame.equality
thf(fact_230_AxB_Ocases,axiom,
! [A2: epistemic_fm_o] :
( ( epistemic_AxB_o @ A2 )
=> ~ ! [P2: epistemic_fm_o,I3: $o] :
( A2
!= ( epistemic_Imp_o @ P2 @ ( epistemic_K_o @ I3 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I3 @ ( epistemic_Imp_o @ P2 @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ) ).
% AxB.cases
thf(fact_231_AxB_Ocases,axiom,
! [A2: epistemic_fm_b] :
( ( epistemic_AxB_b @ A2 )
=> ~ ! [P2: epistemic_fm_b,I3: b] :
( A2
!= ( epistemic_Imp_b @ P2 @ ( epistemic_K_b @ I3 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I3 @ ( epistemic_Imp_b @ P2 @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ) ).
% AxB.cases
thf(fact_232_AxB_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_AxB_a @ A2 )
=> ~ ! [P2: epistemic_fm_a,I3: a] :
( A2
!= ( epistemic_Imp_a @ P2 @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ).
% AxB.cases
thf(fact_233_refl__Euclid__equiv,axiom,
! [M2: episte3224730989885420256t_unit] :
( ( episte3426825378771607259t_unit @ M2 )
=> ( ( episte6418945320598494989t_unit @ M2 )
=> ( ( episte3426825378771607259t_unit @ M2 )
& ( episte4899516349225340390t_unit @ M2 )
& ( episte9018110198556821958t_unit @ M2 ) ) ) ) ).
% refl_Euclid_equiv
thf(fact_234_refl__Euclid__equiv,axiom,
! [M2: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M2 )
=> ( ( episte2449151000174023629t_unit @ M2 )
=> ( ( episte5648423998891577755t_unit @ M2 )
& ( episte5478016696552465318t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) ) ) ) ).
% refl_Euclid_equiv
thf(fact_235_symm__trans__Euclid,axiom,
! [M2: episte3224730989885420256t_unit] :
( ( episte4899516349225340390t_unit @ M2 )
=> ( ( episte9018110198556821958t_unit @ M2 )
=> ( episte6418945320598494989t_unit @ M2 ) ) ) ).
% symm_trans_Euclid
thf(fact_236_symm__trans__Euclid,axiom,
! [M2: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M2 )
=> ( ( episte8364071018013720454t_unit @ M2 )
=> ( episte2449151000174023629t_unit @ M2 ) ) ) ).
% symm_trans_Euclid
thf(fact_237_soundness__AxB,axiom,
! [P: epistemic_fm_a,M2: episte3259645218793129376t_unit,W: $o] :
( ( epistemic_AxB_a @ P )
=> ( ( episte2022334575264998182t_unit @ M2 )
=> ( ( member_o @ W @ ( episte876160520765907443t_unit @ M2 ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_238_soundness__AxB,axiom,
! [P: epistemic_fm_a,M2: episte4442589092411611552t_unit,W: set_o] :
( ( epistemic_AxB_a @ P )
=> ( ( episte1938654652935425830t_unit @ M2 )
=> ( ( member_set_o @ W @ ( episte7994720633578350067t_unit @ M2 ) )
=> ( episte5787648899026124830_set_o @ M2 @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_239_soundness__AxB,axiom,
! [P: epistemic_fm_a,M2: episte8559422309061589728t_unit,W: set_b] :
( ( epistemic_AxB_a @ P )
=> ( ( episte3701050113606757862t_unit @ M2 )
=> ( ( member_set_b @ W @ ( episte2726209618768418739t_unit @ M2 ) )
=> ( episte586216548244424325_set_b @ M2 @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_240_soundness__AxB,axiom,
! [P: epistemic_fm_a,M2: episte6182337868402532512t_unit,W: a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte820475350133869606t_unit @ M2 )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_241_soundness__AxB,axiom,
! [P: epistemic_fm_a,M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte5478016696552465318t_unit @ M2 )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_242_soundness__AxB,axiom,
! [P: epistemic_fm_a,M2: episte3224730989885420256t_unit,W: b] :
( ( epistemic_AxB_a @ P )
=> ( ( episte4899516349225340390t_unit @ M2 )
=> ( ( member_b @ W @ ( episte1782384855165018035t_unit @ M2 ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_243_neg__introspection,axiom,
! [M2: episte3224730989885420256t_unit,W: b,I2: a,P: epistemic_fm_a] :
( ( episte4899516349225340390t_unit @ M2 )
=> ( ( episte9018110198556821958t_unit @ M2 )
=> ( ( member_b @ W @ ( episte1782384855165018035t_unit @ M2 ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_244_neg__introspection,axiom,
! [M2: episte2543566300658443718t_unit,W: b,I2: $o,P: epistemic_fm_o] :
( ( episte242693516716275276t_unit @ M2 )
=> ( ( episte7950819053531357740t_unit @ M2 )
=> ( ( member_b @ W @ ( episte9142089827432095385t_unit @ M2 ) )
=> ( episte1021116911515925003cs_o_b @ M2 @ W @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ epistemic_FF_o ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_245_neg__introspection,axiom,
! [M2: episte8859593475650739846t_unit,W: $o,I2: $o,P: epistemic_fm_o] :
( ( episte2818820560449417612t_unit @ M2 )
=> ( ( episte4142360316505390444t_unit @ M2 )
=> ( ( member_o @ W @ ( episte1038022579713443545t_unit @ M2 ) )
=> ( episte4695001281782364324cs_o_o @ M2 @ W @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ epistemic_FF_o ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_246_neg__introspection,axiom,
! [M2: episte5501173179175555974t_unit,W: a,I2: $o,P: epistemic_fm_o] :
( ( episte5387024554479580300t_unit @ M2 )
=> ( ( episte3871778054439886956t_unit @ M2 )
=> ( ( member_a @ W @ ( episte5063048828340624601t_unit @ M2 ) )
=> ( episte1021116911515925002cs_o_a @ M2 @ W @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ epistemic_FF_o ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_247_neg__introspection,axiom,
! [M2: episte3259645218793129376t_unit,W: $o,I2: a,P: epistemic_fm_a] :
( ( episte2022334575264998182t_unit @ M2 )
=> ( ( episte1265558571371986694t_unit @ M2 )
=> ( ( member_o @ W @ ( episte876160520765907443t_unit @ M2 ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_248_neg__introspection,axiom,
! [M2: episte6182337868402532512t_unit,W: a,I2: a,P: epistemic_fm_a] :
( ( episte820475350133869606t_unit @ M2 )
=> ( ( episte4939069199465351174t_unit @ M2 )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_249_neg__introspection,axiom,
! [M2: episte4519467831615171103t_unit,W: b,I2: b,P: epistemic_fm_b] :
( ( episte990440275078780325t_unit @ M2 )
=> ( ( episte5109034124410261893t_unit @ M2 )
=> ( ( member_b @ W @ ( episte7096680817873233778t_unit @ M2 ) )
=> ( episte6731534340014680036cs_b_b @ M2 @ W @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ epistemic_FF_b ) @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ epistemic_FF_b ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_250_neg__introspection,axiom,
! [M2: episte6655095512717024223t_unit,W: $o,I2: b,P: epistemic_fm_b] :
( ( episte79107156119931365t_unit @ M2 )
=> ( ( episte8545703189081695685t_unit @ M2 )
=> ( ( member_o @ W @ ( episte8156305138475616434t_unit @ M2 ) )
=> ( episte7424746705372054269cs_b_o @ M2 @ W @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ epistemic_FF_b ) @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ epistemic_FF_b ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_251_neg__introspection,axiom,
! [M2: episte7477074710132283359t_unit,W: a,I2: b,P: epistemic_fm_b] :
( ( episte6134771312842085349t_unit @ M2 )
=> ( ( episte1029993125318791109t_unit @ M2 )
=> ( ( member_a @ W @ ( episte3017639818781762994t_unit @ M2 ) )
=> ( episte6731534340014680035cs_b_a @ M2 @ W @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ epistemic_FF_b ) @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ epistemic_FF_b ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_252_neg__introspection,axiom,
! [M2: episte399968703042273926t_unit,W: set_o,I2: $o,P: epistemic_fm_o] :
( ( episte6049797970692746636t_unit @ M2 )
=> ( ( episte6499144448312146284t_unit @ M2 )
=> ( ( member_set_o @ W @ ( episte7065225365757467353t_unit @ M2 ) )
=> ( episte1745421503759077252_set_o @ M2 @ W @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ epistemic_FF_o ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_253_AxB_Ointros,axiom,
! [P: epistemic_fm_o,I2: $o] : ( epistemic_AxB_o @ ( epistemic_Imp_o @ P @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ P @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ).
% AxB.intros
thf(fact_254_AxB_Ointros,axiom,
! [P: epistemic_fm_b,I2: b] : ( epistemic_AxB_b @ ( epistemic_Imp_b @ P @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ P @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ).
% AxB.intros
thf(fact_255_AxB_Ointros,axiom,
! [P: epistemic_fm_a,I2: a] : ( epistemic_AxB_a @ ( epistemic_Imp_a @ P @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% AxB.intros
thf(fact_256_AxB_Osimps,axiom,
( epistemic_AxB_o
= ( ^ [A3: epistemic_fm_o] :
? [P3: epistemic_fm_o,I: $o] :
( A3
= ( epistemic_Imp_o @ P3 @ ( epistemic_K_o @ I @ ( epistemic_Imp_o @ ( epistemic_K_o @ I @ ( epistemic_Imp_o @ P3 @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ) ) ).
% AxB.simps
thf(fact_257_AxB_Osimps,axiom,
( epistemic_AxB_b
= ( ^ [A3: epistemic_fm_b] :
? [P3: epistemic_fm_b,I: b] :
( A3
= ( epistemic_Imp_b @ P3 @ ( epistemic_K_b @ I @ ( epistemic_Imp_b @ ( epistemic_K_b @ I @ ( epistemic_Imp_b @ P3 @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ) ) ).
% AxB.simps
thf(fact_258_AxB_Osimps,axiom,
( epistemic_AxB_a
= ( ^ [A3: epistemic_fm_a] :
? [P3: epistemic_fm_a,I: a] :
( A3
= ( epistemic_Imp_a @ P3 @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% AxB.simps
thf(fact_259_frame_Osurjective,axiom,
! [R: episte1560738328020401952t_unit] :
( R
= ( episte2888590659910966568t_unit @ ( episte8072386903178013299t_unit @ R ) @ ( episte6250069432388174439t_unit @ R ) @ ( episte3309513806868946049t_unit @ R ) ) ) ).
% frame.surjective
thf(fact_260_frame_Osurjective,axiom,
! [R: episte3224730989885420256t_unit] :
( R
= ( episte4938218340073005086t_unit @ ( episte1782384855165018035t_unit @ R ) @ ( episte1071096959727133607t_unit @ R ) @ ( episte6669210725420356545t_unit @ R ) ) ) ).
% frame.surjective
thf(fact_261_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M2: episte3259645218793129376t_unit,W: $o] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte1265558571371986694t_unit @ M2 )
=> ( ( member_o @ W @ ( episte876160520765907443t_unit @ M2 ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_262_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M2: episte4442589092411611552t_unit,W: set_o] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte3979770203462150918t_unit @ M2 )
=> ( ( member_set_o @ W @ ( episte7994720633578350067t_unit @ M2 ) )
=> ( episte5787648899026124830_set_o @ M2 @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_263_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M2: episte8559422309061589728t_unit,W: set_b] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte1388884068568345030t_unit @ M2 )
=> ( ( member_set_b @ W @ ( episte2726209618768418739t_unit @ M2 ) )
=> ( episte586216548244424325_set_b @ M2 @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_264_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M2: episte6182337868402532512t_unit,W: a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte4939069199465351174t_unit @ M2 )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_265_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte8364071018013720454t_unit @ M2 )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_266_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M2: episte3224730989885420256t_unit,W: b] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte9018110198556821958t_unit @ M2 )
=> ( ( member_b @ W @ ( episte1782384855165018035t_unit @ M2 ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_267_Ax5_Ointros,axiom,
! [I2: $o,P: epistemic_fm_o] : ( epistemic_Ax5_o @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ P @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ P @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ).
% Ax5.intros
thf(fact_268_Ax5_Ointros,axiom,
! [I2: b,P: epistemic_fm_b] : ( epistemic_Ax5_b @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ P @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ P @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ).
% Ax5.intros
thf(fact_269_Ax5_Ointros,axiom,
! [I2: a,P: epistemic_fm_a] : ( epistemic_Ax5_a @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% Ax5.intros
thf(fact_270_Ax5_Osimps,axiom,
( epistemic_Ax5_o
= ( ^ [A3: epistemic_fm_o] :
? [I: $o,P3: epistemic_fm_o] :
( A3
= ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I @ ( epistemic_Imp_o @ P3 @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I @ ( epistemic_Imp_o @ ( epistemic_K_o @ I @ ( epistemic_Imp_o @ P3 @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_271_Ax5_Osimps,axiom,
( epistemic_Ax5_b
= ( ^ [A3: epistemic_fm_b] :
? [I: b,P3: epistemic_fm_b] :
( A3
= ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I @ ( epistemic_Imp_b @ P3 @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ ( epistemic_K_b @ I @ ( epistemic_Imp_b @ ( epistemic_K_b @ I @ ( epistemic_Imp_b @ P3 @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_272_Ax5_Osimps,axiom,
( epistemic_Ax5_a
= ( ^ [A3: epistemic_fm_a] :
? [I: a,P3: epistemic_fm_a] :
( A3
= ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_273_Ax5_Ocases,axiom,
! [A2: epistemic_fm_o] :
( ( epistemic_Ax5_o @ A2 )
=> ~ ! [I3: $o,P2: epistemic_fm_o] :
( A2
!= ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I3 @ ( epistemic_Imp_o @ P2 @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I3 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I3 @ ( epistemic_Imp_o @ P2 @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ) ).
% Ax5.cases
thf(fact_274_Ax5_Ocases,axiom,
! [A2: epistemic_fm_b] :
( ( epistemic_Ax5_b @ A2 )
=> ~ ! [I3: b,P2: epistemic_fm_b] :
( A2
!= ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I3 @ ( epistemic_Imp_b @ P2 @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ ( epistemic_K_b @ I3 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I3 @ ( epistemic_Imp_b @ P2 @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ) ).
% Ax5.cases
thf(fact_275_Ax5_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_Ax5_a @ A2 )
=> ~ ! [I3: a,P2: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ).
% Ax5.cases
thf(fact_276_soundness__AxT,axiom,
! [P: epistemic_fm_a,M2: episte3259645218793129376t_unit,W: $o] :
( ( epistemic_AxT_a @ P )
=> ( ( episte2671743178186770715t_unit @ M2 )
=> ( ( member_o @ W @ ( episte876160520765907443t_unit @ M2 ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_277_soundness__AxT,axiom,
! [P: epistemic_fm_a,M2: episte4442589092411611552t_unit,W: set_o] :
( ( epistemic_AxT_a @ P )
=> ( ( episte1753692356642032923t_unit @ M2 )
=> ( ( member_set_o @ W @ ( episte7994720633578350067t_unit @ M2 ) )
=> ( episte5787648899026124830_set_o @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_278_soundness__AxT,axiom,
! [P: epistemic_fm_a,M2: episte8559422309061589728t_unit,W: set_b] :
( ( epistemic_AxT_a @ P )
=> ( ( episte4299101107980113627t_unit @ M2 )
=> ( ( member_set_b @ W @ ( episte2726209618768418739t_unit @ M2 ) )
=> ( episte586216548244424325_set_b @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_279_soundness__AxT,axiom,
! [P: epistemic_fm_a,M2: episte6182337868402532512t_unit,W: a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte8571156416534912283t_unit @ M2 )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_280_soundness__AxT,axiom,
! [P: epistemic_fm_a,M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte5648423998891577755t_unit @ M2 )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_281_soundness__AxT,axiom,
! [P: epistemic_fm_a,M2: episte3224730989885420256t_unit,W: b] :
( ( epistemic_AxT_a @ P )
=> ( ( episte3426825378771607259t_unit @ M2 )
=> ( ( member_b @ W @ ( episte1782384855165018035t_unit @ M2 ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_282_duality__taut,axiom,
! [I2: $o,P: epistemic_fm_o,Q: epistemic_fm_o,G2: list_char > $o,H: epistemic_fm_o > $o] : ( epistemic_eval_o @ G2 @ H @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ Q @ epistemic_FF_o ) ) ) @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ Q @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ epistemic_FF_o ) ) ) ) ).
% duality_taut
thf(fact_283_duality__taut,axiom,
! [I2: b,P: epistemic_fm_b,Q: epistemic_fm_b,G2: list_char > $o,H: epistemic_fm_b > $o] : ( epistemic_eval_b @ G2 @ H @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ Q @ epistemic_FF_b ) ) ) @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ Q @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ epistemic_FF_b ) ) ) ) ).
% duality_taut
thf(fact_284_duality__taut,axiom,
! [I2: a,P: epistemic_fm_a,Q: epistemic_fm_a,G2: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ H @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ epistemic_FF_a ) ) ) ) ).
% duality_taut
thf(fact_285_frame_Ofold__congs_I3_J,axiom,
! [R: episte3224730989885420256t_unit,R2: episte3224730989885420256t_unit,V: episte2291185870209430742t_unit,F2: episte2291185870209430742t_unit > episte2291185870209430742t_unit,F3: episte2291185870209430742t_unit > episte2291185870209430742t_unit] :
( ( R = R2 )
=> ( ( ( episte6669210725420356545t_unit @ R2 )
= V )
=> ( ! [V2: episte2291185870209430742t_unit] :
( ( V = V2 )
=> ( ( F2 @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte2553589567909738282it_a_b @ F2 @ R )
= ( episte2553589567909738282it_a_b @ F3 @ R2 ) ) ) ) ) ).
% frame.fold_congs(3)
thf(fact_286_frame_Ounfold__congs_I3_J,axiom,
! [R: episte3224730989885420256t_unit,R2: episte3224730989885420256t_unit,V: episte2291185870209430742t_unit,F2: episte2291185870209430742t_unit > episte2291185870209430742t_unit,F3: episte2291185870209430742t_unit > episte2291185870209430742t_unit] :
( ( R = R2 )
=> ( ( ( episte6669210725420356545t_unit @ R2 )
= V )
=> ( ! [V2: episte2291185870209430742t_unit] :
( ( V2 = V )
=> ( ( F2 @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte2553589567909738282it_a_b @ F2 @ R )
= ( episte2553589567909738282it_a_b @ F3 @ R2 ) ) ) ) ) ).
% frame.unfold_congs(3)
thf(fact_287_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M2: episte3259645218793129376t_unit,W: $o] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte2671743178186770715t_unit @ M2 )
& ( episte2022334575264998182t_unit @ M2 )
& ( episte1265558571371986694t_unit @ M2 ) )
=> ( ( member_o @ W @ ( episte876160520765907443t_unit @ M2 ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_288_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M2: episte4442589092411611552t_unit,W: set_o] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte1753692356642032923t_unit @ M2 )
& ( episte1938654652935425830t_unit @ M2 )
& ( episte3979770203462150918t_unit @ M2 ) )
=> ( ( member_set_o @ W @ ( episte7994720633578350067t_unit @ M2 ) )
=> ( episte5787648899026124830_set_o @ M2 @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_289_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M2: episte8559422309061589728t_unit,W: set_b] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte4299101107980113627t_unit @ M2 )
& ( episte3701050113606757862t_unit @ M2 )
& ( episte1388884068568345030t_unit @ M2 ) )
=> ( ( member_set_b @ W @ ( episte2726209618768418739t_unit @ M2 ) )
=> ( episte586216548244424325_set_b @ M2 @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_290_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M2: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M2 )
& ( episte820475350133869606t_unit @ M2 )
& ( episte4939069199465351174t_unit @ M2 ) )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_291_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte5478016696552465318t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_292_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M2: episte3224730989885420256t_unit,W: b] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte3426825378771607259t_unit @ M2 )
& ( episte4899516349225340390t_unit @ M2 )
& ( episte9018110198556821958t_unit @ M2 ) )
=> ( ( member_b @ W @ ( episte1782384855165018035t_unit @ M2 ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_293_frame_Oext__inject,axiom,
! [W2: set_b,K: a > b > set_b,More: episte2291185870209430742t_unit,W3: set_b,K2: a > b > set_b,More2: episte2291185870209430742t_unit] :
( ( ( episte4938218340073005086t_unit @ W2 @ K @ More )
= ( episte4938218340073005086t_unit @ W3 @ K2 @ More2 ) )
= ( ( W2 = W3 )
& ( K = K2 )
& ( More = More2 ) ) ) ).
% frame.ext_inject
thf(fact_294_frame_Oext__inject,axiom,
! [W2: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit,W3: set_se5208064806568342746c_fm_a,K2: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More2: episte1193835314949844379t_unit] :
( ( ( episte2888590659910966568t_unit @ W2 @ K @ More )
= ( episte2888590659910966568t_unit @ W3 @ K2 @ More2 ) )
= ( ( W2 = W3 )
& ( K = K2 )
& ( More = More2 ) ) ) ).
% frame.ext_inject
thf(fact_295_frame_Ocases__scheme,axiom,
! [R: episte3224730989885420256t_unit] :
~ ! [W4: set_b,K3: a > b > set_b,More3: episte2291185870209430742t_unit] :
( R
!= ( episte4938218340073005086t_unit @ W4 @ K3 @ More3 ) ) ).
% frame.cases_scheme
thf(fact_296_frame_Ocases__scheme,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W4: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More3: episte1193835314949844379t_unit] :
( R
!= ( episte2888590659910966568t_unit @ W4 @ K3 @ More3 ) ) ).
% frame.cases_scheme
thf(fact_297_frame_Oupdate__convs_I3_J,axiom,
! [More2: episte2291185870209430742t_unit > episte2291185870209430742t_unit,W2: set_b,K: a > b > set_b,More: episte2291185870209430742t_unit] :
( ( episte2553589567909738282it_a_b @ More2 @ ( episte4938218340073005086t_unit @ W2 @ K @ More ) )
= ( episte4938218340073005086t_unit @ W2 @ K @ ( More2 @ More ) ) ) ).
% frame.update_convs(3)
thf(fact_298_frame_Oupdate__convs_I3_J,axiom,
! [More2: episte1193835314949844379t_unit > episte1193835314949844379t_unit,W2: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte7964699311158398698c_fm_a @ More2 @ ( episte2888590659910966568t_unit @ W2 @ K @ More ) )
= ( episte2888590659910966568t_unit @ W2 @ K @ ( More2 @ More ) ) ) ).
% frame.update_convs(3)
thf(fact_299_eval_Osimps_I5_J,axiom,
! [G3: list_char > $o,H2: epistemic_fm_b > $o,P: epistemic_fm_b,Q: epistemic_fm_b] :
( ( epistemic_eval_b @ G3 @ H2 @ ( epistemic_Imp_b @ P @ Q ) )
= ( ( epistemic_eval_b @ G3 @ H2 @ P )
=> ( epistemic_eval_b @ G3 @ H2 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_300_eval_Osimps_I5_J,axiom,
! [G3: list_char > $o,H2: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Imp_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G3 @ H2 @ P )
=> ( epistemic_eval_a @ G3 @ H2 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_301_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H2: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b] :
( ( epistemic_eval_b @ Ux @ H2 @ ( epistemic_K_b @ I2 @ P ) )
= ( H2 @ ( epistemic_K_b @ I2 @ P ) ) ) ).
% eval.simps(6)
thf(fact_302_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H2: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o] :
( ( epistemic_eval_o @ Ux @ H2 @ ( epistemic_K_o @ I2 @ P ) )
= ( H2 @ ( epistemic_K_o @ I2 @ P ) ) ) ).
% eval.simps(6)
thf(fact_303_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H2: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a] :
( ( epistemic_eval_a @ Ux @ H2 @ ( epistemic_K_a @ I2 @ P ) )
= ( H2 @ ( epistemic_K_a @ I2 @ P ) ) ) ).
% eval.simps(6)
thf(fact_304_eval_Osimps_I1_J,axiom,
! [Uu: list_char > $o,Uv: epistemic_fm_o > $o] :
~ ( epistemic_eval_o @ Uu @ Uv @ epistemic_FF_o ) ).
% eval.simps(1)
thf(fact_305_eval_Osimps_I1_J,axiom,
! [Uu: list_char > $o,Uv: epistemic_fm_a > $o] :
~ ( epistemic_eval_a @ Uu @ Uv @ epistemic_FF_a ) ).
% eval.simps(1)
thf(fact_306_tautology,axiom,
! [P: epistemic_fm_a,M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ P )
=> ( episte7081087998767065248c_fm_a @ M2 @ W @ P ) ) ).
% tautology
thf(fact_307_tautology,axiom,
! [P: epistemic_fm_a,M2: episte3224730989885420256t_unit,W: b] :
( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ P )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ P ) ) ).
% tautology
thf(fact_308_frame_Oselect__convs_I1_J,axiom,
! [W2: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte8072386903178013299t_unit @ ( episte2888590659910966568t_unit @ W2 @ K @ More ) )
= W2 ) ).
% frame.select_convs(1)
thf(fact_309_frame_Oselect__convs_I1_J,axiom,
! [W2: set_b,K: a > b > set_b,More: episte2291185870209430742t_unit] :
( ( episte1782384855165018035t_unit @ ( episte4938218340073005086t_unit @ W2 @ K @ More ) )
= W2 ) ).
% frame.select_convs(1)
thf(fact_310_frame_Oselect__convs_I2_J,axiom,
! [W2: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte6250069432388174439t_unit @ ( episte2888590659910966568t_unit @ W2 @ K @ More ) )
= K ) ).
% frame.select_convs(2)
thf(fact_311_frame_Oselect__convs_I2_J,axiom,
! [W2: set_b,K: a > b > set_b,More: episte2291185870209430742t_unit] :
( ( episte1071096959727133607t_unit @ ( episte4938218340073005086t_unit @ W2 @ K @ More ) )
= K ) ).
% frame.select_convs(2)
thf(fact_312_frame_Oselect__convs_I3_J,axiom,
! [W2: set_b,K: a > b > set_b,More: episte2291185870209430742t_unit] :
( ( episte6669210725420356545t_unit @ ( episte4938218340073005086t_unit @ W2 @ K @ More ) )
= More ) ).
% frame.select_convs(3)
thf(fact_313_frame_Oselect__convs_I3_J,axiom,
! [W2: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte3309513806868946049t_unit @ ( episte2888590659910966568t_unit @ W2 @ K @ More ) )
= More ) ).
% frame.select_convs(3)
thf(fact_314_AxT_Ocases,axiom,
! [A2: epistemic_fm_o] :
( ( epistemic_AxT_o @ A2 )
=> ~ ! [I3: $o,P2: epistemic_fm_o] :
( A2
!= ( epistemic_Imp_o @ ( epistemic_K_o @ I3 @ P2 ) @ P2 ) ) ) ).
% AxT.cases
thf(fact_315_AxT_Ocases,axiom,
! [A2: epistemic_fm_b] :
( ( epistemic_AxT_b @ A2 )
=> ~ ! [I3: b,P2: epistemic_fm_b] :
( A2
!= ( epistemic_Imp_b @ ( epistemic_K_b @ I3 @ P2 ) @ P2 ) ) ) ).
% AxT.cases
thf(fact_316_AxT_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_AxT_a @ A2 )
=> ~ ! [I3: a,P2: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P2 ) @ P2 ) ) ) ).
% AxT.cases
thf(fact_317_AxT_Osimps,axiom,
( epistemic_AxT_o
= ( ^ [A3: epistemic_fm_o] :
? [I: $o,P3: epistemic_fm_o] :
( A3
= ( epistemic_Imp_o @ ( epistemic_K_o @ I @ P3 ) @ P3 ) ) ) ) ).
% AxT.simps
thf(fact_318_AxT_Osimps,axiom,
( epistemic_AxT_b
= ( ^ [A3: epistemic_fm_b] :
? [I: b,P3: epistemic_fm_b] :
( A3
= ( epistemic_Imp_b @ ( epistemic_K_b @ I @ P3 ) @ P3 ) ) ) ) ).
% AxT.simps
thf(fact_319_AxT_Osimps,axiom,
( epistemic_AxT_a
= ( ^ [A3: epistemic_fm_a] :
? [I: a,P3: epistemic_fm_a] :
( A3
= ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P3 ) @ P3 ) ) ) ) ).
% AxT.simps
thf(fact_320_AxT_Ointros,axiom,
! [I2: $o,P: epistemic_fm_o] : ( epistemic_AxT_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ P ) ) ).
% AxT.intros
thf(fact_321_AxT_Ointros,axiom,
! [I2: b,P: epistemic_fm_b] : ( epistemic_AxT_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ P ) ) ).
% AxT.intros
thf(fact_322_AxT_Ointros,axiom,
! [I2: a,P: epistemic_fm_a] : ( epistemic_AxT_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ P ) ) ).
% AxT.intros
thf(fact_323_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M2: episte3259645218793129376t_unit,W: $o] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte2671743178186770715t_unit @ M2 )
& ( episte1265558571371986694t_unit @ M2 ) )
=> ( ( member_o @ W @ ( episte876160520765907443t_unit @ M2 ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_324_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M2: episte4442589092411611552t_unit,W: set_o] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte1753692356642032923t_unit @ M2 )
& ( episte3979770203462150918t_unit @ M2 ) )
=> ( ( member_set_o @ W @ ( episte7994720633578350067t_unit @ M2 ) )
=> ( episte5787648899026124830_set_o @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_325_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M2: episte8559422309061589728t_unit,W: set_b] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte4299101107980113627t_unit @ M2 )
& ( episte1388884068568345030t_unit @ M2 ) )
=> ( ( member_set_b @ W @ ( episte2726209618768418739t_unit @ M2 ) )
=> ( episte586216548244424325_set_b @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_326_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M2: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M2 )
& ( episte4939069199465351174t_unit @ M2 ) )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_327_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_328_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M2: episte3224730989885420256t_unit,W: b] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte3426825378771607259t_unit @ M2 )
& ( episte9018110198556821958t_unit @ M2 ) )
=> ( ( member_b @ W @ ( episte1782384855165018035t_unit @ M2 ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_329_Ax4_Ocases,axiom,
! [A2: epistemic_fm_o] :
( ( epistemic_Ax4_o @ A2 )
=> ~ ! [I3: $o,P2: epistemic_fm_o] :
( A2
!= ( epistemic_Imp_o @ ( epistemic_K_o @ I3 @ P2 ) @ ( epistemic_K_o @ I3 @ ( epistemic_K_o @ I3 @ P2 ) ) ) ) ) ).
% Ax4.cases
thf(fact_330_Ax4_Ocases,axiom,
! [A2: epistemic_fm_b] :
( ( epistemic_Ax4_b @ A2 )
=> ~ ! [I3: b,P2: epistemic_fm_b] :
( A2
!= ( epistemic_Imp_b @ ( epistemic_K_b @ I3 @ P2 ) @ ( epistemic_K_b @ I3 @ ( epistemic_K_b @ I3 @ P2 ) ) ) ) ) ).
% Ax4.cases
thf(fact_331_Ax4_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_Ax4_a @ A2 )
=> ~ ! [I3: a,P2: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P2 ) @ ( epistemic_K_a @ I3 @ ( epistemic_K_a @ I3 @ P2 ) ) ) ) ) ).
% Ax4.cases
thf(fact_332_Ax4_Osimps,axiom,
( epistemic_Ax4_o
= ( ^ [A3: epistemic_fm_o] :
? [I: $o,P3: epistemic_fm_o] :
( A3
= ( epistemic_Imp_o @ ( epistemic_K_o @ I @ P3 ) @ ( epistemic_K_o @ I @ ( epistemic_K_o @ I @ P3 ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_333_Ax4_Osimps,axiom,
( epistemic_Ax4_b
= ( ^ [A3: epistemic_fm_b] :
? [I: b,P3: epistemic_fm_b] :
( A3
= ( epistemic_Imp_b @ ( epistemic_K_b @ I @ P3 ) @ ( epistemic_K_b @ I @ ( epistemic_K_b @ I @ P3 ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_334_Ax4_Osimps,axiom,
( epistemic_Ax4_a
= ( ^ [A3: epistemic_fm_a] :
? [I: a,P3: epistemic_fm_a] :
( A3
= ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P3 ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P3 ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_335_Ax4_Ointros,axiom,
! [I2: $o,P: epistemic_fm_o] : ( epistemic_Ax4_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ ( epistemic_K_o @ I2 @ P ) ) ) ) ).
% Ax4.intros
thf(fact_336_Ax4_Ointros,axiom,
! [I2: b,P: epistemic_fm_b] : ( epistemic_Ax4_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_K_b @ I2 @ ( epistemic_K_b @ I2 @ P ) ) ) ) ).
% Ax4.intros
thf(fact_337_Ax4_Ointros,axiom,
! [I2: a,P: epistemic_fm_a] : ( epistemic_Ax4_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ ( epistemic_K_a @ I2 @ P ) ) ) ) ).
% Ax4.intros
thf(fact_338_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M2: episte3259645218793129376t_unit,W: $o] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte2671743178186770715t_unit @ M2 )
& ( episte2022334575264998182t_unit @ M2 )
& ( episte1265558571371986694t_unit @ M2 ) )
=> ( ( member_o @ W @ ( episte876160520765907443t_unit @ M2 ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_339_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M2: episte4442589092411611552t_unit,W: set_o] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte1753692356642032923t_unit @ M2 )
& ( episte1938654652935425830t_unit @ M2 )
& ( episte3979770203462150918t_unit @ M2 ) )
=> ( ( member_set_o @ W @ ( episte7994720633578350067t_unit @ M2 ) )
=> ( episte5787648899026124830_set_o @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_340_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M2: episte8559422309061589728t_unit,W: set_b] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte4299101107980113627t_unit @ M2 )
& ( episte3701050113606757862t_unit @ M2 )
& ( episte1388884068568345030t_unit @ M2 ) )
=> ( ( member_set_b @ W @ ( episte2726209618768418739t_unit @ M2 ) )
=> ( episte586216548244424325_set_b @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_341_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M2: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M2 )
& ( episte820475350133869606t_unit @ M2 )
& ( episte4939069199465351174t_unit @ M2 ) )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_342_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte5478016696552465318t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_343_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M2: episte3224730989885420256t_unit,W: b] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte3426825378771607259t_unit @ M2 )
& ( episte4899516349225340390t_unit @ M2 )
& ( episte9018110198556821958t_unit @ M2 ) )
=> ( ( member_b @ W @ ( episte1782384855165018035t_unit @ M2 ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_344_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M2: episte3259645218793129376t_unit,W: $o] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte2052337558702062413t_unit @ M2 )
=> ( ( member_o @ W @ ( episte876160520765907443t_unit @ M2 ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_345_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M2: episte4442589092411611552t_unit,W: set_o] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte905247779971886925t_unit @ M2 )
=> ( ( member_set_o @ W @ ( episte7994720633578350067t_unit @ M2 ) )
=> ( episte5787648899026124830_set_o @ M2 @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_346_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M2: episte8559422309061589728t_unit,W: set_b] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte3789892348602042125t_unit @ M2 )
=> ( ( member_set_b @ W @ ( episte2726209618768418739t_unit @ M2 ) )
=> ( episte586216548244424325_set_b @ M2 @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_347_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M2: episte6182337868402532512t_unit,W: a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte2339904321507024205t_unit @ M2 )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_348_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte2449151000174023629t_unit @ M2 )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_349_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M2: episte3224730989885420256t_unit,W: b] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte6418945320598494989t_unit @ M2 )
=> ( ( member_b @ W @ ( episte1782384855165018035t_unit @ M2 ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_350_distribution,axiom,
! [M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I2: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( episte7081087998767065248c_fm_a @ M2 @ W @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ Q ) ) ) @ ( epistemic_K_a @ I2 @ Q ) ) ) ).
% distribution
thf(fact_351_distribution,axiom,
! [M2: episte3224730989885420256t_unit,W: b,I2: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( episte295617885132580261cs_a_b @ M2 @ W @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ Q ) ) ) @ ( epistemic_K_a @ I2 @ Q ) ) ) ).
% distribution
thf(fact_352_boolean__algebra__cancel_Oinf2,axiom,
! [B: b > $o,K4: b > $o,B2: b > $o,A2: b > $o] :
( ( B
= ( inf_inf_b_o @ K4 @ B2 ) )
=> ( ( inf_inf_b_o @ A2 @ B )
= ( inf_inf_b_o @ K4 @ ( inf_inf_b_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_353_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_o,K4: set_o,B2: set_o,A2: set_o] :
( ( B
= ( inf_inf_set_o @ K4 @ B2 ) )
=> ( ( inf_inf_set_o @ A2 @ B )
= ( inf_inf_set_o @ K4 @ ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_354_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_set_b,K4: set_set_b,B2: set_set_b,A2: set_set_b] :
( ( B
= ( inf_inf_set_set_b @ K4 @ B2 ) )
=> ( ( inf_inf_set_set_b @ A2 @ B )
= ( inf_inf_set_set_b @ K4 @ ( inf_inf_set_set_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_355_boolean__algebra__cancel_Oinf2,axiom,
! [B: $o,K4: $o,B2: $o,A2: $o] :
( ( B
= ( inf_inf_o @ K4 @ B2 ) )
=> ( ( inf_inf_o @ A2 @ B )
= ( inf_inf_o @ K4 @ ( inf_inf_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_356_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_b,K4: set_b,B2: set_b,A2: set_b] :
( ( B
= ( inf_inf_set_b @ K4 @ B2 ) )
=> ( ( inf_inf_set_b @ A2 @ B )
= ( inf_inf_set_b @ K4 @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_357_boolean__algebra__cancel_Oinf1,axiom,
! [A: b > $o,K4: b > $o,A2: b > $o,B2: b > $o] :
( ( A
= ( inf_inf_b_o @ K4 @ A2 ) )
=> ( ( inf_inf_b_o @ A @ B2 )
= ( inf_inf_b_o @ K4 @ ( inf_inf_b_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_358_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_o,K4: set_o,A2: set_o,B2: set_o] :
( ( A
= ( inf_inf_set_o @ K4 @ A2 ) )
=> ( ( inf_inf_set_o @ A @ B2 )
= ( inf_inf_set_o @ K4 @ ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_359_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_set_b,K4: set_set_b,A2: set_set_b,B2: set_set_b] :
( ( A
= ( inf_inf_set_set_b @ K4 @ A2 ) )
=> ( ( inf_inf_set_set_b @ A @ B2 )
= ( inf_inf_set_set_b @ K4 @ ( inf_inf_set_set_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_360_boolean__algebra__cancel_Oinf1,axiom,
! [A: $o,K4: $o,A2: $o,B2: $o] :
( ( A
= ( inf_inf_o @ K4 @ A2 ) )
=> ( ( inf_inf_o @ A @ B2 )
= ( inf_inf_o @ K4 @ ( inf_inf_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_361_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_b,K4: set_b,A2: set_b,B2: set_b] :
( ( A
= ( inf_inf_set_b @ K4 @ A2 ) )
=> ( ( inf_inf_set_b @ A @ B2 )
= ( inf_inf_set_b @ K4 @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_362_frame_Oupdate__convs_I1_J,axiom,
! [W3: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a,W2: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte8842944213534508900t_unit @ W3 @ ( episte2888590659910966568t_unit @ W2 @ K @ More ) )
= ( episte2888590659910966568t_unit @ ( W3 @ W2 ) @ K @ More ) ) ).
% frame.update_convs(1)
thf(fact_363_frame_Oupdate__convs_I1_J,axiom,
! [W3: set_b > set_b,W2: set_b,K: a > b > set_b,More: episte2291185870209430742t_unit] :
( ( episte3904078566835727962t_unit @ W3 @ ( episte4938218340073005086t_unit @ W2 @ K @ More ) )
= ( episte4938218340073005086t_unit @ ( W3 @ W2 ) @ K @ More ) ) ).
% frame.update_convs(1)
thf(fact_364_frame_Ounfold__congs_I2_J,axiom,
! [R: episte3224730989885420256t_unit,R2: episte3224730989885420256t_unit,V: a > b > set_b,F2: ( a > b > set_b ) > a > b > set_b,F3: ( a > b > set_b ) > a > b > set_b] :
( ( R = R2 )
=> ( ( ( episte1071096959727133607t_unit @ R2 )
= V )
=> ( ! [V2: a > b > set_b] :
( ( V2 = V )
=> ( ( F2 @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte5760896158387330320t_unit @ F2 @ R )
= ( episte5760896158387330320t_unit @ F3 @ R2 ) ) ) ) ) ).
% frame.unfold_congs(2)
thf(fact_365_frame_Ofold__congs_I2_J,axiom,
! [R: episte3224730989885420256t_unit,R2: episte3224730989885420256t_unit,V: a > b > set_b,F2: ( a > b > set_b ) > a > b > set_b,F3: ( a > b > set_b ) > a > b > set_b] :
( ( R = R2 )
=> ( ( ( episte1071096959727133607t_unit @ R2 )
= V )
=> ( ! [V2: a > b > set_b] :
( ( V = V2 )
=> ( ( F2 @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte5760896158387330320t_unit @ F2 @ R )
= ( episte5760896158387330320t_unit @ F3 @ R2 ) ) ) ) ) ).
% frame.fold_congs(2)
thf(fact_366_frame_Ounfold__congs_I1_J,axiom,
! [R: episte1560738328020401952t_unit,R2: episte1560738328020401952t_unit,V: set_se5208064806568342746c_fm_a,F2: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a,F3: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a] :
( ( R = R2 )
=> ( ( ( episte8072386903178013299t_unit @ R2 )
= V )
=> ( ! [V2: set_se5208064806568342746c_fm_a] :
( ( V2 = V )
=> ( ( F2 @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte8842944213534508900t_unit @ F2 @ R )
= ( episte8842944213534508900t_unit @ F3 @ R2 ) ) ) ) ) ).
% frame.unfold_congs(1)
thf(fact_367_frame_Ounfold__congs_I1_J,axiom,
! [R: episte3224730989885420256t_unit,R2: episte3224730989885420256t_unit,V: set_b,F2: set_b > set_b,F3: set_b > set_b] :
( ( R = R2 )
=> ( ( ( episte1782384855165018035t_unit @ R2 )
= V )
=> ( ! [V2: set_b] :
( ( V2 = V )
=> ( ( F2 @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte3904078566835727962t_unit @ F2 @ R )
= ( episte3904078566835727962t_unit @ F3 @ R2 ) ) ) ) ) ).
% frame.unfold_congs(1)
thf(fact_368_frame_Ofold__congs_I1_J,axiom,
! [R: episte1560738328020401952t_unit,R2: episte1560738328020401952t_unit,V: set_se5208064806568342746c_fm_a,F2: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a,F3: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a] :
( ( R = R2 )
=> ( ( ( episte8072386903178013299t_unit @ R2 )
= V )
=> ( ! [V2: set_se5208064806568342746c_fm_a] :
( ( V = V2 )
=> ( ( F2 @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte8842944213534508900t_unit @ F2 @ R )
= ( episte8842944213534508900t_unit @ F3 @ R2 ) ) ) ) ) ).
% frame.fold_congs(1)
thf(fact_369_frame_Ofold__congs_I1_J,axiom,
! [R: episte3224730989885420256t_unit,R2: episte3224730989885420256t_unit,V: set_b,F2: set_b > set_b,F3: set_b > set_b] :
( ( R = R2 )
=> ( ( ( episte1782384855165018035t_unit @ R2 )
= V )
=> ( ! [V2: set_b] :
( ( V = V2 )
=> ( ( F2 @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte3904078566835727962t_unit @ F2 @ R )
= ( episte3904078566835727962t_unit @ F3 @ R2 ) ) ) ) ) ).
% frame.fold_congs(1)
thf(fact_370_frame_Oupdate__convs_I2_J,axiom,
! [K2: ( a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a ) > a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,W2: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte7087520964475219152t_unit @ K2 @ ( episte2888590659910966568t_unit @ W2 @ K @ More ) )
= ( episte2888590659910966568t_unit @ W2 @ ( K2 @ K ) @ More ) ) ).
% frame.update_convs(2)
thf(fact_371_frame_Oupdate__convs_I2_J,axiom,
! [K2: ( a > b > set_b ) > a > b > set_b,W2: set_b,K: a > b > set_b,More: episte2291185870209430742t_unit] :
( ( episte5760896158387330320t_unit @ K2 @ ( episte4938218340073005086t_unit @ W2 @ K @ More ) )
= ( episte4938218340073005086t_unit @ W2 @ ( K2 @ K ) @ More ) ) ).
% frame.update_convs(2)
thf(fact_372_kripke_Oequality,axiom,
! [R: episte1560738328020401952t_unit,R2: episte1560738328020401952t_unit] :
( ( ( episte8072386903178013299t_unit @ R )
= ( episte8072386903178013299t_unit @ R2 ) )
=> ( ( ( episte6250069432388174439t_unit @ R )
= ( episte6250069432388174439t_unit @ R2 ) )
=> ( ( ( episte2398645135750866164t_unit @ R )
= ( episte2398645135750866164t_unit @ R2 ) )
=> ( ( ( episte5479201149095757850t_unit @ R )
= ( episte5479201149095757850t_unit @ R2 ) )
=> ( R = R2 ) ) ) ) ) ).
% kripke.equality
thf(fact_373_kripke_Oequality,axiom,
! [R: episte3224730989885420256t_unit,R2: episte3224730989885420256t_unit] :
( ( ( episte1782384855165018035t_unit @ R )
= ( episte1782384855165018035t_unit @ R2 ) )
=> ( ( ( episte1071096959727133607t_unit @ R )
= ( episte1071096959727133607t_unit @ R2 ) )
=> ( ( ( episte5693316124195166255t_unit @ R )
= ( episte5693316124195166255t_unit @ R2 ) )
=> ( ( ( episte7270682542085591125t_unit @ R )
= ( episte7270682542085591125t_unit @ R2 ) )
=> ( R = R2 ) ) ) ) ) ).
% kripke.equality
thf(fact_374_fm_Oinject_I3_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( ( epistemic_Con_a @ X41 @ X42 )
= ( epistemic_Con_a @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_375_fm_Oinject_I3_J,axiom,
! [X41: epistemic_fm_b,X42: epistemic_fm_b,Y41: epistemic_fm_b,Y42: epistemic_fm_b] :
( ( ( epistemic_Con_b @ X41 @ X42 )
= ( epistemic_Con_b @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_376_fm_Odistinct_I25_J,axiom,
! [X41: epistemic_fm_b,X42: epistemic_fm_b,X51: epistemic_fm_b,X52: epistemic_fm_b] :
( ( epistemic_Con_b @ X41 @ X42 )
!= ( epistemic_Imp_b @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_377_fm_Odistinct_I25_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Con_a @ X41 @ X42 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_378_fm_Odistinct_I27_J,axiom,
! [X41: epistemic_fm_o,X42: epistemic_fm_o,X61: $o,X62: epistemic_fm_o] :
( ( epistemic_Con_o @ X41 @ X42 )
!= ( epistemic_K_o @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_379_fm_Odistinct_I27_J,axiom,
! [X41: epistemic_fm_b,X42: epistemic_fm_b,X61: b,X62: epistemic_fm_b] :
( ( epistemic_Con_b @ X41 @ X42 )
!= ( epistemic_K_b @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_380_fm_Odistinct_I27_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Con_a @ X41 @ X42 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_381_fm_Odistinct_I5_J,axiom,
! [X41: epistemic_fm_o,X42: epistemic_fm_o] :
( epistemic_FF_o
!= ( epistemic_Con_o @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_382_fm_Odistinct_I5_J,axiom,
! [X41: epistemic_fm_b,X42: epistemic_fm_b] :
( epistemic_FF_b
!= ( epistemic_Con_b @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_383_fm_Odistinct_I5_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_384_semantics_Osimps_I4_J,axiom,
! [M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M2 @ W @ ( epistemic_Con_a @ P @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M2 @ W @ P )
& ( episte7081087998767065248c_fm_a @ M2 @ W @ Q ) ) ) ).
% semantics.simps(4)
thf(fact_385_semantics_Osimps_I4_J,axiom,
! [M2: episte3224730989885420256t_unit,W: b,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte295617885132580261cs_a_b @ M2 @ W @ ( epistemic_Con_a @ P @ Q ) )
= ( ( episte295617885132580261cs_a_b @ M2 @ W @ P )
& ( episte295617885132580261cs_a_b @ M2 @ W @ Q ) ) ) ).
% semantics.simps(4)
thf(fact_386_eval_Osimps_I4_J,axiom,
! [G3: list_char > $o,H2: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Con_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G3 @ H2 @ P )
& ( epistemic_eval_a @ G3 @ H2 @ Q ) ) ) ).
% eval.simps(4)
thf(fact_387_eval_Osimps_I4_J,axiom,
! [G3: list_char > $o,H2: epistemic_fm_b > $o,P: epistemic_fm_b,Q: epistemic_fm_b] :
( ( epistemic_eval_b @ G3 @ H2 @ ( epistemic_Con_b @ P @ Q ) )
= ( ( epistemic_eval_b @ G3 @ H2 @ P )
& ( epistemic_eval_b @ G3 @ H2 @ Q ) ) ) ).
% eval.simps(4)
thf(fact_388_kripke_Osurjective,axiom,
! [R: episte1560738328020401952t_unit] :
( R
= ( episte2888590659910966568t_unit @ ( episte8072386903178013299t_unit @ R ) @ ( episte6250069432388174439t_unit @ R ) @ ( episte8239586592105053771t_unit @ ( episte2398645135750866164t_unit @ R ) @ ( episte5479201149095757850t_unit @ R ) ) ) ) ).
% kripke.surjective
thf(fact_389_kripke_Osurjective,axiom,
! [R: episte3224730989885420256t_unit] :
( R
= ( episte4938218340073005086t_unit @ ( episte1782384855165018035t_unit @ R ) @ ( episte1071096959727133607t_unit @ R ) @ ( episte6143282164508887558t_unit @ ( episte5693316124195166255t_unit @ R ) @ ( episte7270682542085591125t_unit @ R ) ) ) ) ).
% kripke.surjective
thf(fact_390_kripke_Ofold__congs_I4_J,axiom,
! [R: episte3224730989885420256t_unit,R2: episte3224730989885420256t_unit,V: product_unit,F2: product_unit > product_unit,F3: product_unit > product_unit] :
( ( R = R2 )
=> ( ( ( episte7270682542085591125t_unit @ R2 )
= V )
=> ( ! [V2: product_unit] :
( ( V = V2 )
=> ( ( F2 @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte5119575119384494574it_a_b @ F2 @ R )
= ( episte5119575119384494574it_a_b @ F3 @ R2 ) ) ) ) ) ).
% kripke.fold_congs(4)
thf(fact_391_kripke_Ounfold__congs_I4_J,axiom,
! [R: episte3224730989885420256t_unit,R2: episte3224730989885420256t_unit,V: product_unit,F2: product_unit > product_unit,F3: product_unit > product_unit] :
( ( R = R2 )
=> ( ( ( episte7270682542085591125t_unit @ R2 )
= V )
=> ( ! [V2: product_unit] :
( ( V2 = V )
=> ( ( F2 @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte5119575119384494574it_a_b @ F2 @ R )
= ( episte5119575119384494574it_a_b @ F3 @ R2 ) ) ) ) ) ).
% kripke.unfold_congs(4)
thf(fact_392_AK_Osimps,axiom,
( epistemic_AK_o
= ( ^ [A4: epistemic_fm_o > $o,A3: epistemic_fm_o] :
( ? [P3: epistemic_fm_o] :
( ( A3 = P3 )
& ! [G: list_char > $o,H4: epistemic_fm_o > $o] : ( epistemic_eval_o @ G @ H4 @ P3 ) )
| ? [I: $o,P3: epistemic_fm_o,Q3: epistemic_fm_o] :
( A3
= ( epistemic_Imp_o @ ( epistemic_Con_o @ ( epistemic_K_o @ I @ P3 ) @ ( epistemic_K_o @ I @ ( epistemic_Imp_o @ P3 @ Q3 ) ) ) @ ( epistemic_K_o @ I @ Q3 ) ) )
| ? [P3: epistemic_fm_o] :
( ( A3 = P3 )
& ( A4 @ P3 ) )
| ? [P3: epistemic_fm_o,Q3: epistemic_fm_o] :
( ( A3 = Q3 )
& ( epistemic_AK_o @ A4 @ P3 )
& ( epistemic_AK_o @ A4 @ ( epistemic_Imp_o @ P3 @ Q3 ) ) )
| ? [P3: epistemic_fm_o,I: $o] :
( ( A3
= ( epistemic_K_o @ I @ P3 ) )
& ( epistemic_AK_o @ A4 @ P3 ) ) ) ) ) ).
% AK.simps
thf(fact_393_AK_Osimps,axiom,
( epistemic_AK_b
= ( ^ [A4: epistemic_fm_b > $o,A3: epistemic_fm_b] :
( ? [P3: epistemic_fm_b] :
( ( A3 = P3 )
& ! [G: list_char > $o,H4: epistemic_fm_b > $o] : ( epistemic_eval_b @ G @ H4 @ P3 ) )
| ? [I: b,P3: epistemic_fm_b,Q3: epistemic_fm_b] :
( A3
= ( epistemic_Imp_b @ ( epistemic_Con_b @ ( epistemic_K_b @ I @ P3 ) @ ( epistemic_K_b @ I @ ( epistemic_Imp_b @ P3 @ Q3 ) ) ) @ ( epistemic_K_b @ I @ Q3 ) ) )
| ? [P3: epistemic_fm_b] :
( ( A3 = P3 )
& ( A4 @ P3 ) )
| ? [P3: epistemic_fm_b,Q3: epistemic_fm_b] :
( ( A3 = Q3 )
& ( epistemic_AK_b @ A4 @ P3 )
& ( epistemic_AK_b @ A4 @ ( epistemic_Imp_b @ P3 @ Q3 ) ) )
| ? [P3: epistemic_fm_b,I: b] :
( ( A3
= ( epistemic_K_b @ I @ P3 ) )
& ( epistemic_AK_b @ A4 @ P3 ) ) ) ) ) ).
% AK.simps
thf(fact_394_AK_Osimps,axiom,
( epistemic_AK_a
= ( ^ [A4: epistemic_fm_a > $o,A3: epistemic_fm_a] :
( ? [P3: epistemic_fm_a] :
( ( A3 = P3 )
& ! [G: list_char > $o,H4: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H4 @ P3 ) )
| ? [I: a,P3: epistemic_fm_a,Q3: epistemic_fm_a] :
( A3
= ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P3 ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P3 @ Q3 ) ) ) @ ( epistemic_K_a @ I @ Q3 ) ) )
| ? [P3: epistemic_fm_a] :
( ( A3 = P3 )
& ( A4 @ P3 ) )
| ? [P3: epistemic_fm_a,Q3: epistemic_fm_a] :
( ( A3 = Q3 )
& ( epistemic_AK_a @ A4 @ P3 )
& ( epistemic_AK_a @ A4 @ ( epistemic_Imp_a @ P3 @ Q3 ) ) )
| ? [P3: epistemic_fm_a,I: a] :
( ( A3
= ( epistemic_K_a @ I @ P3 ) )
& ( epistemic_AK_a @ A4 @ P3 ) ) ) ) ) ).
% AK.simps
thf(fact_395_AK_Ocases,axiom,
! [A: epistemic_fm_o > $o,A2: epistemic_fm_o] :
( ( epistemic_AK_o @ A @ A2 )
=> ( ~ ! [G2: list_char > $o,H: epistemic_fm_o > $o] : ( epistemic_eval_o @ G2 @ H @ A2 )
=> ( ! [I3: $o,P2: epistemic_fm_o,Q4: epistemic_fm_o] :
( A2
!= ( epistemic_Imp_o @ ( epistemic_Con_o @ ( epistemic_K_o @ I3 @ P2 ) @ ( epistemic_K_o @ I3 @ ( epistemic_Imp_o @ P2 @ Q4 ) ) ) @ ( epistemic_K_o @ I3 @ Q4 ) ) )
=> ( ~ ( A @ A2 )
=> ( ! [P2: epistemic_fm_o] :
( ( epistemic_AK_o @ A @ P2 )
=> ~ ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ P2 @ A2 ) ) )
=> ~ ! [P2: epistemic_fm_o] :
( ? [I3: $o] :
( A2
= ( epistemic_K_o @ I3 @ P2 ) )
=> ~ ( epistemic_AK_o @ A @ P2 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_396_AK_Ocases,axiom,
! [A: epistemic_fm_b > $o,A2: epistemic_fm_b] :
( ( epistemic_AK_b @ A @ A2 )
=> ( ~ ! [G2: list_char > $o,H: epistemic_fm_b > $o] : ( epistemic_eval_b @ G2 @ H @ A2 )
=> ( ! [I3: b,P2: epistemic_fm_b,Q4: epistemic_fm_b] :
( A2
!= ( epistemic_Imp_b @ ( epistemic_Con_b @ ( epistemic_K_b @ I3 @ P2 ) @ ( epistemic_K_b @ I3 @ ( epistemic_Imp_b @ P2 @ Q4 ) ) ) @ ( epistemic_K_b @ I3 @ Q4 ) ) )
=> ( ~ ( A @ A2 )
=> ( ! [P2: epistemic_fm_b] :
( ( epistemic_AK_b @ A @ P2 )
=> ~ ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ P2 @ A2 ) ) )
=> ~ ! [P2: epistemic_fm_b] :
( ? [I3: b] :
( A2
= ( epistemic_K_b @ I3 @ P2 ) )
=> ~ ( epistemic_AK_b @ A @ P2 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_397_AK_Ocases,axiom,
! [A: epistemic_fm_a > $o,A2: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ A2 )
=> ( ~ ! [G2: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ H @ A2 )
=> ( ! [I3: a,P2: epistemic_fm_a,Q4: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I3 @ P2 ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P2 @ Q4 ) ) ) @ ( epistemic_K_a @ I3 @ Q4 ) ) )
=> ( ~ ( A @ A2 )
=> ( ! [P2: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ P2 )
=> ~ ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P2 @ A2 ) ) )
=> ~ ! [P2: epistemic_fm_a] :
( ? [I3: a] :
( A2
= ( epistemic_K_a @ I3 @ P2 ) )
=> ~ ( epistemic_AK_a @ A @ P2 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_398_K__thm,axiom,
! [A: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o,Q: epistemic_fm_o] : ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_Con_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ Q @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_Con_o @ P @ Q ) @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ).
% K_thm
thf(fact_399_K__thm,axiom,
! [A: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b,Q: epistemic_fm_b] : ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_Con_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ Q @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_Con_b @ P @ Q ) @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ).
% K_thm
thf(fact_400_K__thm,axiom,
! [A: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_Con_a @ P @ Q ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ).
% K_thm
thf(fact_401_kripke_Ofold__congs_I3_J,axiom,
! [R: episte3224730989885420256t_unit,R2: episte3224730989885420256t_unit,V: b > list_char > $o,F2: ( b > list_char > $o ) > b > list_char > $o,F3: ( b > list_char > $o ) > b > list_char > $o] :
( ( R = R2 )
=> ( ( ( episte5693316124195166255t_unit @ R2 )
= V )
=> ( ! [V2: b > list_char > $o] :
( ( V = V2 )
=> ( ( F2 @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte7164994875842154440t_unit @ F2 @ R )
= ( episte7164994875842154440t_unit @ F3 @ R2 ) ) ) ) ) ).
% kripke.fold_congs(3)
thf(fact_402_kripke_Ounfold__congs_I3_J,axiom,
! [R: episte3224730989885420256t_unit,R2: episte3224730989885420256t_unit,V: b > list_char > $o,F2: ( b > list_char > $o ) > b > list_char > $o,F3: ( b > list_char > $o ) > b > list_char > $o] :
( ( R = R2 )
=> ( ( ( episte5693316124195166255t_unit @ R2 )
= V )
=> ( ! [V2: b > list_char > $o] :
( ( V2 = V )
=> ( ( F2 @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte7164994875842154440t_unit @ F2 @ R )
= ( episte7164994875842154440t_unit @ F3 @ R2 ) ) ) ) ) ).
% kripke.unfold_congs(3)
thf(fact_403_kripke_Oext__inject,axiom,
! [Pi: b > list_char > $o,More: product_unit,Pi2: b > list_char > $o,More2: product_unit] :
( ( ( episte6143282164508887558t_unit @ Pi @ More )
= ( episte6143282164508887558t_unit @ Pi2 @ More2 ) )
= ( ( Pi = Pi2 )
& ( More = More2 ) ) ) ).
% kripke.ext_inject
thf(fact_404_kripke_Oext__inject,axiom,
! [Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit,Pi2: set_Epistemic_fm_a > list_char > $o,More2: product_unit] :
( ( ( episte8239586592105053771t_unit @ Pi @ More )
= ( episte8239586592105053771t_unit @ Pi2 @ More2 ) )
= ( ( Pi = Pi2 )
& ( More = More2 ) ) ) ).
% kripke.ext_inject
thf(fact_405_kripke_Oext__induct,axiom,
! [P4: episte2291185870209430742t_unit > $o,R: episte2291185870209430742t_unit] :
( ! [Pi3: b > list_char > $o,More3: product_unit] : ( P4 @ ( episte6143282164508887558t_unit @ Pi3 @ More3 ) )
=> ( P4 @ R ) ) ).
% kripke.ext_induct
thf(fact_406_kripke_Oext__induct,axiom,
! [P4: episte1193835314949844379t_unit > $o,R: episte1193835314949844379t_unit] :
( ! [Pi3: set_Epistemic_fm_a > list_char > $o,More3: product_unit] : ( P4 @ ( episte8239586592105053771t_unit @ Pi3 @ More3 ) )
=> ( P4 @ R ) ) ).
% kripke.ext_induct
thf(fact_407_Ax,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a] :
( ( A @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% Ax
thf(fact_408_kripke_Oupdate__convs_I2_J,axiom,
! [More2: product_unit > product_unit,W2: set_b,K: a > b > set_b,Pi: b > list_char > $o,More: product_unit] :
( ( episte5119575119384494574it_a_b @ More2 @ ( episte4938218340073005086t_unit @ W2 @ K @ ( episte6143282164508887558t_unit @ Pi @ More ) ) )
= ( episte4938218340073005086t_unit @ W2 @ K @ ( episte6143282164508887558t_unit @ Pi @ ( More2 @ More ) ) ) ) ).
% kripke.update_convs(2)
thf(fact_409_kripke_Oupdate__convs_I2_J,axiom,
! [More2: product_unit > product_unit,W2: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte9120385895580347753c_fm_a @ More2 @ ( episte2888590659910966568t_unit @ W2 @ K @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= ( episte2888590659910966568t_unit @ W2 @ K @ ( episte8239586592105053771t_unit @ Pi @ ( More2 @ More ) ) ) ) ).
% kripke.update_convs(2)
thf(fact_410_kripke_Oupdate__convs_I1_J,axiom,
! [Pi2: ( b > list_char > $o ) > b > list_char > $o,W2: set_b,K: a > b > set_b,Pi: b > list_char > $o,More: product_unit] :
( ( episte7164994875842154440t_unit @ Pi2 @ ( episte4938218340073005086t_unit @ W2 @ K @ ( episte6143282164508887558t_unit @ Pi @ More ) ) )
= ( episte4938218340073005086t_unit @ W2 @ K @ ( episte6143282164508887558t_unit @ ( Pi2 @ Pi ) @ More ) ) ) ).
% kripke.update_convs(1)
thf(fact_411_kripke_Oupdate__convs_I1_J,axiom,
! [Pi2: ( set_Epistemic_fm_a > list_char > $o ) > set_Epistemic_fm_a > list_char > $o,W2: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte1857908288096881731t_unit @ Pi2 @ ( episte2888590659910966568t_unit @ W2 @ K @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= ( episte2888590659910966568t_unit @ W2 @ K @ ( episte8239586592105053771t_unit @ ( Pi2 @ Pi ) @ More ) ) ) ).
% kripke.update_convs(1)
thf(fact_412_kripke_Ocases__scheme,axiom,
! [R: episte3224730989885420256t_unit] :
~ ! [W4: set_b,K3: a > b > set_b,Pi3: b > list_char > $o,More3: product_unit] :
( R
!= ( episte4938218340073005086t_unit @ W4 @ K3 @ ( episte6143282164508887558t_unit @ Pi3 @ More3 ) ) ) ).
% kripke.cases_scheme
thf(fact_413_kripke_Ocases__scheme,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W4: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi3: set_Epistemic_fm_a > list_char > $o,More3: product_unit] :
( R
!= ( episte2888590659910966568t_unit @ W4 @ K3 @ ( episte8239586592105053771t_unit @ Pi3 @ More3 ) ) ) ).
% kripke.cases_scheme
thf(fact_414_R1,axiom,
! [A: epistemic_fm_b > $o,P: epistemic_fm_b,Q: epistemic_fm_b] :
( ( epistemic_AK_b @ A @ P )
=> ( ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ P @ Q ) )
=> ( epistemic_AK_b @ A @ Q ) ) ) ).
% R1
thf(fact_415_R1,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ P )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( epistemic_AK_a @ A @ Q ) ) ) ).
% R1
thf(fact_416_K__trans,axiom,
! [A: epistemic_fm_b > $o,P: epistemic_fm_b,Q: epistemic_fm_b,R: epistemic_fm_b] : ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ P @ Q ) @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ Q @ R ) @ ( epistemic_Imp_b @ P @ R ) ) ) ) ).
% K_trans
thf(fact_417_K__trans,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,R: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ Q ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ Q @ R ) @ ( epistemic_Imp_a @ P @ R ) ) ) ) ).
% K_trans
thf(fact_418_K__imp__trans,axiom,
! [A: epistemic_fm_b > $o,P: epistemic_fm_b,Q: epistemic_fm_b,R: epistemic_fm_b] :
( ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ P @ Q ) )
=> ( ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ Q @ R ) )
=> ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ P @ R ) ) ) ) ).
% K_imp_trans
thf(fact_419_K__imp__trans,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,R: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ Q @ R ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ R ) ) ) ) ).
% K_imp_trans
thf(fact_420_K__imp__trans_H,axiom,
! [A: epistemic_fm_b > $o,Q: epistemic_fm_b,R: epistemic_fm_b,P: epistemic_fm_b] :
( ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ Q @ R ) )
=> ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ P @ Q ) @ ( epistemic_Imp_b @ P @ R ) ) ) ) ).
% K_imp_trans'
thf(fact_421_K__imp__trans_H,axiom,
! [A: epistemic_fm_a > $o,Q: epistemic_fm_a,R: epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ Q @ R ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ Q ) @ ( epistemic_Imp_a @ P @ R ) ) ) ) ).
% K_imp_trans'
thf(fact_422_R2,axiom,
! [A: epistemic_fm_b > $o,P: epistemic_fm_b,I2: b] :
( ( epistemic_AK_b @ A @ P )
=> ( epistemic_AK_b @ A @ ( epistemic_K_b @ I2 @ P ) ) ) ).
% R2
thf(fact_423_R2,axiom,
! [A: epistemic_fm_o > $o,P: epistemic_fm_o,I2: $o] :
( ( epistemic_AK_o @ A @ P )
=> ( epistemic_AK_o @ A @ ( epistemic_K_o @ I2 @ P ) ) ) ).
% R2
thf(fact_424_R2,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,I2: a] :
( ( epistemic_AK_a @ A @ P )
=> ( epistemic_AK_a @ A @ ( epistemic_K_a @ I2 @ P ) ) ) ).
% R2
thf(fact_425_A1,axiom,
! [P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% A1
thf(fact_426_K__A2_H,axiom,
! [A: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o,Q: epistemic_fm_o] : ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ P @ Q ) ) @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ Q ) ) ) ) ).
% K_A2'
thf(fact_427_K__A2_H,axiom,
! [A: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b,Q: epistemic_fm_b] : ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ P @ Q ) ) @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_K_b @ I2 @ Q ) ) ) ) ).
% K_A2'
thf(fact_428_K__A2_H,axiom,
! [A: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ Q ) ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ Q ) ) ) ) ).
% K_A2'
thf(fact_429_K__map,axiom,
! [A: epistemic_fm_o > $o,P: epistemic_fm_o,Q: epistemic_fm_o,I2: $o] :
( ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ P @ Q ) )
=> ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ Q ) ) ) ) ).
% K_map
thf(fact_430_K__map,axiom,
! [A: epistemic_fm_b > $o,P: epistemic_fm_b,Q: epistemic_fm_b,I2: b] :
( ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ P @ Q ) )
=> ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_K_b @ I2 @ Q ) ) ) ) ).
% K_map
thf(fact_431_K__map,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,I2: a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ Q ) ) ) ) ).
% K_map
thf(fact_432_K__multi__imply,axiom,
! [A: epistemic_fm_b > $o,A2: epistemic_fm_b,B2: epistemic_fm_b,C: epistemic_fm_b] :
( ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ A2 @ ( epistemic_Imp_b @ B2 @ C ) ) )
=> ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_Con_b @ A2 @ B2 ) @ C ) ) ) ).
% K_multi_imply
thf(fact_433_K__multi__imply,axiom,
! [A: epistemic_fm_a > $o,A2: epistemic_fm_a,B2: epistemic_fm_a,C: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ A2 @ ( epistemic_Imp_a @ B2 @ C ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Con_a @ A2 @ B2 ) @ C ) ) ) ).
% K_multi_imply
thf(fact_434_K__imply__multi,axiom,
! [A: epistemic_fm_b > $o,A2: epistemic_fm_b,B2: epistemic_fm_b,C: epistemic_fm_b] :
( ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ A2 @ B2 ) )
=> ( ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ A2 @ C ) )
=> ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ A2 @ ( epistemic_Con_b @ B2 @ C ) ) ) ) ) ).
% K_imply_multi
thf(fact_435_K__imply__multi,axiom,
! [A: epistemic_fm_a > $o,A2: epistemic_fm_a,B2: epistemic_fm_a,C: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ A2 @ B2 ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ A2 @ C ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ A2 @ ( epistemic_Con_a @ B2 @ C ) ) ) ) ) ).
% K_imply_multi
thf(fact_436_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte3259645218793129376t_unit > $o,P: epistemic_fm_a,M2: episte3259645218793129376t_unit,W: $o] :
( ! [M3: episte3259645218793129376t_unit,W5: $o,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member_o @ W5 @ ( episte876160520765907443t_unit @ M3 ) )
=> ( episte1100597067981672766cs_a_o @ M3 @ W5 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M2 )
=> ( ( member_o @ W @ ( episte876160520765907443t_unit @ M2 ) )
=> ( episte1100597067981672766cs_a_o @ M2 @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_437_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte4442589092411611552t_unit > $o,P: epistemic_fm_a,M2: episte4442589092411611552t_unit,W: set_o] :
( ! [M3: episte4442589092411611552t_unit,W5: set_o,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member_set_o @ W5 @ ( episte7994720633578350067t_unit @ M3 ) )
=> ( episte5787648899026124830_set_o @ M3 @ W5 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M2 )
=> ( ( member_set_o @ W @ ( episte7994720633578350067t_unit @ M2 ) )
=> ( episte5787648899026124830_set_o @ M2 @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_438_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte8559422309061589728t_unit > $o,P: epistemic_fm_a,M2: episte8559422309061589728t_unit,W: set_b] :
( ! [M3: episte8559422309061589728t_unit,W5: set_b,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member_set_b @ W5 @ ( episte2726209618768418739t_unit @ M3 ) )
=> ( episte586216548244424325_set_b @ M3 @ W5 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M2 )
=> ( ( member_set_b @ W @ ( episte2726209618768418739t_unit @ M2 ) )
=> ( episte586216548244424325_set_b @ M2 @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_439_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,P: epistemic_fm_a,M2: episte6182337868402532512t_unit,W: a] :
( ! [M3: episte6182337868402532512t_unit,W5: a,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member_a @ W5 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ W5 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M2 )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_440_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,P: epistemic_fm_a,M2: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit,W5: set_Epistemic_fm_a,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member536094252920883875c_fm_a @ W5 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ W5 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M2 )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_441_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte3224730989885420256t_unit > $o,P: epistemic_fm_a,M2: episte3224730989885420256t_unit,W: b] :
( ! [M3: episte3224730989885420256t_unit,W5: b,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member_b @ W5 @ ( episte1782384855165018035t_unit @ M3 ) )
=> ( episte295617885132580261cs_a_b @ M3 @ W5 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M2 )
=> ( ( member_b @ W @ ( episte1782384855165018035t_unit @ M2 ) )
=> ( episte295617885132580261cs_a_b @ M2 @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_442_kripke_Oselect__convs_I1_J,axiom,
! [W2: set_b,K: a > b > set_b,Pi: b > list_char > $o,More: product_unit] :
( ( episte5693316124195166255t_unit @ ( episte4938218340073005086t_unit @ W2 @ K @ ( episte6143282164508887558t_unit @ Pi @ More ) ) )
= Pi ) ).
% kripke.select_convs(1)
thf(fact_443_kripke_Oselect__convs_I1_J,axiom,
! [W2: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte2398645135750866164t_unit @ ( episte2888590659910966568t_unit @ W2 @ K @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= Pi ) ).
% kripke.select_convs(1)
thf(fact_444_kripke_Oselect__convs_I2_J,axiom,
! [W2: set_b,K: a > b > set_b,Pi: b > list_char > $o,More: product_unit] :
( ( episte7270682542085591125t_unit @ ( episte4938218340073005086t_unit @ W2 @ K @ ( episte6143282164508887558t_unit @ Pi @ More ) ) )
= More ) ).
% kripke.select_convs(2)
thf(fact_445_kripke_Oselect__convs_I2_J,axiom,
! [W2: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte5479201149095757850t_unit @ ( episte2888590659910966568t_unit @ W2 @ K @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= More ) ).
% kripke.select_convs(2)
thf(fact_446_K__LK,axiom,
! [A: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o] : ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ P @ epistemic_FF_o ) @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ epistemic_FF_o ) ) ) ).
% K_LK
thf(fact_447_K__LK,axiom,
! [A: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b] : ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ P @ epistemic_FF_b ) @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ epistemic_FF_b ) ) ) ).
% K_LK
thf(fact_448_K__LK,axiom,
! [A: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ epistemic_FF_a ) ) ) ).
% K_LK
thf(fact_449_K__L__dual,axiom,
! [A: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o] : ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ P @ epistemic_FF_o ) @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I2 @ P ) ) ) ).
% K_L_dual
thf(fact_450_K__L__dual,axiom,
! [A: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b] : ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ P @ epistemic_FF_b ) @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ epistemic_FF_b ) @ ( epistemic_K_b @ I2 @ P ) ) ) ).
% K_L_dual
thf(fact_451_K__L__dual,axiom,
! [A: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I2 @ P ) ) ) ).
% K_L_dual
thf(fact_452_A2,axiom,
! [A: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o,Q: epistemic_fm_o] : ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_Con_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ P @ Q ) ) ) @ ( epistemic_K_o @ I2 @ Q ) ) ) ).
% A2
thf(fact_453_A2,axiom,
! [A: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b,Q: epistemic_fm_b] : ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_Con_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ P @ Q ) ) ) @ ( epistemic_K_b @ I2 @ Q ) ) ) ).
% A2
thf(fact_454_A2,axiom,
! [A: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ Q ) ) ) @ ( epistemic_K_a @ I2 @ Q ) ) ) ).
% A2
thf(fact_455_K__conjunction__in,axiom,
! [A: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o,Q: epistemic_fm_o] : ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Con_o @ P @ Q ) ) @ ( epistemic_Con_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ Q ) ) ) ) ).
% K_conjunction_in
thf(fact_456_K__conjunction__in,axiom,
! [A: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b,Q: epistemic_fm_b] : ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Con_b @ P @ Q ) ) @ ( epistemic_Con_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_K_b @ I2 @ Q ) ) ) ) ).
% K_conjunction_in
thf(fact_457_K__conjunction__in,axiom,
! [A: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Con_a @ P @ Q ) ) @ ( epistemic_Con_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ Q ) ) ) ) ).
% K_conjunction_in
thf(fact_458_K__conjunction__out,axiom,
! [A: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o,Q: epistemic_fm_o] : ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_Con_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ Q ) ) @ ( epistemic_K_o @ I2 @ ( epistemic_Con_o @ P @ Q ) ) ) ) ).
% K_conjunction_out
thf(fact_459_K__conjunction__out,axiom,
! [A: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b,Q: epistemic_fm_b] : ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_Con_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_K_b @ I2 @ Q ) ) @ ( epistemic_K_b @ I2 @ ( epistemic_Con_b @ P @ Q ) ) ) ) ).
% K_conjunction_out
thf(fact_460_K__conjunction__out,axiom,
! [A: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ Q ) ) @ ( epistemic_K_a @ I2 @ ( epistemic_Con_a @ P @ Q ) ) ) ) ).
% K_conjunction_out
thf(fact_461_K__conj__imply__factor,axiom,
! [A: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o,Q: epistemic_fm_o,R: epistemic_fm_o] : ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_Con_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ Q ) ) @ R ) @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Con_o @ P @ Q ) ) @ R ) ) ) ).
% K_conj_imply_factor
thf(fact_462_K__conj__imply__factor,axiom,
! [A: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b,Q: epistemic_fm_b,R: epistemic_fm_b] : ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_Con_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_K_b @ I2 @ Q ) ) @ R ) @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Con_b @ P @ Q ) ) @ R ) ) ) ).
% K_conj_imply_factor
thf(fact_463_K__conj__imply__factor,axiom,
! [A: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a,Q: epistemic_fm_a,R: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ Q ) ) @ R ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Con_a @ P @ Q ) ) @ R ) ) ) ).
% K_conj_imply_factor
thf(fact_464_KB4__5,axiom,
! [A: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o] :
( ( ord_le1608183570384977833fm_o_o @ epistemic_AxB_o @ A )
=> ( ( ord_le1608183570384977833fm_o_o @ epistemic_Ax4_o @ A )
=> ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ P @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ P @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ) ) ).
% KB4_5
thf(fact_465_KB4__5,axiom,
! [A: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b] :
( ( ord_le1256275114586606850fm_b_o @ epistemic_AxB_b @ A )
=> ( ( ord_le1256275114586606850fm_b_o @ epistemic_Ax4_b @ A )
=> ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ P @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ P @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ) ) ).
% KB4_5
thf(fact_466_KB4__5,axiom,
! [A: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% KB4_5
thf(fact_467_S5_H__B,axiom,
! [A: epistemic_fm_o > $o,P: epistemic_fm_o,I2: $o] :
( ( ord_le1608183570384977833fm_o_o @ epistemic_AxT_o @ A )
=> ( ( ord_le1608183570384977833fm_o_o @ epistemic_Ax5_o @ A )
=> ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ P @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ P @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ) ) ).
% S5'_B
thf(fact_468_S5_H__B,axiom,
! [A: epistemic_fm_b > $o,P: epistemic_fm_b,I2: b] :
( ( ord_le1256275114586606850fm_b_o @ epistemic_AxT_b @ A )
=> ( ( ord_le1256275114586606850fm_b_o @ epistemic_Ax5_b @ A )
=> ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ P @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ P @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ) ) ).
% S5'_B
thf(fact_469_S5_H__B,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,I2: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% S5'_B
thf(fact_470_S5_H__4,axiom,
! [A: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o] :
( ( ord_le1608183570384977833fm_o_o @ epistemic_AxT_o @ A )
=> ( ( ord_le1608183570384977833fm_o_o @ epistemic_Ax5_o @ A )
=> ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ ( epistemic_K_o @ I2 @ ( epistemic_K_o @ I2 @ P ) ) ) ) ) ) ).
% S5'_4
thf(fact_471_S5_H__4,axiom,
! [A: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b] :
( ( ord_le1256275114586606850fm_b_o @ epistemic_AxT_b @ A )
=> ( ( ord_le1256275114586606850fm_b_o @ epistemic_Ax5_b @ A )
=> ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ ( epistemic_K_b @ I2 @ ( epistemic_K_b @ I2 @ P ) ) ) ) ) ) ).
% S5'_4
thf(fact_472_S5_H__4,axiom,
! [A: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ ( epistemic_K_a @ I2 @ ( epistemic_K_a @ I2 @ P ) ) ) ) ) ) ).
% S5'_4
thf(fact_473_K5__L,axiom,
! [A: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o] :
( ( ord_le1608183570384977833fm_o_o @ epistemic_Ax5_o @ A )
=> ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ P ) @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ ( epistemic_K_o @ I2 @ P ) ) ) ) ).
% K5_L
thf(fact_474_K5__L,axiom,
! [A: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b] :
( ( ord_le1256275114586606850fm_b_o @ epistemic_Ax5_b @ A )
=> ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ P ) @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ ( epistemic_K_b @ I2 @ P ) ) ) ) ).
% K5_L
thf(fact_475_K5__L,axiom,
! [A: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I2 @ P ) ) ) ) ).
% K5_L
thf(fact_476_K4__L,axiom,
! [A: epistemic_fm_o > $o,I2: $o,P: epistemic_fm_o] :
( ( ord_le1608183570384977833fm_o_o @ epistemic_Ax4_o @ A )
=> ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ P @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ epistemic_FF_o ) ) @ epistemic_FF_o ) @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ P @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ).
% K4_L
thf(fact_477_K4__L,axiom,
! [A: epistemic_fm_b > $o,I2: b,P: epistemic_fm_b] :
( ( ord_le1256275114586606850fm_b_o @ epistemic_Ax4_b @ A )
=> ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ P @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ epistemic_FF_b ) ) @ epistemic_FF_b ) @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ P @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ).
% K4_L
thf(fact_478_K4__L,axiom,
! [A: epistemic_fm_a > $o,I2: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% K4_L
thf(fact_479_T__L,axiom,
! [A: epistemic_fm_o > $o,P: epistemic_fm_o,I2: $o] :
( ( ord_le1608183570384977833fm_o_o @ epistemic_AxT_o @ A )
=> ( epistemic_AK_o @ A @ ( epistemic_Imp_o @ P @ ( epistemic_Imp_o @ ( epistemic_K_o @ I2 @ ( epistemic_Imp_o @ P @ epistemic_FF_o ) ) @ epistemic_FF_o ) ) ) ) ).
% T_L
thf(fact_480_T__L,axiom,
! [A: epistemic_fm_b > $o,P: epistemic_fm_b,I2: b] :
( ( ord_le1256275114586606850fm_b_o @ epistemic_AxT_b @ A )
=> ( epistemic_AK_b @ A @ ( epistemic_Imp_b @ P @ ( epistemic_Imp_b @ ( epistemic_K_b @ I2 @ ( epistemic_Imp_b @ P @ epistemic_FF_b ) ) @ epistemic_FF_b ) ) ) ) ).
% T_L
thf(fact_481_T__L,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,I2: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% T_L
thf(fact_482_S5_H__S5,axiom,
! [P: epistemic_fm_a] :
( ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax5_a @ P3 ) )
@ P )
=> ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_AxB_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ P ) ) ).
% S5'_S5
thf(fact_483_S5__S5_H,axiom,
! [P: epistemic_fm_a] :
( ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_AxB_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ P )
=> ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax5_a @ P3 ) )
@ P ) ) ).
% S5_S5'
thf(fact_484_fm_Oinject_I1_J,axiom,
! [X22: list_char,Y22: list_char] :
( ( ( epistemic_Pro_a @ X22 )
= ( epistemic_Pro_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% fm.inject(1)
thf(fact_485_fm_Oinject_I1_J,axiom,
! [X22: list_char,Y22: list_char] :
( ( ( epistemic_Pro_o @ X22 )
= ( epistemic_Pro_o @ Y22 ) )
= ( X22 = Y22 ) ) ).
% fm.inject(1)
thf(fact_486_fm_Oinject_I2_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( ( epistemic_Dis_a @ X31 @ X32 )
= ( epistemic_Dis_a @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fm.inject(2)
thf(fact_487_fm_Oinject_I2_J,axiom,
! [X31: epistemic_fm_b,X32: epistemic_fm_b,Y31: epistemic_fm_b,Y32: epistemic_fm_b] :
( ( ( epistemic_Dis_b @ X31 @ X32 )
= ( epistemic_Dis_b @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fm.inject(2)
thf(fact_488_le__inf__iff,axiom,
! [X3: $o,Y2: $o,Z2: $o] :
( ( ord_less_eq_o @ X3 @ ( inf_inf_o @ Y2 @ Z2 ) )
= ( ( ord_less_eq_o @ X3 @ Y2 )
& ( ord_less_eq_o @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_489_le__inf__iff,axiom,
! [X3: set_o,Y2: set_o,Z2: set_o] :
( ( ord_less_eq_set_o @ X3 @ ( inf_inf_set_o @ Y2 @ Z2 ) )
= ( ( ord_less_eq_set_o @ X3 @ Y2 )
& ( ord_less_eq_set_o @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_490_le__inf__iff,axiom,
! [X3: set_set_b,Y2: set_set_b,Z2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ ( inf_inf_set_set_b @ Y2 @ Z2 ) )
= ( ( ord_le3795704787696855135_set_b @ X3 @ Y2 )
& ( ord_le3795704787696855135_set_b @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_491_le__inf__iff,axiom,
! [X3: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ ( inf_in3606484609122063093fm_a_o @ Y2 @ Z2 ) )
= ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 )
& ( ord_le4043730696559282883fm_a_o @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_492_le__inf__iff,axiom,
! [X3: b > $o,Y2: b > $o,Z2: b > $o] :
( ( ord_less_eq_b_o @ X3 @ ( inf_inf_b_o @ Y2 @ Z2 ) )
= ( ( ord_less_eq_b_o @ X3 @ Y2 )
& ( ord_less_eq_b_o @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_493_le__inf__iff,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X3 @ ( inf_inf_set_b @ Y2 @ Z2 ) )
= ( ( ord_less_eq_set_b @ X3 @ Y2 )
& ( ord_less_eq_set_b @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_494_inf_Obounded__iff,axiom,
! [A2: $o,B2: $o,C: $o] :
( ( ord_less_eq_o @ A2 @ ( inf_inf_o @ B2 @ C ) )
= ( ( ord_less_eq_o @ A2 @ B2 )
& ( ord_less_eq_o @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_495_inf_Obounded__iff,axiom,
! [A2: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( inf_inf_set_o @ B2 @ C ) )
= ( ( ord_less_eq_set_o @ A2 @ B2 )
& ( ord_less_eq_set_o @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_496_inf_Obounded__iff,axiom,
! [A2: set_set_b,B2: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ ( inf_inf_set_set_b @ B2 @ C ) )
= ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
& ( ord_le3795704787696855135_set_b @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_497_inf_Obounded__iff,axiom,
! [A2: epistemic_fm_a > $o,B2: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ ( inf_in3606484609122063093fm_a_o @ B2 @ C ) )
= ( ( ord_le4043730696559282883fm_a_o @ A2 @ B2 )
& ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_498_inf_Obounded__iff,axiom,
! [A2: b > $o,B2: b > $o,C: b > $o] :
( ( ord_less_eq_b_o @ A2 @ ( inf_inf_b_o @ B2 @ C ) )
= ( ( ord_less_eq_b_o @ A2 @ B2 )
& ( ord_less_eq_b_o @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_499_inf_Obounded__iff,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) )
= ( ( ord_less_eq_set_b @ A2 @ B2 )
& ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_500_fm_Odistinct_I11_J,axiom,
! [X22: list_char,X31: epistemic_fm_b,X32: epistemic_fm_b] :
( ( epistemic_Pro_b @ X22 )
!= ( epistemic_Dis_b @ X31 @ X32 ) ) ).
% fm.distinct(11)
thf(fact_501_fm_Odistinct_I11_J,axiom,
! [X22: list_char,X31: epistemic_fm_a,X32: epistemic_fm_a] :
( ( epistemic_Pro_a @ X22 )
!= ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.distinct(11)
thf(fact_502_fm_Odistinct_I11_J,axiom,
! [X22: list_char,X31: epistemic_fm_o,X32: epistemic_fm_o] :
( ( epistemic_Pro_o @ X22 )
!= ( epistemic_Dis_o @ X31 @ X32 ) ) ).
% fm.distinct(11)
thf(fact_503_Int__def,axiom,
( inf_inf_set_set_o
= ( ^ [A4: set_set_o,B4: set_set_o] :
( collect_set_o
@ ^ [X2: set_o] :
( ( member_set_o @ X2 @ A4 )
& ( member_set_o @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_504_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A4 )
& ( member_a @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_505_Int__def,axiom,
( inf_in1884693029477671368c_fm_a
= ( ^ [A4: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A4 )
& ( member536094252920883875c_fm_a @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_506_Int__def,axiom,
( inf_in2882328776738922472c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A4 )
& ( member6642669571620171971c_fm_a @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_507_Int__def,axiom,
( inf_inf_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A4 )
& ( member_o @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_508_Int__def,axiom,
( inf_inf_set_set_b
= ( ^ [A4: set_set_b,B4: set_set_b] :
( collect_set_b
@ ^ [X2: set_b] :
( ( member_set_b @ X2 @ A4 )
& ( member_set_b @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_509_Int__def,axiom,
( inf_inf_set_b
= ( ^ [A4: set_b,B4: set_b] :
( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A4 )
& ( member_b @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_510_Int__Collect,axiom,
! [X3: set_o,A: set_set_o,P4: set_o > $o] :
( ( member_set_o @ X3 @ ( inf_inf_set_set_o @ A @ ( collect_set_o @ P4 ) ) )
= ( ( member_set_o @ X3 @ A )
& ( P4 @ X3 ) ) ) ).
% Int_Collect
thf(fact_511_Int__Collect,axiom,
! [X3: a,A: set_a,P4: a > $o] :
( ( member_a @ X3 @ ( inf_inf_set_a @ A @ ( collect_a @ P4 ) ) )
= ( ( member_a @ X3 @ A )
& ( P4 @ X3 ) ) ) ).
% Int_Collect
thf(fact_512_Int__Collect,axiom,
! [X3: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ X3 @ ( inf_in1884693029477671368c_fm_a @ A @ ( collec2519470961442302949c_fm_a @ P4 ) ) )
= ( ( member536094252920883875c_fm_a @ X3 @ A )
& ( P4 @ X3 ) ) ) ).
% Int_Collect
thf(fact_513_Int__Collect,axiom,
! [X3: epistemic_fm_a,A: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ X3 @ ( inf_in2882328776738922472c_fm_a @ A @ ( collec4904205152690461189c_fm_a @ P4 ) ) )
= ( ( member6642669571620171971c_fm_a @ X3 @ A )
& ( P4 @ X3 ) ) ) ).
% Int_Collect
thf(fact_514_Int__Collect,axiom,
! [X3: $o,A: set_o,P4: $o > $o] :
( ( member_o @ X3 @ ( inf_inf_set_o @ A @ ( collect_o @ P4 ) ) )
= ( ( member_o @ X3 @ A )
& ( P4 @ X3 ) ) ) ).
% Int_Collect
thf(fact_515_Int__Collect,axiom,
! [X3: set_b,A: set_set_b,P4: set_b > $o] :
( ( member_set_b @ X3 @ ( inf_inf_set_set_b @ A @ ( collect_set_b @ P4 ) ) )
= ( ( member_set_b @ X3 @ A )
& ( P4 @ X3 ) ) ) ).
% Int_Collect
thf(fact_516_Int__Collect,axiom,
! [X3: b,A: set_b,P4: b > $o] :
( ( member_b @ X3 @ ( inf_inf_set_b @ A @ ( collect_b @ P4 ) ) )
= ( ( member_b @ X3 @ A )
& ( P4 @ X3 ) ) ) ).
% Int_Collect
thf(fact_517_inf__set__def,axiom,
( inf_inf_set_set_o
= ( ^ [A4: set_set_o,B4: set_set_o] :
( collect_set_o
@ ( inf_inf_set_o_o
@ ^ [X2: set_o] : ( member_set_o @ X2 @ A4 )
@ ^ [X2: set_o] : ( member_set_o @ X2 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_518_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A4 )
@ ^ [X2: a] : ( member_a @ X2 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_519_inf__set__def,axiom,
( inf_in1884693029477671368c_fm_a
= ( ^ [A4: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( collec2519470961442302949c_fm_a
@ ( inf_in5938521974262266133fm_a_o
@ ^ [X2: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X2 @ A4 )
@ ^ [X2: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X2 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_520_inf__set__def,axiom,
( inf_in2882328776738922472c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ( inf_in3606484609122063093fm_a_o
@ ^ [X2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X2 @ A4 )
@ ^ [X2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X2 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_521_inf__set__def,axiom,
( inf_inf_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ( inf_inf_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A4 )
@ ^ [X2: $o] : ( member_o @ X2 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_522_inf__set__def,axiom,
( inf_inf_set_set_b
= ( ^ [A4: set_set_b,B4: set_set_b] :
( collect_set_b
@ ( inf_inf_set_b_o
@ ^ [X2: set_b] : ( member_set_b @ X2 @ A4 )
@ ^ [X2: set_b] : ( member_set_b @ X2 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_523_inf__set__def,axiom,
( inf_inf_set_b
= ( ^ [A4: set_b,B4: set_b] :
( collect_b
@ ( inf_inf_b_o
@ ^ [X2: b] : ( member_b @ X2 @ A4 )
@ ^ [X2: b] : ( member_b @ X2 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_524_Collect__conj__eq,axiom,
! [P4: set_Epistemic_fm_a > $o,Q2: set_Epistemic_fm_a > $o] :
( ( collec2519470961442302949c_fm_a
@ ^ [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
& ( Q2 @ X2 ) ) )
= ( inf_in1884693029477671368c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_525_Collect__conj__eq,axiom,
! [P4: epistemic_fm_a > $o,Q2: epistemic_fm_a > $o] :
( ( collec4904205152690461189c_fm_a
@ ^ [X2: epistemic_fm_a] :
( ( P4 @ X2 )
& ( Q2 @ X2 ) ) )
= ( inf_in2882328776738922472c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_526_Collect__conj__eq,axiom,
! [P4: $o > $o,Q2: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( P4 @ X2 )
& ( Q2 @ X2 ) ) )
= ( inf_inf_set_o @ ( collect_o @ P4 ) @ ( collect_o @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_527_Collect__conj__eq,axiom,
! [P4: set_b > $o,Q2: set_b > $o] :
( ( collect_set_b
@ ^ [X2: set_b] :
( ( P4 @ X2 )
& ( Q2 @ X2 ) ) )
= ( inf_inf_set_set_b @ ( collect_set_b @ P4 ) @ ( collect_set_b @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_528_Collect__conj__eq,axiom,
! [P4: b > $o,Q2: b > $o] :
( ( collect_b
@ ^ [X2: b] :
( ( P4 @ X2 )
& ( Q2 @ X2 ) ) )
= ( inf_inf_set_b @ ( collect_b @ P4 ) @ ( collect_b @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_529_inf__sup__ord_I2_J,axiom,
! [X3: $o,Y2: $o] : ( ord_less_eq_o @ ( inf_inf_o @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_530_inf__sup__ord_I2_J,axiom,
! [X3: set_o,Y2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_531_inf__sup__ord_I2_J,axiom,
! [X3: set_set_b,Y2: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_532_inf__sup__ord_I2_J,axiom,
! [X3: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_533_inf__sup__ord_I2_J,axiom,
! [X3: b > $o,Y2: b > $o] : ( ord_less_eq_b_o @ ( inf_inf_b_o @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_534_inf__sup__ord_I2_J,axiom,
! [X3: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_535_inf__sup__ord_I1_J,axiom,
! [X3: $o,Y2: $o] : ( ord_less_eq_o @ ( inf_inf_o @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_536_inf__sup__ord_I1_J,axiom,
! [X3: set_o,Y2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_537_inf__sup__ord_I1_J,axiom,
! [X3: set_set_b,Y2: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_538_inf__sup__ord_I1_J,axiom,
! [X3: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_539_inf__sup__ord_I1_J,axiom,
! [X3: b > $o,Y2: b > $o] : ( ord_less_eq_b_o @ ( inf_inf_b_o @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_540_inf__sup__ord_I1_J,axiom,
! [X3: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_541_inf__le1,axiom,
! [X3: $o,Y2: $o] : ( ord_less_eq_o @ ( inf_inf_o @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_542_inf__le1,axiom,
! [X3: set_o,Y2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_543_inf__le1,axiom,
! [X3: set_set_b,Y2: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_544_inf__le1,axiom,
! [X3: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_545_inf__le1,axiom,
! [X3: b > $o,Y2: b > $o] : ( ord_less_eq_b_o @ ( inf_inf_b_o @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_546_inf__le1,axiom,
! [X3: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_547_inf__le2,axiom,
! [X3: $o,Y2: $o] : ( ord_less_eq_o @ ( inf_inf_o @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_548_inf__le2,axiom,
! [X3: set_o,Y2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_549_inf__le2,axiom,
! [X3: set_set_b,Y2: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_550_inf__le2,axiom,
! [X3: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_551_inf__le2,axiom,
! [X3: b > $o,Y2: b > $o] : ( ord_less_eq_b_o @ ( inf_inf_b_o @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_552_inf__le2,axiom,
! [X3: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_553_le__infE,axiom,
! [X3: $o,A2: $o,B2: $o] :
( ( ord_less_eq_o @ X3 @ ( inf_inf_o @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_o @ X3 @ A2 )
=> ~ ( ord_less_eq_o @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_554_le__infE,axiom,
! [X3: set_o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ X3 @ ( inf_inf_set_o @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_o @ X3 @ A2 )
=> ~ ( ord_less_eq_set_o @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_555_le__infE,axiom,
! [X3: set_set_b,A2: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ ( inf_inf_set_set_b @ A2 @ B2 ) )
=> ~ ( ( ord_le3795704787696855135_set_b @ X3 @ A2 )
=> ~ ( ord_le3795704787696855135_set_b @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_556_le__infE,axiom,
! [X3: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ ( inf_in3606484609122063093fm_a_o @ A2 @ B2 ) )
=> ~ ( ( ord_le4043730696559282883fm_a_o @ X3 @ A2 )
=> ~ ( ord_le4043730696559282883fm_a_o @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_557_le__infE,axiom,
! [X3: b > $o,A2: b > $o,B2: b > $o] :
( ( ord_less_eq_b_o @ X3 @ ( inf_inf_b_o @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_b_o @ X3 @ A2 )
=> ~ ( ord_less_eq_b_o @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_558_le__infE,axiom,
! [X3: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ X3 @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_b @ X3 @ A2 )
=> ~ ( ord_less_eq_set_b @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_559_le__infI,axiom,
! [X3: $o,A2: $o,B2: $o] :
( ( ord_less_eq_o @ X3 @ A2 )
=> ( ( ord_less_eq_o @ X3 @ B2 )
=> ( ord_less_eq_o @ X3 @ ( inf_inf_o @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_560_le__infI,axiom,
! [X3: set_o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ X3 @ A2 )
=> ( ( ord_less_eq_set_o @ X3 @ B2 )
=> ( ord_less_eq_set_o @ X3 @ ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_561_le__infI,axiom,
! [X3: set_set_b,A2: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ A2 )
=> ( ( ord_le3795704787696855135_set_b @ X3 @ B2 )
=> ( ord_le3795704787696855135_set_b @ X3 @ ( inf_inf_set_set_b @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_562_le__infI,axiom,
! [X3: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ B2 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( inf_in3606484609122063093fm_a_o @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_563_le__infI,axiom,
! [X3: b > $o,A2: b > $o,B2: b > $o] :
( ( ord_less_eq_b_o @ X3 @ A2 )
=> ( ( ord_less_eq_b_o @ X3 @ B2 )
=> ( ord_less_eq_b_o @ X3 @ ( inf_inf_b_o @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_564_le__infI,axiom,
! [X3: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ X3 @ A2 )
=> ( ( ord_less_eq_set_b @ X3 @ B2 )
=> ( ord_less_eq_set_b @ X3 @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_565_inf__mono,axiom,
! [A2: $o,C: $o,B2: $o,D: $o] :
( ( ord_less_eq_o @ A2 @ C )
=> ( ( ord_less_eq_o @ B2 @ D )
=> ( ord_less_eq_o @ ( inf_inf_o @ A2 @ B2 ) @ ( inf_inf_o @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_566_inf__mono,axiom,
! [A2: set_o,C: set_o,B2: set_o,D: set_o] :
( ( ord_less_eq_set_o @ A2 @ C )
=> ( ( ord_less_eq_set_o @ B2 @ D )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ ( inf_inf_set_o @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_567_inf__mono,axiom,
! [A2: set_set_b,C: set_set_b,B2: set_set_b,D: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ C )
=> ( ( ord_le3795704787696855135_set_b @ B2 @ D )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ ( inf_inf_set_set_b @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_568_inf__mono,axiom,
! [A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o,B2: epistemic_fm_a > $o,D: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ C )
=> ( ( ord_le4043730696559282883fm_a_o @ B2 @ D )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B2 ) @ ( inf_in3606484609122063093fm_a_o @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_569_inf__mono,axiom,
! [A2: b > $o,C: b > $o,B2: b > $o,D: b > $o] :
( ( ord_less_eq_b_o @ A2 @ C )
=> ( ( ord_less_eq_b_o @ B2 @ D )
=> ( ord_less_eq_b_o @ ( inf_inf_b_o @ A2 @ B2 ) @ ( inf_inf_b_o @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_570_inf__mono,axiom,
! [A2: set_b,C: set_b,B2: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ( ord_less_eq_set_b @ B2 @ D )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( inf_inf_set_b @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_571_le__infI1,axiom,
! [A2: $o,X3: $o,B2: $o] :
( ( ord_less_eq_o @ A2 @ X3 )
=> ( ord_less_eq_o @ ( inf_inf_o @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_572_le__infI1,axiom,
! [A2: set_o,X3: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ X3 )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_573_le__infI1,axiom,
! [A2: set_set_b,X3: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ X3 )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_574_le__infI1,axiom,
! [A2: epistemic_fm_a > $o,X3: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ X3 )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_575_le__infI1,axiom,
! [A2: b > $o,X3: b > $o,B2: b > $o] :
( ( ord_less_eq_b_o @ A2 @ X3 )
=> ( ord_less_eq_b_o @ ( inf_inf_b_o @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_576_le__infI1,axiom,
! [A2: set_b,X3: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ X3 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_577_le__infI2,axiom,
! [B2: $o,X3: $o,A2: $o] :
( ( ord_less_eq_o @ B2 @ X3 )
=> ( ord_less_eq_o @ ( inf_inf_o @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_578_le__infI2,axiom,
! [B2: set_o,X3: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ X3 )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_579_le__infI2,axiom,
! [B2: set_set_b,X3: set_set_b,A2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B2 @ X3 )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_580_le__infI2,axiom,
! [B2: epistemic_fm_a > $o,X3: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B2 @ X3 )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_581_le__infI2,axiom,
! [B2: b > $o,X3: b > $o,A2: b > $o] :
( ( ord_less_eq_b_o @ B2 @ X3 )
=> ( ord_less_eq_b_o @ ( inf_inf_b_o @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_582_le__infI2,axiom,
! [B2: set_b,X3: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ X3 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_583_inf_OorderE,axiom,
! [A2: $o,B2: $o] :
( ( ord_less_eq_o @ A2 @ B2 )
=> ( A2
= ( inf_inf_o @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_584_inf_OorderE,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_o @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_585_inf_OorderE,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_set_b @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_586_inf_OorderE,axiom,
! [A2: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B2 )
=> ( A2
= ( inf_in3606484609122063093fm_a_o @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_587_inf_OorderE,axiom,
! [A2: b > $o,B2: b > $o] :
( ( ord_less_eq_b_o @ A2 @ B2 )
=> ( A2
= ( inf_inf_b_o @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_588_inf_OorderE,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_b @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_589_inf_OorderI,axiom,
! [A2: $o,B2: $o] :
( ( A2
= ( inf_inf_o @ A2 @ B2 ) )
=> ( ord_less_eq_o @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_590_inf_OorderI,axiom,
! [A2: set_o,B2: set_o] :
( ( A2
= ( inf_inf_set_o @ A2 @ B2 ) )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_591_inf_OorderI,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( A2
= ( inf_inf_set_set_b @ A2 @ B2 ) )
=> ( ord_le3795704787696855135_set_b @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_592_inf_OorderI,axiom,
! [A2: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( A2
= ( inf_in3606484609122063093fm_a_o @ A2 @ B2 ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_593_inf_OorderI,axiom,
! [A2: b > $o,B2: b > $o] :
( ( A2
= ( inf_inf_b_o @ A2 @ B2 ) )
=> ( ord_less_eq_b_o @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_594_inf_OorderI,axiom,
! [A2: set_b,B2: set_b] :
( ( A2
= ( inf_inf_set_b @ A2 @ B2 ) )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_595_inf__unique,axiom,
! [F2: $o > $o > $o,X3: $o,Y2: $o] :
( ! [X: $o,Y3: $o] : ( ord_less_eq_o @ ( F2 @ X @ Y3 ) @ X )
=> ( ! [X: $o,Y3: $o] : ( ord_less_eq_o @ ( F2 @ X @ Y3 ) @ Y3 )
=> ( ! [X: $o,Y3: $o,Z3: $o] :
( ( ord_less_eq_o @ X @ Y3 )
=> ( ( ord_less_eq_o @ X @ Z3 )
=> ( ord_less_eq_o @ X @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_o @ X3 @ Y2 )
= ( F2 @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_596_inf__unique,axiom,
! [F2: set_o > set_o > set_o,X3: set_o,Y2: set_o] :
( ! [X: set_o,Y3: set_o] : ( ord_less_eq_set_o @ ( F2 @ X @ Y3 ) @ X )
=> ( ! [X: set_o,Y3: set_o] : ( ord_less_eq_set_o @ ( F2 @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_o,Y3: set_o,Z3: set_o] :
( ( ord_less_eq_set_o @ X @ Y3 )
=> ( ( ord_less_eq_set_o @ X @ Z3 )
=> ( ord_less_eq_set_o @ X @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_o @ X3 @ Y2 )
= ( F2 @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_597_inf__unique,axiom,
! [F2: set_set_b > set_set_b > set_set_b,X3: set_set_b,Y2: set_set_b] :
( ! [X: set_set_b,Y3: set_set_b] : ( ord_le3795704787696855135_set_b @ ( F2 @ X @ Y3 ) @ X )
=> ( ! [X: set_set_b,Y3: set_set_b] : ( ord_le3795704787696855135_set_b @ ( F2 @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_set_b,Y3: set_set_b,Z3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X @ Y3 )
=> ( ( ord_le3795704787696855135_set_b @ X @ Z3 )
=> ( ord_le3795704787696855135_set_b @ X @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_b @ X3 @ Y2 )
= ( F2 @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_598_inf__unique,axiom,
! [F2: ( epistemic_fm_a > $o ) > ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X3: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ! [X: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( F2 @ X @ Y3 ) @ X )
=> ( ! [X: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( F2 @ X @ Y3 ) @ Y3 )
=> ( ! [X: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o,Z3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y3 )
=> ( ( ord_le4043730696559282883fm_a_o @ X @ Z3 )
=> ( ord_le4043730696559282883fm_a_o @ X @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_in3606484609122063093fm_a_o @ X3 @ Y2 )
= ( F2 @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_599_inf__unique,axiom,
! [F2: ( b > $o ) > ( b > $o ) > b > $o,X3: b > $o,Y2: b > $o] :
( ! [X: b > $o,Y3: b > $o] : ( ord_less_eq_b_o @ ( F2 @ X @ Y3 ) @ X )
=> ( ! [X: b > $o,Y3: b > $o] : ( ord_less_eq_b_o @ ( F2 @ X @ Y3 ) @ Y3 )
=> ( ! [X: b > $o,Y3: b > $o,Z3: b > $o] :
( ( ord_less_eq_b_o @ X @ Y3 )
=> ( ( ord_less_eq_b_o @ X @ Z3 )
=> ( ord_less_eq_b_o @ X @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_b_o @ X3 @ Y2 )
= ( F2 @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_600_inf__unique,axiom,
! [F2: set_b > set_b > set_b,X3: set_b,Y2: set_b] :
( ! [X: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( F2 @ X @ Y3 ) @ X )
=> ( ! [X: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( F2 @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_b,Y3: set_b,Z3: set_b] :
( ( ord_less_eq_set_b @ X @ Y3 )
=> ( ( ord_less_eq_set_b @ X @ Z3 )
=> ( ord_less_eq_set_b @ X @ ( F2 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_b @ X3 @ Y2 )
= ( F2 @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_601_le__iff__inf,axiom,
( ord_less_eq_o
= ( ^ [X2: $o,Y: $o] :
( ( inf_inf_o @ X2 @ Y )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_602_le__iff__inf,axiom,
( ord_less_eq_set_o
= ( ^ [X2: set_o,Y: set_o] :
( ( inf_inf_set_o @ X2 @ Y )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_603_le__iff__inf,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [X2: set_set_b,Y: set_set_b] :
( ( inf_inf_set_set_b @ X2 @ Y )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_604_le__iff__inf,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [X2: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( inf_in3606484609122063093fm_a_o @ X2 @ Y )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_605_le__iff__inf,axiom,
( ord_less_eq_b_o
= ( ^ [X2: b > $o,Y: b > $o] :
( ( inf_inf_b_o @ X2 @ Y )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_606_le__iff__inf,axiom,
( ord_less_eq_set_b
= ( ^ [X2: set_b,Y: set_b] :
( ( inf_inf_set_b @ X2 @ Y )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_607_inf_Oabsorb1,axiom,
! [A2: $o,B2: $o] :
( ( ord_less_eq_o @ A2 @ B2 )
=> ( ( inf_inf_o @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_608_inf_Oabsorb1,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( inf_inf_set_o @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_609_inf_Oabsorb1,axiom,
! [A2: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ( ( inf_inf_set_set_b @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_610_inf_Oabsorb1,axiom,
! [A2: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B2 )
=> ( ( inf_in3606484609122063093fm_a_o @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_611_inf_Oabsorb1,axiom,
! [A2: b > $o,B2: b > $o] :
( ( ord_less_eq_b_o @ A2 @ B2 )
=> ( ( inf_inf_b_o @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_612_inf_Oabsorb1,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_613_inf_Oabsorb2,axiom,
! [B2: $o,A2: $o] :
( ( ord_less_eq_o @ B2 @ A2 )
=> ( ( inf_inf_o @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_614_inf_Oabsorb2,axiom,
! [B2: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ( inf_inf_set_o @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_615_inf_Oabsorb2,axiom,
! [B2: set_set_b,A2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B2 @ A2 )
=> ( ( inf_inf_set_set_b @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_616_inf_Oabsorb2,axiom,
! [B2: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B2 @ A2 )
=> ( ( inf_in3606484609122063093fm_a_o @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_617_inf_Oabsorb2,axiom,
! [B2: b > $o,A2: b > $o] :
( ( ord_less_eq_b_o @ B2 @ A2 )
=> ( ( inf_inf_b_o @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_618_inf_Oabsorb2,axiom,
! [B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_619_inf__absorb1,axiom,
! [X3: $o,Y2: $o] :
( ( ord_less_eq_o @ X3 @ Y2 )
=> ( ( inf_inf_o @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_620_inf__absorb1,axiom,
! [X3: set_o,Y2: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y2 )
=> ( ( inf_inf_set_o @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_621_inf__absorb1,axiom,
! [X3: set_set_b,Y2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y2 )
=> ( ( inf_inf_set_set_b @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_622_inf__absorb1,axiom,
! [X3: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 )
=> ( ( inf_in3606484609122063093fm_a_o @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_623_inf__absorb1,axiom,
! [X3: b > $o,Y2: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y2 )
=> ( ( inf_inf_b_o @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_624_inf__absorb1,axiom,
! [X3: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ( inf_inf_set_b @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_625_inf__absorb2,axiom,
! [Y2: $o,X3: $o] :
( ( ord_less_eq_o @ Y2 @ X3 )
=> ( ( inf_inf_o @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_626_inf__absorb2,axiom,
! [Y2: set_o,X3: set_o] :
( ( ord_less_eq_set_o @ Y2 @ X3 )
=> ( ( inf_inf_set_o @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_627_inf__absorb2,axiom,
! [Y2: set_set_b,X3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ Y2 @ X3 )
=> ( ( inf_inf_set_set_b @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_628_inf__absorb2,axiom,
! [Y2: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 )
=> ( ( inf_in3606484609122063093fm_a_o @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_629_inf__absorb2,axiom,
! [Y2: b > $o,X3: b > $o] :
( ( ord_less_eq_b_o @ Y2 @ X3 )
=> ( ( inf_inf_b_o @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_630_inf__absorb2,axiom,
! [Y2: set_b,X3: set_b] :
( ( ord_less_eq_set_b @ Y2 @ X3 )
=> ( ( inf_inf_set_b @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_631_inf_OboundedE,axiom,
! [A2: $o,B2: $o,C: $o] :
( ( ord_less_eq_o @ A2 @ ( inf_inf_o @ B2 @ C ) )
=> ~ ( ( ord_less_eq_o @ A2 @ B2 )
=> ~ ( ord_less_eq_o @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_632_inf_OboundedE,axiom,
! [A2: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( inf_inf_set_o @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_o @ A2 @ B2 )
=> ~ ( ord_less_eq_set_o @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_633_inf_OboundedE,axiom,
! [A2: set_set_b,B2: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ ( inf_inf_set_set_b @ B2 @ C ) )
=> ~ ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ~ ( ord_le3795704787696855135_set_b @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_634_inf_OboundedE,axiom,
! [A2: epistemic_fm_a > $o,B2: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ ( inf_in3606484609122063093fm_a_o @ B2 @ C ) )
=> ~ ( ( ord_le4043730696559282883fm_a_o @ A2 @ B2 )
=> ~ ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_635_inf_OboundedE,axiom,
! [A2: b > $o,B2: b > $o,C: b > $o] :
( ( ord_less_eq_b_o @ A2 @ ( inf_inf_b_o @ B2 @ C ) )
=> ~ ( ( ord_less_eq_b_o @ A2 @ B2 )
=> ~ ( ord_less_eq_b_o @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_636_inf_OboundedE,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ~ ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_637_inf_OboundedI,axiom,
! [A2: $o,B2: $o,C: $o] :
( ( ord_less_eq_o @ A2 @ B2 )
=> ( ( ord_less_eq_o @ A2 @ C )
=> ( ord_less_eq_o @ A2 @ ( inf_inf_o @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_638_inf_OboundedI,axiom,
! [A2: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_o @ A2 @ C )
=> ( ord_less_eq_set_o @ A2 @ ( inf_inf_set_o @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_639_inf_OboundedI,axiom,
! [A2: set_set_b,B2: set_set_b,C: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
=> ( ( ord_le3795704787696855135_set_b @ A2 @ C )
=> ( ord_le3795704787696855135_set_b @ A2 @ ( inf_inf_set_set_b @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_640_inf_OboundedI,axiom,
! [A2: epistemic_fm_a > $o,B2: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B2 )
=> ( ( ord_le4043730696559282883fm_a_o @ A2 @ C )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( inf_in3606484609122063093fm_a_o @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_641_inf_OboundedI,axiom,
! [A2: b > $o,B2: b > $o,C: b > $o] :
( ( ord_less_eq_b_o @ A2 @ B2 )
=> ( ( ord_less_eq_b_o @ A2 @ C )
=> ( ord_less_eq_b_o @ A2 @ ( inf_inf_b_o @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_642_inf_OboundedI,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ C )
=> ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_643_inf__greatest,axiom,
! [X3: $o,Y2: $o,Z2: $o] :
( ( ord_less_eq_o @ X3 @ Y2 )
=> ( ( ord_less_eq_o @ X3 @ Z2 )
=> ( ord_less_eq_o @ X3 @ ( inf_inf_o @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_644_inf__greatest,axiom,
! [X3: set_o,Y2: set_o,Z2: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y2 )
=> ( ( ord_less_eq_set_o @ X3 @ Z2 )
=> ( ord_less_eq_set_o @ X3 @ ( inf_inf_set_o @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_645_inf__greatest,axiom,
! [X3: set_set_b,Y2: set_set_b,Z2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ X3 @ Y2 )
=> ( ( ord_le3795704787696855135_set_b @ X3 @ Z2 )
=> ( ord_le3795704787696855135_set_b @ X3 @ ( inf_inf_set_set_b @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_646_inf__greatest,axiom,
! [X3: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Z2 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( inf_in3606484609122063093fm_a_o @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_647_inf__greatest,axiom,
! [X3: b > $o,Y2: b > $o,Z2: b > $o] :
( ( ord_less_eq_b_o @ X3 @ Y2 )
=> ( ( ord_less_eq_b_o @ X3 @ Z2 )
=> ( ord_less_eq_b_o @ X3 @ ( inf_inf_b_o @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_648_inf__greatest,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y2 )
=> ( ( ord_less_eq_set_b @ X3 @ Z2 )
=> ( ord_less_eq_set_b @ X3 @ ( inf_inf_set_b @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_649_inf_Oorder__iff,axiom,
( ord_less_eq_o
= ( ^ [A3: $o,B3: $o] :
( A3
= ( inf_inf_o @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_650_inf_Oorder__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
( A3
= ( inf_inf_set_o @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_651_inf_Oorder__iff,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A3: set_set_b,B3: set_set_b] :
( A3
= ( inf_inf_set_set_b @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_652_inf_Oorder__iff,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [A3: epistemic_fm_a > $o,B3: epistemic_fm_a > $o] :
( A3
= ( inf_in3606484609122063093fm_a_o @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_653_inf_Oorder__iff,axiom,
( ord_less_eq_b_o
= ( ^ [A3: b > $o,B3: b > $o] :
( A3
= ( inf_inf_b_o @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_654_inf_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
( A3
= ( inf_inf_set_b @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_655_inf_Ocobounded1,axiom,
! [A2: $o,B2: $o] : ( ord_less_eq_o @ ( inf_inf_o @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_656_inf_Ocobounded1,axiom,
! [A2: set_o,B2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_657_inf_Ocobounded1,axiom,
! [A2: set_set_b,B2: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_658_inf_Ocobounded1,axiom,
! [A2: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_659_inf_Ocobounded1,axiom,
! [A2: b > $o,B2: b > $o] : ( ord_less_eq_b_o @ ( inf_inf_b_o @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_660_inf_Ocobounded1,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_661_inf_Ocobounded2,axiom,
! [A2: $o,B2: $o] : ( ord_less_eq_o @ ( inf_inf_o @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_662_inf_Ocobounded2,axiom,
! [A2: set_o,B2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_663_inf_Ocobounded2,axiom,
! [A2: set_set_b,B2: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_664_inf_Ocobounded2,axiom,
! [A2: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_665_inf_Ocobounded2,axiom,
! [A2: b > $o,B2: b > $o] : ( ord_less_eq_b_o @ ( inf_inf_b_o @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_666_inf_Ocobounded2,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_667_inf_Oabsorb__iff1,axiom,
( ord_less_eq_o
= ( ^ [A3: $o,B3: $o] :
( ( inf_inf_o @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_668_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
( ( inf_inf_set_o @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_669_inf_Oabsorb__iff1,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A3: set_set_b,B3: set_set_b] :
( ( inf_inf_set_set_b @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_670_inf_Oabsorb__iff1,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [A3: epistemic_fm_a > $o,B3: epistemic_fm_a > $o] :
( ( inf_in3606484609122063093fm_a_o @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_671_inf_Oabsorb__iff1,axiom,
( ord_less_eq_b_o
= ( ^ [A3: b > $o,B3: b > $o] :
( ( inf_inf_b_o @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_672_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( inf_inf_set_b @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_673_inf_Oabsorb__iff2,axiom,
( ord_less_eq_o
= ( ^ [B3: $o,A3: $o] :
( ( inf_inf_o @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_674_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_o
= ( ^ [B3: set_o,A3: set_o] :
( ( inf_inf_set_o @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_675_inf_Oabsorb__iff2,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [B3: set_set_b,A3: set_set_b] :
( ( inf_inf_set_set_b @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_676_inf_Oabsorb__iff2,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [B3: epistemic_fm_a > $o,A3: epistemic_fm_a > $o] :
( ( inf_in3606484609122063093fm_a_o @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_677_inf_Oabsorb__iff2,axiom,
( ord_less_eq_b_o
= ( ^ [B3: b > $o,A3: b > $o] :
( ( inf_inf_b_o @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_678_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [B3: set_b,A3: set_b] :
( ( inf_inf_set_b @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_679_inf_OcoboundedI1,axiom,
! [A2: $o,C: $o,B2: $o] :
( ( ord_less_eq_o @ A2 @ C )
=> ( ord_less_eq_o @ ( inf_inf_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_680_inf_OcoboundedI1,axiom,
! [A2: set_o,C: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ C )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_681_inf_OcoboundedI1,axiom,
! [A2: set_set_b,C: set_set_b,B2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ C )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_682_inf_OcoboundedI1,axiom,
! [A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ C )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_683_inf_OcoboundedI1,axiom,
! [A2: b > $o,C: b > $o,B2: b > $o] :
( ( ord_less_eq_b_o @ A2 @ C )
=> ( ord_less_eq_b_o @ ( inf_inf_b_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_684_inf_OcoboundedI1,axiom,
! [A2: set_b,C: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_685_inf_OcoboundedI2,axiom,
! [B2: set_set_b,C: set_set_b,A2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B2 @ C )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_686_inf_OcoboundedI2,axiom,
! [B2: epistemic_fm_a > $o,C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B2 @ C )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_687_inf_OcoboundedI2,axiom,
! [B2: b > $o,C: b > $o,A2: b > $o] :
( ( ord_less_eq_b_o @ B2 @ C )
=> ( ord_less_eq_b_o @ ( inf_inf_b_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_688_inf_OcoboundedI2,axiom,
! [B2: set_b,C: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_689_fm_Odistinct_I15_J,axiom,
! [X22: list_char,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Pro_a @ X22 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_690_fm_Odistinct_I21_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Dis_a @ X31 @ X32 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(21)
thf(fact_691_fm_Odistinct_I17_J,axiom,
! [X22: list_char,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Pro_a @ X22 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_692_fm_Odistinct_I23_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Dis_a @ X31 @ X32 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(23)
thf(fact_693_fm_Odistinct_I1_J,axiom,
! [X22: list_char] :
( epistemic_FF_a
!= ( epistemic_Pro_a @ X22 ) ) ).
% fm.distinct(1)
thf(fact_694_fm_Odistinct_I3_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.distinct(3)
thf(fact_695_semantics_Osimps_I2_J,axiom,
! [M2: episte3224730989885420256t_unit,W: b,X3: list_char] :
( ( episte295617885132580261cs_a_b @ M2 @ W @ ( epistemic_Pro_a @ X3 ) )
= ( episte5693316124195166255t_unit @ M2 @ W @ X3 ) ) ).
% semantics.simps(2)
thf(fact_696_semantics_Osimps_I3_J,axiom,
! [M2: episte3224730989885420256t_unit,W: b,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte295617885132580261cs_a_b @ M2 @ W @ ( epistemic_Dis_a @ P @ Q ) )
= ( ( episte295617885132580261cs_a_b @ M2 @ W @ P )
| ( episte295617885132580261cs_a_b @ M2 @ W @ Q ) ) ) ).
% semantics.simps(3)
thf(fact_697_fm_Oexhaust,axiom,
! [Y2: epistemic_fm_a] :
( ( Y2 != epistemic_FF_a )
=> ( ! [X23: list_char] :
( Y2
!= ( epistemic_Pro_a @ X23 ) )
=> ( ! [X312: epistemic_fm_a,X322: epistemic_fm_a] :
( Y2
!= ( epistemic_Dis_a @ X312 @ X322 ) )
=> ( ! [X412: epistemic_fm_a,X422: epistemic_fm_a] :
( Y2
!= ( epistemic_Con_a @ X412 @ X422 ) )
=> ( ! [X512: epistemic_fm_a,X522: epistemic_fm_a] :
( Y2
!= ( epistemic_Imp_a @ X512 @ X522 ) )
=> ~ ! [X612: a,X622: epistemic_fm_a] :
( Y2
!= ( epistemic_K_a @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_698_fm_Orel__induct,axiom,
! [R3: a > a > $o,X3: epistemic_fm_a,Y2: epistemic_fm_a,Q2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( epistemic_rel_fm_a_a @ R3 @ X3 @ Y2 )
=> ( ( Q2 @ epistemic_FF_a @ epistemic_FF_a )
=> ( ! [A22: list_char,B22: list_char] :
( ( A22 = B22 )
=> ( Q2 @ ( epistemic_Pro_a @ A22 ) @ ( epistemic_Pro_a @ B22 ) ) )
=> ( ! [A31: epistemic_fm_a,A32: epistemic_fm_a,B31: epistemic_fm_a,B32: epistemic_fm_a] :
( ( Q2 @ A31 @ B31 )
=> ( ( Q2 @ A32 @ B32 )
=> ( Q2 @ ( epistemic_Dis_a @ A31 @ A32 ) @ ( epistemic_Dis_a @ B31 @ B32 ) ) ) )
=> ( ! [A41: epistemic_fm_a,A42: epistemic_fm_a,B41: epistemic_fm_a,B42: epistemic_fm_a] :
( ( Q2 @ A41 @ B41 )
=> ( ( Q2 @ A42 @ B42 )
=> ( Q2 @ ( epistemic_Con_a @ A41 @ A42 ) @ ( epistemic_Con_a @ B41 @ B42 ) ) ) )
=> ( ! [A51: epistemic_fm_a,A52: epistemic_fm_a,B51: epistemic_fm_a,B52: epistemic_fm_a] :
( ( Q2 @ A51 @ B51 )
=> ( ( Q2 @ A52 @ B52 )
=> ( Q2 @ ( epistemic_Imp_a @ A51 @ A52 ) @ ( epistemic_Imp_a @ B51 @ B52 ) ) ) )
=> ( ! [A61: a,A62: epistemic_fm_a,B61: a,B62: epistemic_fm_a] :
( ( R3 @ A61 @ B61 )
=> ( ( Q2 @ A62 @ B62 )
=> ( Q2 @ ( epistemic_K_a @ A61 @ A62 ) @ ( epistemic_K_a @ B61 @ B62 ) ) ) )
=> ( Q2 @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_699_fm_Orel__cases,axiom,
! [R3: a > a > $o,A2: epistemic_fm_a,B2: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ A2 @ B2 )
=> ( ( ( A2 = epistemic_FF_a )
=> ( B2 != epistemic_FF_a ) )
=> ( ! [X: list_char] :
( ( A2
= ( epistemic_Pro_a @ X ) )
=> ! [Y3: list_char] :
( ( B2
= ( epistemic_Pro_a @ Y3 ) )
=> ( X != Y3 ) ) )
=> ( ! [X1: epistemic_fm_a,X2a: epistemic_fm_a] :
( ( A2
= ( epistemic_Dis_a @ X1 @ X2a ) )
=> ! [Y1: epistemic_fm_a,Y2a: epistemic_fm_a] :
( ( B2
= ( epistemic_Dis_a @ Y1 @ Y2a ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2a @ Y2a ) ) ) )
=> ( ! [X1a: epistemic_fm_a,X2b: epistemic_fm_a] :
( ( A2
= ( epistemic_Con_a @ X1a @ X2b ) )
=> ! [Y1a: epistemic_fm_a,Y2b: epistemic_fm_a] :
( ( B2
= ( epistemic_Con_a @ Y1a @ Y2b ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2b @ Y2b ) ) ) )
=> ( ! [X1b: epistemic_fm_a,X2c: epistemic_fm_a] :
( ( A2
= ( epistemic_Imp_a @ X1b @ X2c ) )
=> ! [Y1b: epistemic_fm_a,Y2c: epistemic_fm_a] :
( ( B2
= ( epistemic_Imp_a @ Y1b @ Y2c ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2c @ Y2c ) ) ) )
=> ~ ! [X1c: a,X2d: epistemic_fm_a] :
( ( A2
= ( epistemic_K_a @ X1c @ X2d ) )
=> ! [Y1c: a,Y2d: epistemic_fm_a] :
( ( B2
= ( epistemic_K_a @ Y1c @ Y2d ) )
=> ( ( R3 @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_700_inf__Int__eq,axiom,
! [R3: set_b,S: set_b] :
( ( inf_inf_b_o
@ ^ [X2: b] : ( member_b @ X2 @ R3 )
@ ^ [X2: b] : ( member_b @ X2 @ S ) )
= ( ^ [X2: b] : ( member_b @ X2 @ ( inf_inf_set_b @ R3 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_701_fm_Oset__cases,axiom,
! [E: b,A2: epistemic_fm_b] :
( ( member_b @ E @ ( epistemic_set_fm_b @ A2 ) )
=> ( ! [Z1: epistemic_fm_b] :
( ? [Z22: epistemic_fm_b] :
( A2
= ( epistemic_Dis_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( epistemic_set_fm_b @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_b,Z22: epistemic_fm_b] :
( ( A2
= ( epistemic_Dis_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( epistemic_set_fm_b @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_b] :
( ? [Z22: epistemic_fm_b] :
( A2
= ( epistemic_Con_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( epistemic_set_fm_b @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_b,Z22: epistemic_fm_b] :
( ( A2
= ( epistemic_Con_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( epistemic_set_fm_b @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_b] :
( ? [Z22: epistemic_fm_b] :
( A2
= ( epistemic_Imp_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( epistemic_set_fm_b @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_b,Z22: epistemic_fm_b] :
( ( A2
= ( epistemic_Imp_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( epistemic_set_fm_b @ Z22 ) ) )
=> ( ! [Z22: epistemic_fm_b] :
( A2
!= ( epistemic_K_b @ E @ Z22 ) )
=> ~ ! [Z1: b,Z22: epistemic_fm_b] :
( ( A2
= ( epistemic_K_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( epistemic_set_fm_b @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_702_fm_Oset__cases,axiom,
! [E: a,A2: epistemic_fm_a] :
( ( member_a @ E @ ( epistemic_set_fm_a @ A2 ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Dis_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Dis_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Con_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Con_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Imp_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Imp_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z22: epistemic_fm_a] :
( A2
!= ( epistemic_K_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_K_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_703_strong__soundness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_AxB_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte3224730989885420256t_unit] :
( ( ( episte3426825378771607259t_unit @ M4 )
& ( episte4899516349225340390t_unit @ M4 )
& ( episte9018110198556821958t_unit @ M4 ) )
=> ! [X4: b] :
( ( member_b @ X4 @ ( episte1782384855165018035t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ Xa ) )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5
thf(fact_704_strong__soundness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax5_a @ P3 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte3224730989885420256t_unit] :
( ( ( episte3426825378771607259t_unit @ M4 )
& ( episte4899516349225340390t_unit @ M4 )
& ( episte9018110198556821958t_unit @ M4 ) )
=> ! [X4: b] :
( ( member_b @ X4 @ ( episte1782384855165018035t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ Xa ) )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5'
thf(fact_705_eval__semantics,axiom,
! [Pi4: b > list_char > $o,W: b,W6: set_b,R: a > b > set_b,P: epistemic_fm_a] :
( ( epistemic_eval_a @ ( Pi4 @ W ) @ ( episte295617885132580261cs_a_b @ ( episte4938218340073005086t_unit @ W6 @ R @ ( episte6143282164508887558t_unit @ Pi4 @ product_Unity ) ) @ W ) @ P )
= ( episte295617885132580261cs_a_b @ ( episte4938218340073005086t_unit @ W6 @ R @ ( episte6143282164508887558t_unit @ Pi4 @ product_Unity ) ) @ W @ P ) ) ).
% eval_semantics
thf(fact_706_strong__soundness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P3: epistemic_fm_a] :
( ( epistemic_AxT_a @ P3 )
| ( epistemic_Ax4_a @ P3 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte3224730989885420256t_unit] :
( ( ( episte3426825378771607259t_unit @ M4 )
& ( episte9018110198556821958t_unit @ M4 ) )
=> ! [X4: b] :
( ( member_b @ X4 @ ( episte1782384855165018035t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ Xa ) )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>4
thf(fact_707_subsetI,axiom,
! [A: set_b,B: set_b] :
( ! [X: b] :
( ( member_b @ X @ A )
=> ( member_b @ X @ B ) )
=> ( ord_less_eq_set_b @ A @ B ) ) ).
% subsetI
thf(fact_708_Int__subset__iff,axiom,
! [C2: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B ) )
= ( ( ord_less_eq_set_b @ C2 @ A )
& ( ord_less_eq_set_b @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_709_fm_Opred__inject_I5_J,axiom,
! [P4: a > $o,A2: epistemic_fm_a,Aa2: epistemic_fm_a] :
( ( epistemic_pred_fm_a @ P4 @ ( epistemic_Imp_a @ A2 @ Aa2 ) )
= ( ( epistemic_pred_fm_a @ P4 @ A2 )
& ( epistemic_pred_fm_a @ P4 @ Aa2 ) ) ) ).
% fm.pred_inject(5)
thf(fact_710_fm_Opred__inject_I6_J,axiom,
! [P4: a > $o,A2: a,Aa2: epistemic_fm_a] :
( ( epistemic_pred_fm_a @ P4 @ ( epistemic_K_a @ A2 @ Aa2 ) )
= ( ( P4 @ A2 )
& ( epistemic_pred_fm_a @ P4 @ Aa2 ) ) ) ).
% fm.pred_inject(6)
thf(fact_711_in__mono,axiom,
! [A: set_b,B: set_b,X3: b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( member_b @ X3 @ A )
=> ( member_b @ X3 @ B ) ) ) ).
% in_mono
thf(fact_712_subsetD,axiom,
! [A: set_b,B: set_b,C: b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( member_b @ C @ A )
=> ( member_b @ C @ B ) ) ) ).
% subsetD
thf(fact_713_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
! [X2: b] :
( ( member_b @ X2 @ A4 )
=> ( member_b @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_714_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
! [T: b] :
( ( member_b @ T @ A4 )
=> ( member_b @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_715_Collect__subset,axiom,
! [A: set_b,P4: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A )
& ( P4 @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_716_fm_Opred__mono__strong,axiom,
! [P4: b > $o,X3: epistemic_fm_b,Pa: b > $o] :
( ( epistemic_pred_fm_b @ P4 @ X3 )
=> ( ! [Z3: b] :
( ( member_b @ Z3 @ ( epistemic_set_fm_b @ X3 ) )
=> ( ( P4 @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( epistemic_pred_fm_b @ Pa @ X3 ) ) ) ).
% fm.pred_mono_strong
thf(fact_717_fm_Orel__refl__strong,axiom,
! [X3: epistemic_fm_b,Ra: b > b > $o] :
( ! [Z3: b] :
( ( member_b @ Z3 @ ( epistemic_set_fm_b @ X3 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( epistemic_rel_fm_b_b @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_718_fm_Orel__mono__strong,axiom,
! [R3: b > b > $o,X3: epistemic_fm_b,Y2: epistemic_fm_b,Ra: b > b > $o] :
( ( epistemic_rel_fm_b_b @ R3 @ X3 @ Y2 )
=> ( ! [Z3: b,Yb: b] :
( ( member_b @ Z3 @ ( epistemic_set_fm_b @ X3 ) )
=> ( ( member_b @ Yb @ ( epistemic_set_fm_b @ Y2 ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( epistemic_rel_fm_b_b @ Ra @ X3 @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_719_fm_Opred__cong,axiom,
! [X3: epistemic_fm_b,Ya: epistemic_fm_b,P4: b > $o,Pa: b > $o] :
( ( X3 = Ya )
=> ( ! [Z3: b] :
( ( member_b @ Z3 @ ( epistemic_set_fm_b @ Ya ) )
=> ( ( P4 @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( epistemic_pred_fm_b @ P4 @ X3 )
= ( epistemic_pred_fm_b @ Pa @ Ya ) ) ) ) ).
% fm.pred_cong
thf(fact_720_fm_Orel__cong,axiom,
! [X3: epistemic_fm_b,Ya: epistemic_fm_b,Y2: epistemic_fm_b,Xa2: epistemic_fm_b,R3: b > b > $o,Ra: b > b > $o] :
( ( X3 = Ya )
=> ( ( Y2 = Xa2 )
=> ( ! [Z3: b,Yb: b] :
( ( member_b @ Z3 @ ( epistemic_set_fm_b @ Ya ) )
=> ( ( member_b @ Yb @ ( epistemic_set_fm_b @ Xa2 ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( epistemic_rel_fm_b_b @ R3 @ X3 @ Y2 )
= ( epistemic_rel_fm_b_b @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_721_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_a,Qs2: list_Epistemic_fm_a,R: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Ps ) @ ( set_Epistemic_fm_a2 @ Qs2 ) )
=> ! [G2: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ H @ ( epistemic_Imp_a @ ( epistemic_imply_a @ Ps @ R ) @ ( epistemic_imply_a @ Qs2 @ R ) ) ) ) ).
% tautology_imply_superset
thf(fact_722_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte3224730989885420256t_unit > $o,G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte3224730989885420256t_unit,W5: b,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member_b @ W5 @ ( episte1782384855165018035t_unit @ M3 ) )
=> ( episte295617885132580261cs_a_b @ M3 @ W5 @ P2 ) ) ) )
=> ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte3224730989885420256t_unit] :
( ( P4 @ M4 )
=> ! [X4: b] :
( ( member_b @ X4 @ ( episte1782384855165018035t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ Xa ) )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_723_fm_Orel__inject_I5_J,axiom,
! [R3: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) )
= ( ( epistemic_rel_fm_a_a @ R3 @ X51 @ Y51 )
& ( epistemic_rel_fm_a_a @ R3 @ X52 @ Y52 ) ) ) ).
% fm.rel_inject(5)
thf(fact_724_fm_Orel__intros_I5_J,axiom,
! [R3: a > a > $o,X51: epistemic_fm_a,Y51: epistemic_fm_a,X52: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ X51 @ Y51 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X52 @ Y52 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ) ) ).
% fm.rel_intros(5)
thf(fact_725_fm_Orel__inject_I6_J,axiom,
! [R3: a > a > $o,X61: a,X62: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) )
= ( ( R3 @ X61 @ Y61 )
& ( epistemic_rel_fm_a_a @ R3 @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_726_fm_Orel__intros_I6_J,axiom,
! [R3: a > a > $o,X61: a,Y61: a,X62: epistemic_fm_a,Y62: epistemic_fm_a] :
( ( R3 @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X62 @ Y62 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_727_fm_Octr__transfer_I1_J,axiom,
! [R3: a > a > $o] : ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ epistemic_FF_a ) ).
% fm.ctr_transfer(1)
thf(fact_728_strong__soundness_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a,P4: episte3224730989885420256t_unit > $o] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte3224730989885420256t_unit] :
( ( P4 @ M4 )
=> ! [X4: b] :
( ( member_b @ X4 @ ( episte1782384855165018035t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ Xa ) )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K
thf(fact_729_Int__Collect__mono,axiom,
! [A: set_b,B: set_b,P4: b > $o,Q2: b > $o] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ! [X: b] :
( ( member_b @ X @ A )
=> ( ( P4 @ X )
=> ( Q2 @ X ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ ( collect_b @ P4 ) ) @ ( inf_inf_set_b @ B @ ( collect_b @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_730_Int__greatest,axiom,
! [C2: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C2 @ A )
=> ( ( ord_less_eq_set_b @ C2 @ B )
=> ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_731_Int__absorb2,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( inf_inf_set_b @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_732_Int__absorb1,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( inf_inf_set_b @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_733_Int__lower2,axiom,
! [A: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_734_Int__lower1,axiom,
! [A: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_735_Int__mono,axiom,
! [A: set_b,C2: set_b,B: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A @ C2 )
=> ( ( ord_less_eq_set_b @ B @ D2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ ( inf_inf_set_b @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_736_fm_Oset__intros_I6_J,axiom,
! [Yf: b,X52: epistemic_fm_b,X51: epistemic_fm_b] :
( ( member_b @ Yf @ ( epistemic_set_fm_b @ X52 ) )
=> ( member_b @ Yf @ ( epistemic_set_fm_b @ ( epistemic_Imp_b @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_737_fm_Oset__intros_I6_J,axiom,
! [Yf: a,X52: epistemic_fm_a,X51: epistemic_fm_a] :
( ( member_a @ Yf @ ( epistemic_set_fm_a @ X52 ) )
=> ( member_a @ Yf @ ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_738_fm_Oset__intros_I5_J,axiom,
! [Ye: b,X51: epistemic_fm_b,X52: epistemic_fm_b] :
( ( member_b @ Ye @ ( epistemic_set_fm_b @ X51 ) )
=> ( member_b @ Ye @ ( epistemic_set_fm_b @ ( epistemic_Imp_b @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_739_fm_Oset__intros_I5_J,axiom,
! [Ye: a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( member_a @ Ye @ ( epistemic_set_fm_a @ X51 ) )
=> ( member_a @ Ye @ ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_740_fm_Oset__intros_I8_J,axiom,
! [Yg: b,X62: epistemic_fm_b,X61: b] :
( ( member_b @ Yg @ ( epistemic_set_fm_b @ X62 ) )
=> ( member_b @ Yg @ ( epistemic_set_fm_b @ ( epistemic_K_b @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_741_fm_Oset__intros_I8_J,axiom,
! [Yg: a,X62: epistemic_fm_a,X61: a] :
( ( member_a @ Yg @ ( epistemic_set_fm_a @ X62 ) )
=> ( member_a @ Yg @ ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_742_fm_Oset__intros_I7_J,axiom,
! [X61: b,X62: epistemic_fm_b] : ( member_b @ X61 @ ( epistemic_set_fm_b @ ( epistemic_K_b @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_743_fm_Oset__intros_I7_J,axiom,
! [X61: a,X62: epistemic_fm_a] : ( member_a @ X61 @ ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_744_fm_Oset__intros_I3_J,axiom,
! [Yc: b,X41: epistemic_fm_b,X42: epistemic_fm_b] :
( ( member_b @ Yc @ ( epistemic_set_fm_b @ X41 ) )
=> ( member_b @ Yc @ ( epistemic_set_fm_b @ ( epistemic_Con_b @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_745_fm_Oset__intros_I4_J,axiom,
! [Yd: b,X42: epistemic_fm_b,X41: epistemic_fm_b] :
( ( member_b @ Yd @ ( epistemic_set_fm_b @ X42 ) )
=> ( member_b @ Yd @ ( epistemic_set_fm_b @ ( epistemic_Con_b @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_746_fm_Oset__intros_I1_J,axiom,
! [Ya: b,X31: epistemic_fm_b,X32: epistemic_fm_b] :
( ( member_b @ Ya @ ( epistemic_set_fm_b @ X31 ) )
=> ( member_b @ Ya @ ( epistemic_set_fm_b @ ( epistemic_Dis_b @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_747_fm_Oset__intros_I2_J,axiom,
! [Yb2: b,X32: epistemic_fm_b,X31: epistemic_fm_b] :
( ( member_b @ Yb2 @ ( epistemic_set_fm_b @ X32 ) )
=> ( member_b @ Yb2 @ ( epistemic_set_fm_b @ ( epistemic_Dis_b @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_748_fm_Opred__inject_I1_J,axiom,
! [P4: a > $o] : ( epistemic_pred_fm_a @ P4 @ epistemic_FF_a ) ).
% fm.pred_inject(1)
thf(fact_749_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte3224730989885420256t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte3224730989885420256t_unit,W5: b,P2: epistemic_fm_a] :
( ( A @ P2 )
=> ( ( P4 @ M3 )
=> ( ( member_b @ W5 @ ( episte1782384855165018035t_unit @ M3 ) )
=> ( episte295617885132580261cs_a_b @ M3 @ W5 @ P2 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M4: episte3224730989885420256t_unit] :
( ( P4 @ M4 )
=> ! [X4: b] :
( ( member_b @ X4 @ ( episte1782384855165018035t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ Xa ) )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_750_less__eq__set__def,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
( ord_less_eq_b_o
@ ^ [X2: b] : ( member_b @ X2 @ A4 )
@ ^ [X2: b] : ( member_b @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_751_strong__soundness_092_060_094sub_062T,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte3224730989885420256t_unit] :
( ( episte3426825378771607259t_unit @ M4 )
=> ! [X4: b] :
( ( member_b @ X4 @ ( episte1782384855165018035t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ Xa ) )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>T
thf(fact_752_strong__soundness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte3224730989885420256t_unit] :
( ( episte9018110198556821958t_unit @ M4 )
=> ! [X4: b] :
( ( member_b @ X4 @ ( episte1782384855165018035t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ Xa ) )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>4
thf(fact_753_strong__soundness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte3224730989885420256t_unit] :
( ( episte4899516349225340390t_unit @ M4 )
=> ! [X4: b] :
( ( member_b @ X4 @ ( episte1782384855165018035t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ Xa ) )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>B
thf(fact_754_fm_Orel__distinct_I30_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.rel_distinct(30)
thf(fact_755_fm_Orel__distinct_I29_J,axiom,
! [R3: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(29)
thf(fact_756_strong__soundness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte3224730989885420256t_unit] :
( ( episte6418945320598494989t_unit @ M4 )
=> ! [X4: b] :
( ( member_b @ X4 @ ( episte1782384855165018035t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ Xa ) )
=> ( episte295617885132580261cs_a_b @ M4 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>5
thf(fact_757_fm_Orel__distinct_I8_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(8)
thf(fact_758_fm_Orel__distinct_I7_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(7)
thf(fact_759_fm_Orel__distinct_I10_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(10)
thf(fact_760_fm_Orel__distinct_I9_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(9)
thf(fact_761_K__right__mp,axiom,
! [A: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ ( epistemic_Imp_a @ P @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ Q ) ) ) ) ).
% K_right_mp
thf(fact_762_conjunct__imply,axiom,
! [G5: list_Epistemic_fm_a,P: epistemic_fm_a,G2: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ H @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G5 ) @ P ) @ ( epistemic_imply_a @ G5 @ P ) ) ) ).
% conjunct_imply
thf(fact_763_imply__conjunct,axiom,
! [G5: list_Epistemic_fm_a,P: epistemic_fm_a,G2: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ H @ ( epistemic_Imp_a @ ( epistemic_imply_a @ G5 @ P ) @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G5 ) @ P ) ) ) ).
% imply_conjunct
thf(fact_764_K__conjunct__imply,axiom,
! [A: epistemic_fm_a > $o,G5: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G5 ) @ P ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ P ) ) ) ).
% K_conjunct_imply
thf(fact_765_K__imply__conjunct,axiom,
! [A: epistemic_fm_a > $o,G5: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ P ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G5 ) @ P ) ) ) ).
% K_imply_conjunct
thf(fact_766_consistent__def,axiom,
( episte2285483198712856226tent_a
= ( ^ [A4: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
~ ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ S2 )
& ( epistemic_AK_a @ A4 @ ( epistemic_imply_a @ Qs3 @ epistemic_FF_a ) ) ) ) ) ).
% consistent_def
thf(fact_767_K__Boole,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G5: list_Epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ G5 ) @ epistemic_FF_a ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ P ) ) ) ).
% K_Boole
thf(fact_768_strong__completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,G5: set_Epistemic_fm_a,P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W7: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W7 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W7 ) ) )
@ ^ [I: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P3 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G5 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% strong_completeness
thf(fact_769_consequent__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V4: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V4 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V4 )
=> ( ( member6642669571620171971c_fm_a @ P @ V4 )
=> ( ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ Q ) @ V4 )
=> ( member6642669571620171971c_fm_a @ Q @ V4 ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_770_exactly__one__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V4: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V4 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V4 )
=> ( ( member6642669571620171971c_fm_a @ P @ V4 )
= ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ V4 ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_771_dual__reach,axiom,
! [A: epistemic_fm_a > $o,V4: set_Epistemic_fm_a,I2: a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V4 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V4 )
=> ( ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ V4 )
=> ? [W8: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ W8
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V4 ) ) ) ) )
& ( member6642669571620171971c_fm_a @ P @ W8 ) ) ) ) ) ).
% dual_reach
thf(fact_772_reach__dualK,axiom,
! [A: epistemic_fm_a > $o,V4: set_Epistemic_fm_a,W6: set_Epistemic_fm_a,I2: a] :
( ( episte2285483198712856226tent_a @ A @ V4 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V4 )
=> ( ( episte2285483198712856226tent_a @ A @ W6 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W6 )
=> ( ( member536094252920883875c_fm_a @ W6
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V4 ) ) ) ) )
=> ! [P5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ P5 @ W6 )
=> ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P5 @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ V4 ) ) ) ) ) ) ) ).
% reach_dualK
thf(fact_773_truth__lemma,axiom,
! [A: epistemic_fm_a > $o,V4: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V4 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V4 )
=> ( ( member6642669571620171971c_fm_a @ P @ V4 )
= ( episte7081087998767065248c_fm_a
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W7: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W7 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W7 ) ) )
@ ^ [I: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P3 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) )
@ V4
@ P ) ) ) ) ).
% truth_lemma
thf(fact_774_imply_Osimps_I2_J,axiom,
! [P: epistemic_fm_a,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ Q )
= ( epistemic_Imp_a @ P @ ( epistemic_imply_a @ Ps @ Q ) ) ) ).
% imply.simps(2)
thf(fact_775_AxT__reflexive,axiom,
! [A: epistemic_fm_a > $o,V4: set_Epistemic_fm_a,I2: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V4 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V4 )
=> ( member536094252920883875c_fm_a @ V4
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V4 ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_776_Ax4__transitive,axiom,
! [A: epistemic_fm_a > $o,V4: set_Epistemic_fm_a,W6: set_Epistemic_fm_a,I2: a,U: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V4 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V4 )
=> ( ( member536094252920883875c_fm_a @ W6
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V4 ) ) ) ) )
=> ( ( member536094252920883875c_fm_a @ U
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ W6 ) ) ) ) )
=> ( member536094252920883875c_fm_a @ U
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V4 ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_777_Ax5__Euclidean,axiom,
! [A: epistemic_fm_a > $o,U: set_Epistemic_fm_a,V4: set_Epistemic_fm_a,W6: set_Epistemic_fm_a,I2: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ U )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ U )
=> ( ( episte2285483198712856226tent_a @ A @ V4 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V4 )
=> ( ( episte2285483198712856226tent_a @ A @ W6 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W6 )
=> ( ( member536094252920883875c_fm_a @ V4
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ U ) ) ) ) )
=> ( ( member536094252920883875c_fm_a @ W6
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ U ) ) ) ) )
=> ( member536094252920883875c_fm_a @ W6
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_778_AxB__symmetric_H,axiom,
! [A: epistemic_fm_a > $o,V4: set_Epistemic_fm_a,W6: set_Epistemic_fm_a,I2: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V4 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V4 )
=> ( ( episte2285483198712856226tent_a @ A @ W6 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W6 )
=> ( ( member536094252920883875c_fm_a @ W6
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ V4 ) ) ) ) )
=> ( member536094252920883875c_fm_a @ V4
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P3 ) @ W6 ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_779_K__ImpI,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G5: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G5 ) @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ ( epistemic_Imp_a @ P @ Q ) ) ) ) ).
% K_ImpI
thf(fact_780_K__mp,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G5: list_Epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P @ Q ) @ G5 ) ) @ Q ) ) ).
% K_mp
thf(fact_781_reflexive_092_060_094sub_062T,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( episte5648423998891577755t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W7: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W7 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W7 ) ) )
@ ^ [I: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P3 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) ) ) ) ).
% reflexive\<^sub>T
thf(fact_782_transitive_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( episte8364071018013720454t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W7: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W7 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W7 ) ) )
@ ^ [I: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P3 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) ) ) ) ).
% transitive\<^sub>K\<^sub>4
thf(fact_783_Euclidean_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( episte2449151000174023629t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W7: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W7 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W7 ) ) )
@ ^ [I: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P3 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) ) ) ) ).
% Euclidean\<^sub>K\<^sub>5
thf(fact_784_symmetric_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( episte5478016696552465318t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W7: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W7 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W7 ) ) )
@ ^ [I: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P3 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) ) ) ) ).
% symmetric\<^sub>K\<^sub>B
thf(fact_785_completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W7: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W7 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W7 ) ) )
@ ^ [I: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P3 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ) ).
% completeness
thf(fact_786_K__conjunction__out__mult,axiom,
! [A: epistemic_fm_a > $o,I2: a,G5: list_Epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ I2 ) @ G5 ) ) @ ( epistemic_K_a @ I2 @ ( stalnaker_conjunct_a @ G5 ) ) ) ) ).
% K_conjunction_out_mult
thf(fact_787_K__conjunction__in__mult,axiom,
! [A: epistemic_fm_a > $o,I2: a,G5: list_Epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( stalnaker_conjunct_a @ G5 ) ) @ ( stalnaker_conjunct_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ I2 ) @ G5 ) ) ) ) ).
% K_conjunction_in_mult
thf(fact_788_canonical__model_I1_J,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( ( member6642669571620171971c_fm_a @ P @ S )
=> ( episte7081087998767065248c_fm_a
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W7: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W7 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W7 ) ) )
@ ^ [I: a,V3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P3: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P3 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) )
@ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S )
@ P ) ) ) ).
% canonical_model(1)
thf(fact_789_UNIV__I,axiom,
! [X3: b] : ( member_b @ X3 @ top_top_set_b ) ).
% UNIV_I
thf(fact_790_UNIV__I,axiom,
! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_I
thf(fact_791_empty__Collect__eq,axiom,
! [P4: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P4 ) )
= ( ! [X2: $o] :
~ ( P4 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_792_Collect__empty__eq,axiom,
! [P4: $o > $o] :
( ( ( collect_o @ P4 )
= bot_bot_set_o )
= ( ! [X2: $o] :
~ ( P4 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_793_all__not__in__conv,axiom,
! [A: set_b] :
( ( ! [X2: b] :
~ ( member_b @ X2 @ A ) )
= ( A = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_794_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X2: $o] :
~ ( member_o @ X2 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_795_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_796_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_797_subset__empty,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_798_empty__subsetI,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% empty_subsetI
thf(fact_799_inf__bot__right,axiom,
! [X3: set_b] :
( ( inf_inf_set_b @ X3 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% inf_bot_right
thf(fact_800_inf__bot__right,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ X3 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% inf_bot_right
thf(fact_801_inf__bot__left,axiom,
! [X3: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X3 )
= bot_bot_set_b ) ).
% inf_bot_left
thf(fact_802_inf__bot__left,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X3 )
= bot_bot_set_o ) ).
% inf_bot_left
thf(fact_803_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_b] :
( ( inf_inf_set_b @ X3 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_804_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ X3 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_805_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X3 )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_806_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X3 )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_807_inf__top_Oright__neutral,axiom,
! [A2: set_b] :
( ( inf_inf_set_b @ A2 @ top_top_set_b )
= A2 ) ).
% inf_top.right_neutral
thf(fact_808_inf__top_Oright__neutral,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ A2 @ top_top_set_o )
= A2 ) ).
% inf_top.right_neutral
thf(fact_809_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( top_top_set_b
= ( inf_inf_set_b @ A2 @ B2 ) )
= ( ( A2 = top_top_set_b )
& ( B2 = top_top_set_b ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_810_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_o,B2: set_o] :
( ( top_top_set_o
= ( inf_inf_set_o @ A2 @ B2 ) )
= ( ( A2 = top_top_set_o )
& ( B2 = top_top_set_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_811_inf__top_Oleft__neutral,axiom,
! [A2: set_b] :
( ( inf_inf_set_b @ top_top_set_b @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_812_inf__top_Oleft__neutral,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_813_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( ( inf_inf_set_b @ A2 @ B2 )
= top_top_set_b )
= ( ( A2 = top_top_set_b )
& ( B2 = top_top_set_b ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_814_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A2 @ B2 )
= top_top_set_o )
= ( ( A2 = top_top_set_o )
& ( B2 = top_top_set_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_815_top__eq__inf__iff,axiom,
! [X3: set_b,Y2: set_b] :
( ( top_top_set_b
= ( inf_inf_set_b @ X3 @ Y2 ) )
= ( ( X3 = top_top_set_b )
& ( Y2 = top_top_set_b ) ) ) ).
% top_eq_inf_iff
thf(fact_816_top__eq__inf__iff,axiom,
! [X3: set_o,Y2: set_o] :
( ( top_top_set_o
= ( inf_inf_set_o @ X3 @ Y2 ) )
= ( ( X3 = top_top_set_o )
& ( Y2 = top_top_set_o ) ) ) ).
% top_eq_inf_iff
thf(fact_817_inf__eq__top__iff,axiom,
! [X3: set_b,Y2: set_b] :
( ( ( inf_inf_set_b @ X3 @ Y2 )
= top_top_set_b )
= ( ( X3 = top_top_set_b )
& ( Y2 = top_top_set_b ) ) ) ).
% inf_eq_top_iff
thf(fact_818_inf__eq__top__iff,axiom,
! [X3: set_o,Y2: set_o] :
( ( ( inf_inf_set_o @ X3 @ Y2 )
= top_top_set_o )
= ( ( X3 = top_top_set_o )
& ( Y2 = top_top_set_o ) ) ) ).
% inf_eq_top_iff
thf(fact_819_inf__top__right,axiom,
! [X3: set_b] :
( ( inf_inf_set_b @ X3 @ top_top_set_b )
= X3 ) ).
% inf_top_right
thf(fact_820_inf__top__right,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ X3 @ top_top_set_o )
= X3 ) ).
% inf_top_right
thf(fact_821_inf__top__left,axiom,
! [X3: set_b] :
( ( inf_inf_set_b @ top_top_set_b @ X3 )
= X3 ) ).
% inf_top_left
thf(fact_822_inf__top__left,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ X3 )
= X3 ) ).
% inf_top_left
thf(fact_823_Int__UNIV,axiom,
! [A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A @ B )
= top_top_set_b )
= ( ( A = top_top_set_b )
& ( B = top_top_set_b ) ) ) ).
% Int_UNIV
thf(fact_824_Int__UNIV,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= top_top_set_o )
= ( ( A = top_top_set_o )
& ( B = top_top_set_o ) ) ) ).
% Int_UNIV
thf(fact_825_Collect__const,axiom,
! [P4: $o] :
( ( P4
=> ( ( collect_o
@ ^ [S3: $o] : P4 )
= top_top_set_o ) )
& ( ~ P4
=> ( ( collect_o
@ ^ [S3: $o] : P4 )
= bot_bot_set_o ) ) ) ).
% Collect_const
thf(fact_826_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X2: $o] : $true ) ) ).
% UNIV_def
thf(fact_827_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X2: $o] : $false ) ) ).
% empty_def
thf(fact_828_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_829_UNIV__witness,axiom,
? [X: b] : ( member_b @ X @ top_top_set_b ) ).
% UNIV_witness
thf(fact_830_UNIV__witness,axiom,
? [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_witness
thf(fact_831_ex__in__conv,axiom,
! [A: set_b] :
( ( ? [X2: b] : ( member_b @ X2 @ A ) )
= ( A != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_832_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_833_UNIV__eq__I,axiom,
! [A: set_b] :
( ! [X: b] : ( member_b @ X @ A )
=> ( top_top_set_b = A ) ) ).
% UNIV_eq_I
thf(fact_834_UNIV__eq__I,axiom,
! [A: set_o] :
( ! [X: $o] : ( member_o @ X @ A )
=> ( top_top_set_o = A ) ) ).
% UNIV_eq_I
thf(fact_835_equals0I,axiom,
! [A: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A )
=> ( A = bot_bot_set_b ) ) ).
% equals0I
thf(fact_836_equals0I,axiom,
! [A: set_o] :
( ! [Y3: $o] :
~ ( member_o @ Y3 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_837_equals0D,axiom,
! [A: set_b,A2: b] :
( ( A = bot_bot_set_b )
=> ~ ( member_b @ A2 @ A ) ) ).
% equals0D
thf(fact_838_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_839_emptyE,axiom,
! [A2: b] :
~ ( member_b @ A2 @ bot_bot_set_b ) ).
% emptyE
thf(fact_840_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_841_boolean__algebra_Oconj__one__right,axiom,
! [X3: set_b] :
( ( inf_inf_set_b @ X3 @ top_top_set_b )
= X3 ) ).
% boolean_algebra.conj_one_right
thf(fact_842_boolean__algebra_Oconj__one__right,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ X3 @ top_top_set_o )
= X3 ) ).
% boolean_algebra.conj_one_right
thf(fact_843_subset__UNIV,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% subset_UNIV
thf(fact_844_Int__UNIV__right,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ top_top_set_b )
= A ) ).
% Int_UNIV_right
thf(fact_845_Int__UNIV__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ top_top_set_o )
= A ) ).
% Int_UNIV_right
thf(fact_846_Int__UNIV__left,axiom,
! [B: set_b] :
( ( inf_inf_set_b @ top_top_set_b @ B )
= B ) ).
% Int_UNIV_left
thf(fact_847_Int__UNIV__left,axiom,
! [B: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ B )
= B ) ).
% Int_UNIV_left
thf(fact_848_disjoint__iff__not__equal,axiom,
! [A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ! [Y: b] :
( ( member_b @ Y @ B )
=> ( X2 != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_849_disjoint__iff__not__equal,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ! [Y: $o] :
( ( member_o @ Y @ B )
=> ( X2 = (~ Y) ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_850_Int__empty__right,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Int_empty_right
thf(fact_851_Int__empty__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_852_Int__empty__left,axiom,
! [B: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_853_Int__empty__left,axiom,
! [B: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_854_disjoint__iff,axiom,
! [A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
= ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ~ ( member_b @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_855_disjoint__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ~ ( member_o @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_856_Int__emptyI,axiom,
! [A: set_b,B: set_b] :
( ! [X: b] :
( ( member_b @ X @ A )
=> ~ ( member_b @ X @ B ) )
=> ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_857_Int__emptyI,axiom,
! [A: set_o,B: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member_o @ X @ B ) )
=> ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_858_fm_Osimps_I90_J,axiom,
( ( epistemic_set_fm_a @ epistemic_FF_a )
= bot_bot_set_a ) ).
% fm.simps(90)
thf(fact_859_fm_Osimps_I90_J,axiom,
( ( epistemic_set_fm_o @ epistemic_FF_o )
= bot_bot_set_o ) ).
% fm.simps(90)
thf(fact_860_fm_Osimps_I91_J,axiom,
! [X22: list_char] :
( ( epistemic_set_fm_o @ ( epistemic_Pro_o @ X22 ) )
= bot_bot_set_o ) ).
% fm.simps(91)
thf(fact_861_K__distrib__K__imp,axiom,
! [A: epistemic_fm_a > $o,I2: a,G5: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_K_a @ I2 @ ( epistemic_imply_a @ G5 @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ I2 ) @ G5 ) @ ( epistemic_K_a @ I2 @ Q ) ) ) ) ).
% K_distrib_K_imp
thf(fact_862_finite__Int,axiom,
! [F4: set_b,G5: set_b] :
( ( ( finite_finite_b @ F4 )
| ( finite_finite_b @ G5 ) )
=> ( finite_finite_b @ ( inf_inf_set_b @ F4 @ G5 ) ) ) ).
% finite_Int
thf(fact_863_Set_Ois__empty__def,axiom,
( is_empty_o
= ( ^ [A4: set_o] : ( A4 = bot_bot_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_864_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_865_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_866_inter__Set__filter,axiom,
! [B: set_b,A: set_b] :
( ( finite_finite_b @ B )
=> ( ( inf_inf_set_b @ A @ B )
= ( filter_b
@ ^ [X2: b] : ( member_b @ X2 @ A )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_867_member__filter,axiom,
! [X3: b,P4: b > $o,A: set_b] :
( ( member_b @ X3 @ ( filter_b @ P4 @ A ) )
= ( ( member_b @ X3 @ A )
& ( P4 @ X3 ) ) ) ).
% member_filter
thf(fact_868_Set_Ofilter__def,axiom,
( filter_b
= ( ^ [P6: b > $o,A4: set_b] :
( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A4 )
& ( P6 @ A3 ) ) ) ) ) ).
% Set.filter_def
thf(fact_869_inconsistent__subset,axiom,
! [A: epistemic_fm_a > $o,V4: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V4 )
=> ( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P @ bot_bo3626323581529592678c_fm_a ) @ V4 ) )
=> ~ ! [V5: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ V5 ) @ V4 )
=> ~ ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ V5 ) @ epistemic_FF_a ) ) ) ) ) ).
% inconsistent_subset
thf(fact_870_insertCI,axiom,
! [A2: $o,B: set_o,B2: $o] :
( ( ~ ( member_o @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).
% insertCI
thf(fact_871_insertCI,axiom,
! [A2: b,B: set_b,B2: b] :
( ( ~ ( member_b @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_872_insert__iff,axiom,
! [A2: $o,B2: $o,A: set_o] :
( ( member_o @ A2 @ ( insert_o @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_o @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_873_insert__iff,axiom,
! [A2: b,B2: b,A: set_b] :
( ( member_b @ A2 @ ( insert_b @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_874_insert__absorb2,axiom,
! [X3: $o,A: set_o] :
( ( insert_o @ X3 @ ( insert_o @ X3 @ A ) )
= ( insert_o @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_875_UnCI,axiom,
! [C: b,B: set_b,A: set_b] :
( ( ~ ( member_b @ C @ B )
=> ( member_b @ C @ A ) )
=> ( member_b @ C @ ( sup_sup_set_b @ A @ B ) ) ) ).
% UnCI
thf(fact_876_Un__iff,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A @ B ) )
= ( ( member_b @ C @ A )
| ( member_b @ C @ B ) ) ) ).
% Un_iff
thf(fact_877_sup__bot__left,axiom,
! [X3: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_878_sup__bot__right,axiom,
! [X3: set_o] :
( ( sup_sup_set_o @ X3 @ bot_bot_set_o )
= X3 ) ).
% sup_bot_right
thf(fact_879_bot__eq__sup__iff,axiom,
! [X3: set_o,Y2: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ X3 @ Y2 ) )
= ( ( X3 = bot_bot_set_o )
& ( Y2 = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_880_sup__eq__bot__iff,axiom,
! [X3: set_o,Y2: set_o] :
( ( ( sup_sup_set_o @ X3 @ Y2 )
= bot_bot_set_o )
= ( ( X3 = bot_bot_set_o )
& ( Y2 = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_881_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_o,B2: set_o] :
( ( ( sup_sup_set_o @ A2 @ B2 )
= bot_bot_set_o )
= ( ( A2 = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_882_sup__bot_Oleft__neutral,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_883_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_o,B2: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_884_sup__bot_Oright__neutral,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_885_sup__top__left,axiom,
! [X3: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X3 )
= top_top_set_o ) ).
% sup_top_left
thf(fact_886_sup__top__right,axiom,
! [X3: set_o] :
( ( sup_sup_set_o @ X3 @ top_top_set_o )
= top_top_set_o ) ).
% sup_top_right
thf(fact_887_boolean__algebra_Odisj__one__left,axiom,
! [X3: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X3 )
= top_top_set_o ) ).
% boolean_algebra.disj_one_left
thf(fact_888_boolean__algebra_Odisj__one__right,axiom,
! [X3: set_o] :
( ( sup_sup_set_o @ X3 @ top_top_set_o )
= top_top_set_o ) ).
% boolean_algebra.disj_one_right
thf(fact_889_singletonI,axiom,
! [A2: b] : ( member_b @ A2 @ ( insert_b @ A2 @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_890_singletonI,axiom,
! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_891_insert__subset,axiom,
! [X3: $o,A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X3 @ A ) @ B )
= ( ( member_o @ X3 @ B )
& ( ord_less_eq_set_o @ A @ B ) ) ) ).
% insert_subset
thf(fact_892_insert__subset,axiom,
! [X3: b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X3 @ A ) @ B )
= ( ( member_b @ X3 @ B )
& ( ord_less_eq_set_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_893_inf__sup__absorb,axiom,
! [X3: set_b,Y2: set_b] :
( ( inf_inf_set_b @ X3 @ ( sup_sup_set_b @ X3 @ Y2 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_894_sup__inf__absorb,axiom,
! [X3: set_b,Y2: set_b] :
( ( sup_sup_set_b @ X3 @ ( inf_inf_set_b @ X3 @ Y2 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_895_Un__empty,axiom,
! [A: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A @ B )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_896_Int__insert__left__if0,axiom,
! [A2: $o,C2: set_o,B: set_o] :
( ~ ( member_o @ A2 @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( inf_inf_set_o @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_897_Int__insert__left__if0,axiom,
! [A2: b,C2: set_b,B: set_b] :
( ~ ( member_b @ A2 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
= ( inf_inf_set_b @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_898_Int__insert__left__if1,axiom,
! [A2: $o,C2: set_o,B: set_o] :
( ( member_o @ A2 @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( insert_o @ A2 @ ( inf_inf_set_o @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_899_Int__insert__left__if1,axiom,
! [A2: b,C2: set_b,B: set_b] :
( ( member_b @ A2 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
= ( insert_b @ A2 @ ( inf_inf_set_b @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_900_insert__inter__insert,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ ( insert_o @ A2 @ B ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_901_insert__inter__insert,axiom,
! [A2: b,A: set_b,B: set_b] :
( ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ ( insert_b @ A2 @ B ) )
= ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_902_Int__insert__right__if0,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ~ ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( inf_inf_set_o @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_903_Int__insert__right__if0,axiom,
! [A2: b,A: set_b,B: set_b] :
( ~ ( member_b @ A2 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_904_Int__insert__right__if1,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_905_Int__insert__right__if1,axiom,
! [A2: b,A: set_b,B: set_b] :
( ( member_b @ A2 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
= ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_906_Un__insert__left,axiom,
! [A2: $o,B: set_o,C2: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( insert_o @ A2 @ ( sup_sup_set_o @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_907_Un__insert__right,axiom,
! [A: set_o,A2: $o,B: set_o] :
( ( sup_sup_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( insert_o @ A2 @ ( sup_sup_set_o @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_908_Int__Un__eq_I4_J,axiom,
! [T2: set_b,S: set_b] :
( ( sup_sup_set_b @ T2 @ ( inf_inf_set_b @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_909_Int__Un__eq_I3_J,axiom,
! [S: set_b,T2: set_b] :
( ( sup_sup_set_b @ S @ ( inf_inf_set_b @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_910_Int__Un__eq_I2_J,axiom,
! [S: set_b,T2: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_911_Int__Un__eq_I1_J,axiom,
! [S: set_b,T2: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_912_Un__Int__eq_I4_J,axiom,
! [T2: set_b,S: set_b] :
( ( inf_inf_set_b @ T2 @ ( sup_sup_set_b @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_913_Un__Int__eq_I3_J,axiom,
! [S: set_b,T2: set_b] :
( ( inf_inf_set_b @ S @ ( sup_sup_set_b @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_914_Un__Int__eq_I2_J,axiom,
! [S: set_b,T2: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_915_Un__Int__eq_I1_J,axiom,
! [S: set_b,T2: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_916_singleton__conv2,axiom,
! [A2: $o] :
( ( collect_o
@ ( ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ A2 ) )
= ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_917_singleton__conv,axiom,
! [A2: $o] :
( ( collect_o
@ ^ [X2: $o] : ( X2 = A2 ) )
= ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_918_singleton__insert__inj__eq_H,axiom,
! [A2: $o,A: set_o,B2: $o] :
( ( ( insert_o @ A2 @ A )
= ( insert_o @ B2 @ bot_bot_set_o ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_919_singleton__insert__inj__eq,axiom,
! [B2: $o,A2: $o,A: set_o] :
( ( ( insert_o @ B2 @ bot_bot_set_o )
= ( insert_o @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_920_insert__disjoint_I1_J,axiom,
! [A2: b,A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ B )
= bot_bot_set_b )
= ( ~ ( member_b @ A2 @ B )
& ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b ) ) ) ).
% insert_disjoint(1)
thf(fact_921_insert__disjoint_I1_J,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B )
= bot_bot_set_o )
= ( ~ ( member_o @ A2 @ B )
& ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_922_insert__disjoint_I2_J,axiom,
! [A2: b,A: set_b,B: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ ( insert_b @ A2 @ A ) @ B ) )
= ( ~ ( member_b @ A2 @ B )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_923_insert__disjoint_I2_J,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B ) )
= ( ~ ( member_o @ A2 @ B )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_924_disjoint__insert_I1_J,axiom,
! [B: set_b,A2: b,A: set_b] :
( ( ( inf_inf_set_b @ B @ ( insert_b @ A2 @ A ) )
= bot_bot_set_b )
= ( ~ ( member_b @ A2 @ B )
& ( ( inf_inf_set_b @ B @ A )
= bot_bot_set_b ) ) ) ).
% disjoint_insert(1)
thf(fact_925_disjoint__insert_I1_J,axiom,
! [B: set_o,A2: $o,A: set_o] :
( ( ( inf_inf_set_o @ B @ ( insert_o @ A2 @ A ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A2 @ B )
& ( ( inf_inf_set_o @ B @ A )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_926_disjoint__insert_I2_J,axiom,
! [A: set_b,B2: b,B: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ A @ ( insert_b @ B2 @ B ) ) )
= ( ~ ( member_b @ B2 @ A )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_927_disjoint__insert_I2_J,axiom,
! [A: set_o,B2: $o,B: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A @ ( insert_o @ B2 @ B ) ) )
= ( ~ ( member_o @ B2 @ A )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_928_fm_Osimps_I95_J,axiom,
! [X61: $o,X62: epistemic_fm_o] :
( ( epistemic_set_fm_o @ ( epistemic_K_o @ X61 @ X62 ) )
= ( insert_o @ X61 @ ( epistemic_set_fm_o @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_929_fm_Osimps_I95_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) )
= ( insert_a @ X61 @ ( epistemic_set_fm_a @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_930_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_o] :
( ( sup_sup_set_o @ X3 @ bot_bot_set_o )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_931_singleton__Un__iff,axiom,
! [X3: $o,A: set_o,B: set_o] :
( ( ( insert_o @ X3 @ bot_bot_set_o )
= ( sup_sup_set_o @ A @ B ) )
= ( ( ( A = bot_bot_set_o )
& ( B
= ( insert_o @ X3 @ bot_bot_set_o ) ) )
| ( ( A
= ( insert_o @ X3 @ bot_bot_set_o ) )
& ( B = bot_bot_set_o ) )
| ( ( A
= ( insert_o @ X3 @ bot_bot_set_o ) )
& ( B
= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_932_Un__singleton__iff,axiom,
! [A: set_o,B: set_o,X3: $o] :
( ( ( sup_sup_set_o @ A @ B )
= ( insert_o @ X3 @ bot_bot_set_o ) )
= ( ( ( A = bot_bot_set_o )
& ( B
= ( insert_o @ X3 @ bot_bot_set_o ) ) )
| ( ( A
= ( insert_o @ X3 @ bot_bot_set_o ) )
& ( B = bot_bot_set_o ) )
| ( ( A
= ( insert_o @ X3 @ bot_bot_set_o ) )
& ( B
= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_933_insert__is__Un,axiom,
( insert_o
= ( ^ [A3: $o] : ( sup_sup_set_o @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ) ).
% insert_is_Un
thf(fact_934_Un__UNIV__left,axiom,
! [B: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ B )
= top_top_set_o ) ).
% Un_UNIV_left
thf(fact_935_Un__UNIV__right,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ top_top_set_o )
= top_top_set_o ) ).
% Un_UNIV_right
thf(fact_936_Un__empty__left,axiom,
! [B: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B )
= B ) ).
% Un_empty_left
thf(fact_937_Un__empty__right,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% Un_empty_right
thf(fact_938_insert__UNIV,axiom,
! [X3: $o] :
( ( insert_o @ X3 @ top_top_set_o )
= top_top_set_o ) ).
% insert_UNIV
thf(fact_939_singletonD,axiom,
! [B2: b,A2: b] :
( ( member_b @ B2 @ ( insert_b @ A2 @ bot_bot_set_b ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_940_singletonD,axiom,
! [B2: $o,A2: $o] :
( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_941_singleton__iff,axiom,
! [B2: b,A2: b] :
( ( member_b @ B2 @ ( insert_b @ A2 @ bot_bot_set_b ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_942_singleton__iff,axiom,
! [B2: $o,A2: $o] :
( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_943_doubleton__eq__iff,axiom,
! [A2: $o,B2: $o,C: $o,D: $o] :
( ( ( insert_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) )
= ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_944_insert__not__empty,axiom,
! [A2: $o,A: set_o] :
( ( insert_o @ A2 @ A )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_945_singleton__inject,axiom,
! [A2: $o,B2: $o] :
( ( ( insert_o @ A2 @ bot_bot_set_o )
= ( insert_o @ B2 @ bot_bot_set_o ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_946_UnE,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A @ B ) )
=> ( ~ ( member_b @ C @ A )
=> ( member_b @ C @ B ) ) ) ).
% UnE
thf(fact_947_UnI1,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ A )
=> ( member_b @ C @ ( sup_sup_set_b @ A @ B ) ) ) ).
% UnI1
thf(fact_948_UnI2,axiom,
! [C: b,B: set_b,A: set_b] :
( ( member_b @ C @ B )
=> ( member_b @ C @ ( sup_sup_set_b @ A @ B ) ) ) ).
% UnI2
thf(fact_949_insertE,axiom,
! [A2: $o,B2: $o,A: set_o] :
( ( member_o @ A2 @ ( insert_o @ B2 @ A ) )
=> ( ( A2 = (~ B2) )
=> ( member_o @ A2 @ A ) ) ) ).
% insertE
thf(fact_950_insertE,axiom,
! [A2: b,B2: b,A: set_b] :
( ( member_b @ A2 @ ( insert_b @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_951_insertI1,axiom,
! [A2: $o,B: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B ) ) ).
% insertI1
thf(fact_952_insertI1,axiom,
! [A2: b,B: set_b] : ( member_b @ A2 @ ( insert_b @ A2 @ B ) ) ).
% insertI1
thf(fact_953_insertI2,axiom,
! [A2: $o,B: set_o,B2: $o] :
( ( member_o @ A2 @ B )
=> ( member_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).
% insertI2
thf(fact_954_insertI2,axiom,
! [A2: b,B: set_b,B2: b] :
( ( member_b @ A2 @ B )
=> ( member_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_955_Set_Oset__insert,axiom,
! [X3: $o,A: set_o] :
( ( member_o @ X3 @ A )
=> ~ ! [B5: set_o] :
( ( A
= ( insert_o @ X3 @ B5 ) )
=> ( member_o @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_956_Set_Oset__insert,axiom,
! [X3: b,A: set_b] :
( ( member_b @ X3 @ A )
=> ~ ! [B5: set_b] :
( ( A
= ( insert_b @ X3 @ B5 ) )
=> ( member_b @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_957_insert__ident,axiom,
! [X3: $o,A: set_o,B: set_o] :
( ~ ( member_o @ X3 @ A )
=> ( ~ ( member_o @ X3 @ B )
=> ( ( ( insert_o @ X3 @ A )
= ( insert_o @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_958_insert__ident,axiom,
! [X3: b,A: set_b,B: set_b] :
( ~ ( member_b @ X3 @ A )
=> ( ~ ( member_b @ X3 @ B )
=> ( ( ( insert_b @ X3 @ A )
= ( insert_b @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_959_insert__absorb,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ( ( insert_o @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_960_insert__absorb,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ( ( insert_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_961_insert__eq__iff,axiom,
! [A2: $o,A: set_o,B2: $o,B: set_o] :
( ~ ( member_o @ A2 @ A )
=> ( ~ ( member_o @ B2 @ B )
=> ( ( ( insert_o @ A2 @ A )
= ( insert_o @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 = (~ B2) )
=> ? [C3: set_o] :
( ( A
= ( insert_o @ B2 @ C3 ) )
& ~ ( member_o @ B2 @ C3 )
& ( B
= ( insert_o @ A2 @ C3 ) )
& ~ ( member_o @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_962_insert__eq__iff,axiom,
! [A2: b,A: set_b,B2: b,B: set_b] :
( ~ ( member_b @ A2 @ A )
=> ( ~ ( member_b @ B2 @ B )
=> ( ( ( insert_b @ A2 @ A )
= ( insert_b @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_b] :
( ( A
= ( insert_b @ B2 @ C3 ) )
& ~ ( member_b @ B2 @ C3 )
& ( B
= ( insert_b @ A2 @ C3 ) )
& ~ ( member_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_963_insert__commute,axiom,
! [X3: $o,Y2: $o,A: set_o] :
( ( insert_o @ X3 @ ( insert_o @ Y2 @ A ) )
= ( insert_o @ Y2 @ ( insert_o @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_964_mk__disjoint__insert,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ? [B5: set_o] :
( ( A
= ( insert_o @ A2 @ B5 ) )
& ~ ( member_o @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_965_mk__disjoint__insert,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ? [B5: set_b] :
( ( A
= ( insert_b @ A2 @ B5 ) )
& ~ ( member_b @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_966_insert__Collect,axiom,
! [A2: $o,P4: $o > $o] :
( ( insert_o @ A2 @ ( collect_o @ P4 ) )
= ( collect_o
@ ^ [U2: $o] :
( ( U2 != A2 )
=> ( P4 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_967_insert__compr,axiom,
( insert_o
= ( ^ [A3: $o,B4: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( X2 = A3 )
| ( member_o @ X2 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_968_insert__compr,axiom,
( insert_b
= ( ^ [A3: b,B4: set_b] :
( collect_b
@ ^ [X2: b] :
( ( X2 = A3 )
| ( member_b @ X2 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_969_insert__def,axiom,
( insert_o
= ( ^ [A3: $o] :
( sup_sup_set_o
@ ( collect_o
@ ^ [X2: $o] : ( X2 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_970_Un__def,axiom,
( sup_sup_set_b
= ( ^ [A4: set_b,B4: set_b] :
( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A4 )
| ( member_b @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_971_Int__insert__right,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B ) ) ) )
& ( ~ ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( inf_inf_set_o @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_972_Int__insert__right,axiom,
! [A2: b,A: set_b,B: set_b] :
( ( ( member_b @ A2 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
= ( insert_b @ A2 @ ( inf_inf_set_b @ A @ B ) ) ) )
& ( ~ ( member_b @ A2 @ A )
=> ( ( inf_inf_set_b @ A @ ( insert_b @ A2 @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_973_Int__insert__left,axiom,
! [A2: $o,C2: set_o,B: set_o] :
( ( ( member_o @ A2 @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( insert_o @ A2 @ ( inf_inf_set_o @ B @ C2 ) ) ) )
& ( ~ ( member_o @ A2 @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( inf_inf_set_o @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_974_Int__insert__left,axiom,
! [A2: b,C2: set_b,B: set_b] :
( ( ( member_b @ A2 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
= ( insert_b @ A2 @ ( inf_inf_set_b @ B @ C2 ) ) ) )
& ( ~ ( member_b @ A2 @ C2 )
=> ( ( inf_inf_set_b @ ( insert_b @ A2 @ B ) @ C2 )
= ( inf_inf_set_b @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_975_Un__Int__distrib2,axiom,
! [B: set_b,C2: set_b,A: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ B @ C2 ) @ A )
= ( inf_inf_set_b @ ( sup_sup_set_b @ B @ A ) @ ( sup_sup_set_b @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_976_Int__Un__distrib2,axiom,
! [B: set_b,C2: set_b,A: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ B @ C2 ) @ A )
= ( sup_sup_set_b @ ( inf_inf_set_b @ B @ A ) @ ( inf_inf_set_b @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_977_Un__Int__distrib,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( sup_sup_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ A @ B ) @ ( sup_sup_set_b @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_978_Int__Un__distrib,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( inf_inf_set_b @ A @ ( sup_sup_set_b @ B @ C2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ A @ B ) @ ( inf_inf_set_b @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_979_Un__Int__crazy,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ ( inf_inf_set_b @ A @ B ) @ ( inf_inf_set_b @ B @ C2 ) ) @ ( inf_inf_set_b @ C2 @ A ) )
= ( inf_inf_set_b @ ( inf_inf_set_b @ ( sup_sup_set_b @ A @ B ) @ ( sup_sup_set_b @ B @ C2 ) ) @ ( sup_sup_set_b @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_980_insert__mono,axiom,
! [C2: set_o,D2: set_o,A2: $o] :
( ( ord_less_eq_set_o @ C2 @ D2 )
=> ( ord_less_eq_set_o @ ( insert_o @ A2 @ C2 ) @ ( insert_o @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_981_subset__insert,axiom,
! [X3: $o,A: set_o,B: set_o] :
( ~ ( member_o @ X3 @ A )
=> ( ( ord_less_eq_set_o @ A @ ( insert_o @ X3 @ B ) )
= ( ord_less_eq_set_o @ A @ B ) ) ) ).
% subset_insert
thf(fact_982_subset__insert,axiom,
! [X3: b,A: set_b,B: set_b] :
( ~ ( member_b @ X3 @ A )
=> ( ( ord_less_eq_set_b @ A @ ( insert_b @ X3 @ B ) )
= ( ord_less_eq_set_b @ A @ B ) ) ) ).
% subset_insert
thf(fact_983_subset__insertI,axiom,
! [B: set_o,A2: $o] : ( ord_less_eq_set_o @ B @ ( insert_o @ A2 @ B ) ) ).
% subset_insertI
thf(fact_984_subset__insertI2,axiom,
! [A: set_o,B: set_o,B2: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_985_sup__inf__distrib2,axiom,
! [Y2: set_b,Z2: set_b,X3: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ Y2 @ Z2 ) @ X3 )
= ( inf_inf_set_b @ ( sup_sup_set_b @ Y2 @ X3 ) @ ( sup_sup_set_b @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_986_sup__inf__distrib1,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X3 @ ( inf_inf_set_b @ Y2 @ Z2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X3 @ Y2 ) @ ( sup_sup_set_b @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_987_inf__sup__distrib2,axiom,
! [Y2: set_b,Z2: set_b,X3: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ Y2 @ Z2 ) @ X3 )
= ( sup_sup_set_b @ ( inf_inf_set_b @ Y2 @ X3 ) @ ( inf_inf_set_b @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_988_inf__sup__distrib1,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X3 @ ( sup_sup_set_b @ Y2 @ Z2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) @ ( inf_inf_set_b @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_989_distrib__imp2,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ! [X: set_b,Y3: set_b,Z3: set_b] :
( ( sup_sup_set_b @ X @ ( inf_inf_set_b @ Y3 @ Z3 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X @ Y3 ) @ ( sup_sup_set_b @ X @ Z3 ) ) )
=> ( ( inf_inf_set_b @ X3 @ ( sup_sup_set_b @ Y2 @ Z2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) @ ( inf_inf_set_b @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_990_distrib__imp1,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ! [X: set_b,Y3: set_b,Z3: set_b] :
( ( inf_inf_set_b @ X @ ( sup_sup_set_b @ Y3 @ Z3 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X @ Y3 ) @ ( inf_inf_set_b @ X @ Z3 ) ) )
=> ( ( sup_sup_set_b @ X3 @ ( inf_inf_set_b @ Y2 @ Z2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X3 @ Y2 ) @ ( sup_sup_set_b @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_991_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_b,Z2: set_b,X3: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ Y2 @ Z2 ) @ X3 )
= ( inf_inf_set_b @ ( sup_sup_set_b @ Y2 @ X3 ) @ ( sup_sup_set_b @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_992_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_b,Z2: set_b,X3: set_b] :
( ( inf_inf_set_b @ ( sup_sup_set_b @ Y2 @ Z2 ) @ X3 )
= ( sup_sup_set_b @ ( inf_inf_set_b @ Y2 @ X3 ) @ ( inf_inf_set_b @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_993_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ( sup_sup_set_b @ X3 @ ( inf_inf_set_b @ Y2 @ Z2 ) )
= ( inf_inf_set_b @ ( sup_sup_set_b @ X3 @ Y2 ) @ ( sup_sup_set_b @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_994_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ( inf_inf_set_b @ X3 @ ( sup_sup_set_b @ Y2 @ Z2 ) )
= ( sup_sup_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) @ ( inf_inf_set_b @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_995_Collect__conv__if2,axiom,
! [P4: $o > $o,A2: $o] :
( ( ( P4 @ A2 )
=> ( ( collect_o
@ ^ [X2: $o] :
( ( A2 = X2 )
& ( P4 @ X2 ) ) )
= ( insert_o @ A2 @ bot_bot_set_o ) ) )
& ( ~ ( P4 @ A2 )
=> ( ( collect_o
@ ^ [X2: $o] :
( ( A2 = X2 )
& ( P4 @ X2 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_996_Collect__conv__if,axiom,
! [P4: $o > $o,A2: $o] :
( ( ( P4 @ A2 )
=> ( ( collect_o
@ ^ [X2: $o] :
( ( X2 = A2 )
& ( P4 @ X2 ) ) )
= ( insert_o @ A2 @ bot_bot_set_o ) ) )
& ( ~ ( P4 @ A2 )
=> ( ( collect_o
@ ^ [X2: $o] :
( ( X2 = A2 )
& ( P4 @ X2 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_997_distrib__inf__le,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] : ( ord_less_eq_set_b @ ( sup_sup_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) @ ( inf_inf_set_b @ X3 @ Z2 ) ) @ ( inf_inf_set_b @ X3 @ ( sup_sup_set_b @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_998_distrib__sup__le,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] : ( ord_less_eq_set_b @ ( sup_sup_set_b @ X3 @ ( inf_inf_set_b @ Y2 @ Z2 ) ) @ ( inf_inf_set_b @ ( sup_sup_set_b @ X3 @ Y2 ) @ ( sup_sup_set_b @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_999_subset__singleton__iff,axiom,
! [X5: set_o,A2: $o] :
( ( ord_less_eq_set_o @ X5 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( ( X5 = bot_bot_set_o )
| ( X5
= ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_1000_subset__singletonD,axiom,
! [A: set_o,X3: $o] :
( ( ord_less_eq_set_o @ A @ ( insert_o @ X3 @ bot_bot_set_o ) )
=> ( ( A = bot_bot_set_o )
| ( A
= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_1001_Un__Int__assoc__eq,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( ( sup_sup_set_b @ ( inf_inf_set_b @ A @ B ) @ C2 )
= ( inf_inf_set_b @ A @ ( sup_sup_set_b @ B @ C2 ) ) )
= ( ord_less_eq_set_b @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1002_consistent__consequent,axiom,
! [A: epistemic_fm_a > $o,V4: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V4 )
=> ( ( member6642669571620171971c_fm_a @ P @ V4 )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V4 ) ) ) ) ) ).
% consistent_consequent
thf(fact_1003_consistent__consequent_H,axiom,
! [A: epistemic_fm_a > $o,V4: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V4 )
=> ( ( member6642669571620171971c_fm_a @ P @ V4 )
=> ( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ ( epistemic_Imp_a @ P @ Q ) )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V4 ) ) ) ) ) ).
% consistent_consequent'
thf(fact_1004_fm_Osimps_I94_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) )
= ( sup_sup_set_a @ ( epistemic_set_fm_a @ X51 ) @ ( epistemic_set_fm_a @ X52 ) ) ) ).
% fm.simps(94)
thf(fact_1005_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_b,X3: set_b,Y2: set_b] :
( ( ( inf_inf_set_b @ A2 @ X3 )
= bot_bot_set_b )
=> ( ( ( sup_sup_set_b @ A2 @ X3 )
= top_top_set_b )
=> ( ( ( inf_inf_set_b @ A2 @ Y2 )
= bot_bot_set_b )
=> ( ( ( sup_sup_set_b @ A2 @ Y2 )
= top_top_set_b )
=> ( X3 = Y2 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1006_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_o,X3: set_o,Y2: set_o] :
( ( ( inf_inf_set_o @ A2 @ X3 )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A2 @ X3 )
= top_top_set_o )
=> ( ( ( inf_inf_set_o @ A2 @ Y2 )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A2 @ Y2 )
= top_top_set_o )
=> ( X3 = Y2 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1007_exists__finite__inconsistent,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,V4: set_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V4 ) )
=> ~ ! [W8: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W8 ) @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V4 ) )
=> ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ W8 )
=> ( ( finite3304564945125563331c_fm_a @ W8 )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W8 ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_1008_inconsistent__imply,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G5: list_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ ( set_Epistemic_fm_a2 @ G5 ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ P ) ) ) ).
% inconsistent_imply
thf(fact_1009_the__elem__eq,axiom,
! [X3: $o] :
( ( the_elem_o @ ( insert_o @ X3 @ bot_bot_set_o ) )
= X3 ) ).
% the_elem_eq
thf(fact_1010_is__singletonI,axiom,
! [X3: $o] : ( is_singleton_o @ ( insert_o @ X3 @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_1011_subset__singleton__iff__Uniq,axiom,
! [A: set_b] :
( ( ? [A3: b] : ( ord_less_eq_set_b @ A @ ( insert_b @ A3 @ bot_bot_set_b ) ) )
= ( uniq_b
@ ^ [X2: b] : ( member_b @ X2 @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1012_subset__singleton__iff__Uniq,axiom,
! [A: set_o] :
( ( ? [A3: $o] : ( ord_less_eq_set_o @ A @ ( insert_o @ A3 @ bot_bot_set_o ) ) )
= ( uniq_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1013_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A4: set_o] :
? [X2: $o] :
( A4
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_1014_sup__set__def,axiom,
( sup_sup_set_b
= ( ^ [A4: set_b,B4: set_b] :
( collect_b
@ ( sup_sup_b_o
@ ^ [X2: b] : ( member_b @ X2 @ A4 )
@ ^ [X2: b] : ( member_b @ X2 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_1015_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A4: set_o] :
( A4
= ( insert_o @ ( the_elem_o @ A4 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1016_is__singletonI_H,axiom,
! [A: set_b] :
( ( A != bot_bot_set_b )
=> ( ! [X: b,Y3: b] :
( ( member_b @ X @ A )
=> ( ( member_b @ Y3 @ A )
=> ( X = Y3 ) ) )
=> ( is_singleton_b @ A ) ) ) ).
% is_singletonI'
thf(fact_1017_is__singletonI_H,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
=> ( ! [X: $o,Y3: $o] :
( ( member_o @ X @ A )
=> ( ( member_o @ Y3 @ A )
=> ( X = Y3 ) ) )
=> ( is_singleton_o @ A ) ) ) ).
% is_singletonI'
thf(fact_1018_is__singletonE,axiom,
! [A: set_o] :
( ( is_singleton_o @ A )
=> ~ ! [X: $o] :
( A
!= ( insert_o @ X @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_1019_the__elem__def,axiom,
( the_elem_o
= ( ^ [X6: set_o] :
( the_o
@ ^ [X2: $o] :
( X6
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).
% the_elem_def
thf(fact_1020_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X3: set_b,Y2: set_b] :
( ( ( inf_inf_set_b @ X3 @ Y2 )
= bot_bot_set_b )
=> ( ( ( sup_sup_set_b @ X3 @ Y2 )
= top_top_set_b )
=> ( ( uminus_uminus_set_b @ X3 )
= Y2 ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1021_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X3: set_o,Y2: set_o] :
( ( ( inf_inf_set_o @ X3 @ Y2 )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ X3 @ Y2 )
= top_top_set_o )
=> ( ( uminus_uminus_set_o @ X3 )
= Y2 ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1022_Compl__disjoint2,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ ( uminus_uminus_set_b @ A ) @ A )
= bot_bot_set_b ) ).
% Compl_disjoint2
thf(fact_1023_Compl__disjoint2,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A ) @ A )
= bot_bot_set_o ) ).
% Compl_disjoint2
thf(fact_1024_Compl__disjoint,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ ( uminus_uminus_set_b @ A ) )
= bot_bot_set_b ) ).
% Compl_disjoint
thf(fact_1025_Compl__disjoint,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ ( uminus_uminus_set_o @ A ) )
= bot_bot_set_o ) ).
% Compl_disjoint
thf(fact_1026_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.compl_one
thf(fact_1027_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% boolean_algebra.compl_zero
thf(fact_1028_inf__compl__bot__left1,axiom,
! [X3: set_b,Y2: set_b] :
( ( inf_inf_set_b @ ( uminus_uminus_set_b @ X3 ) @ ( inf_inf_set_b @ X3 @ Y2 ) )
= bot_bot_set_b ) ).
% inf_compl_bot_left1
thf(fact_1029_inf__compl__bot__left1,axiom,
! [X3: set_o,Y2: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X3 ) @ ( inf_inf_set_o @ X3 @ Y2 ) )
= bot_bot_set_o ) ).
% inf_compl_bot_left1
thf(fact_1030_inf__compl__bot__left2,axiom,
! [X3: set_b,Y2: set_b] :
( ( inf_inf_set_b @ X3 @ ( inf_inf_set_b @ ( uminus_uminus_set_b @ X3 ) @ Y2 ) )
= bot_bot_set_b ) ).
% inf_compl_bot_left2
thf(fact_1031_inf__compl__bot__left2,axiom,
! [X3: set_o,Y2: set_o] :
( ( inf_inf_set_o @ X3 @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X3 ) @ Y2 ) )
= bot_bot_set_o ) ).
% inf_compl_bot_left2
thf(fact_1032_inf__compl__bot__right,axiom,
! [X3: set_b,Y2: set_b] :
( ( inf_inf_set_b @ X3 @ ( inf_inf_set_b @ Y2 @ ( uminus_uminus_set_b @ X3 ) ) )
= bot_bot_set_b ) ).
% inf_compl_bot_right
thf(fact_1033_inf__compl__bot__right,axiom,
! [X3: set_o,Y2: set_o] :
( ( inf_inf_set_o @ X3 @ ( inf_inf_set_o @ Y2 @ ( uminus_uminus_set_o @ X3 ) ) )
= bot_bot_set_o ) ).
% inf_compl_bot_right
thf(fact_1034_boolean__algebra_Oconj__cancel__left,axiom,
! [X3: set_b] :
( ( inf_inf_set_b @ ( uminus_uminus_set_b @ X3 ) @ X3 )
= bot_bot_set_b ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1035_boolean__algebra_Oconj__cancel__left,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X3 ) @ X3 )
= bot_bot_set_o ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1036_boolean__algebra_Oconj__cancel__right,axiom,
! [X3: set_b] :
( ( inf_inf_set_b @ X3 @ ( uminus_uminus_set_b @ X3 ) )
= bot_bot_set_b ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1037_boolean__algebra_Oconj__cancel__right,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ X3 @ ( uminus_uminus_set_o @ X3 ) )
= bot_bot_set_o ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1038_subset__Compl__singleton,axiom,
! [A: set_b,B2: b] :
( ( ord_less_eq_set_b @ A @ ( uminus_uminus_set_b @ ( insert_b @ B2 @ bot_bot_set_b ) ) )
= ( ~ ( member_b @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1039_subset__Compl__singleton,axiom,
! [A: set_o,B2: $o] :
( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ ( insert_o @ B2 @ bot_bot_set_o ) ) )
= ( ~ ( member_o @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1040_boolean__algebra_Odisj__cancel__right,axiom,
! [X3: set_o] :
( ( sup_sup_set_o @ X3 @ ( uminus_uminus_set_o @ X3 ) )
= top_top_set_o ) ).
% boolean_algebra.disj_cancel_right
thf(fact_1041_boolean__algebra_Odisj__cancel__left,axiom,
! [X3: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ X3 ) @ X3 )
= top_top_set_o ) ).
% boolean_algebra.disj_cancel_left
thf(fact_1042_sup__compl__top__left2,axiom,
! [X3: set_o,Y2: set_o] :
( ( sup_sup_set_o @ X3 @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X3 ) @ Y2 ) )
= top_top_set_o ) ).
% sup_compl_top_left2
thf(fact_1043_sup__compl__top__left1,axiom,
! [X3: set_o,Y2: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ X3 ) @ ( sup_sup_set_o @ X3 @ Y2 ) )
= top_top_set_o ) ).
% sup_compl_top_left1
thf(fact_1044_boolean__algebra_Ode__Morgan__conj,axiom,
! [X3: set_b,Y2: set_b] :
( ( uminus_uminus_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) )
= ( sup_sup_set_b @ ( uminus_uminus_set_b @ X3 ) @ ( uminus_uminus_set_b @ Y2 ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_1045_boolean__algebra_Ode__Morgan__disj,axiom,
! [X3: set_b,Y2: set_b] :
( ( uminus_uminus_set_b @ ( sup_sup_set_b @ X3 @ Y2 ) )
= ( inf_inf_set_b @ ( uminus_uminus_set_b @ X3 ) @ ( uminus_uminus_set_b @ Y2 ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_1046_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% Compl_UNIV_eq
thf(fact_1047_Compl__empty__eq,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% Compl_empty_eq
thf(fact_1048_inf__cancel__left1,axiom,
! [X3: set_b,A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X3 @ A2 ) @ ( inf_inf_set_b @ ( uminus_uminus_set_b @ X3 ) @ B2 ) )
= bot_bot_set_b ) ).
% inf_cancel_left1
thf(fact_1049_inf__cancel__left1,axiom,
! [X3: set_o,A2: set_o,B2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ X3 @ A2 ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X3 ) @ B2 ) )
= bot_bot_set_o ) ).
% inf_cancel_left1
thf(fact_1050_inf__cancel__left2,axiom,
! [X3: set_b,A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ ( uminus_uminus_set_b @ X3 ) @ A2 ) @ ( inf_inf_set_b @ X3 @ B2 ) )
= bot_bot_set_b ) ).
% inf_cancel_left2
thf(fact_1051_inf__cancel__left2,axiom,
! [X3: set_o,A2: set_o,B2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X3 ) @ A2 ) @ ( inf_inf_set_o @ X3 @ B2 ) )
= bot_bot_set_o ) ).
% inf_cancel_left2
thf(fact_1052_sup__cancel__left2,axiom,
! [X3: set_o,A2: set_o,B2: set_o] :
( ( sup_sup_set_o @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X3 ) @ A2 ) @ ( sup_sup_set_o @ X3 @ B2 ) )
= top_top_set_o ) ).
% sup_cancel_left2
thf(fact_1053_sup__cancel__left1,axiom,
! [X3: set_o,A2: set_o,B2: set_o] :
( ( sup_sup_set_o @ ( sup_sup_set_o @ X3 @ A2 ) @ ( sup_sup_set_o @ ( uminus_uminus_set_o @ X3 ) @ B2 ) )
= top_top_set_o ) ).
% sup_cancel_left1
thf(fact_1054_subset__Compl__self__eq,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ A ) )
= ( A = bot_bot_set_o ) ) ).
% subset_Compl_self_eq
thf(fact_1055_Compl__partition2,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ A ) @ A )
= top_top_set_o ) ).
% Compl_partition2
thf(fact_1056_Compl__partition,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ ( uminus_uminus_set_o @ A ) )
= top_top_set_o ) ).
% Compl_partition
thf(fact_1057_Compl__Int,axiom,
! [A: set_b,B: set_b] :
( ( uminus_uminus_set_b @ ( inf_inf_set_b @ A @ B ) )
= ( sup_sup_set_b @ ( uminus_uminus_set_b @ A ) @ ( uminus_uminus_set_b @ B ) ) ) ).
% Compl_Int
thf(fact_1058_Compl__Un,axiom,
! [A: set_b,B: set_b] :
( ( uminus_uminus_set_b @ ( sup_sup_set_b @ A @ B ) )
= ( inf_inf_set_b @ ( uminus_uminus_set_b @ A ) @ ( uminus_uminus_set_b @ B ) ) ) ).
% Compl_Un
thf(fact_1059_inf__shunt,axiom,
! [X3: set_b,Y2: set_b] :
( ( ( inf_inf_set_b @ X3 @ Y2 )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ X3 @ ( uminus_uminus_set_b @ Y2 ) ) ) ).
% inf_shunt
thf(fact_1060_inf__shunt,axiom,
! [X3: set_o,Y2: set_o] :
( ( ( inf_inf_set_o @ X3 @ Y2 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X3 @ ( uminus_uminus_set_o @ Y2 ) ) ) ).
% inf_shunt
thf(fact_1061_sup__shunt,axiom,
! [X3: set_o,Y2: set_o] :
( ( ( sup_sup_set_o @ X3 @ Y2 )
= top_top_set_o )
= ( ord_less_eq_set_o @ ( uminus_uminus_set_o @ X3 ) @ Y2 ) ) ).
% sup_shunt
thf(fact_1062_shunt1,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ ( inf_inf_set_b @ X3 @ Y2 ) @ Z2 )
= ( ord_less_eq_set_b @ X3 @ ( sup_sup_set_b @ ( uminus_uminus_set_b @ Y2 ) @ Z2 ) ) ) ).
% shunt1
thf(fact_1063_shunt2,axiom,
! [X3: set_b,Y2: set_b,Z2: set_b] :
( ( ord_less_eq_set_b @ ( inf_inf_set_b @ X3 @ ( uminus_uminus_set_b @ Y2 ) ) @ Z2 )
= ( ord_less_eq_set_b @ X3 @ ( sup_sup_set_b @ Y2 @ Z2 ) ) ) ).
% shunt2
thf(fact_1064_sup__neg__inf,axiom,
! [P: set_b,Q: set_b,R: set_b] :
( ( ord_less_eq_set_b @ P @ ( sup_sup_set_b @ Q @ R ) )
= ( ord_less_eq_set_b @ ( inf_inf_set_b @ P @ ( uminus_uminus_set_b @ Q ) ) @ R ) ) ).
% sup_neg_inf
thf(fact_1065_disjoint__eq__subset__Compl,axiom,
! [A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ A @ ( uminus_uminus_set_b @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1066_disjoint__eq__subset__Compl,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1067_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea6678413353003181397_set_b @ inf_inf_set_b @ sup_sup_set_b @ uminus_uminus_set_b @ bot_bot_set_b @ top_top_set_b ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1068_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea379910186789422830_set_o @ inf_inf_set_o @ sup_sup_set_o @ uminus_uminus_set_o @ bot_bot_set_o @ top_top_set_o ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1069_range__constant,axiom,
! [X3: $o] :
( ( image_o_o
@ ^ [Uu2: $o] : X3
@ top_top_set_o )
= ( insert_o @ X3 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_1070_image__eqI,axiom,
! [B2: b,F2: b > b,X3: b,A: set_b] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member_b @ X3 @ A )
=> ( member_b @ B2 @ ( image_b_b @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_1071_ComplI,axiom,
! [C: b,A: set_b] :
( ~ ( member_b @ C @ A )
=> ( member_b @ C @ ( uminus_uminus_set_b @ A ) ) ) ).
% ComplI
thf(fact_1072_Compl__iff,axiom,
! [C: b,A: set_b] :
( ( member_b @ C @ ( uminus_uminus_set_b @ A ) )
= ( ~ ( member_b @ C @ A ) ) ) ).
% Compl_iff
thf(fact_1073_image__empty,axiom,
! [F2: $o > $o] :
( ( image_o_o @ F2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_1074_empty__is__image,axiom,
! [F2: $o > $o,A: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F2 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_1075_image__is__empty,axiom,
! [F2: $o > $o,A: set_o] :
( ( ( image_o_o @ F2 @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_1076_image__insert,axiom,
! [F2: $o > $o,A2: $o,B: set_o] :
( ( image_o_o @ F2 @ ( insert_o @ A2 @ B ) )
= ( insert_o @ ( F2 @ A2 ) @ ( image_o_o @ F2 @ B ) ) ) ).
% image_insert
thf(fact_1077_insert__image,axiom,
! [X3: b,A: set_b,F2: b > $o] :
( ( member_b @ X3 @ A )
=> ( ( insert_o @ ( F2 @ X3 ) @ ( image_b_o @ F2 @ A ) )
= ( image_b_o @ F2 @ A ) ) ) ).
% insert_image
thf(fact_1078_ComplD,axiom,
! [C: b,A: set_b] :
( ( member_b @ C @ ( uminus_uminus_set_b @ A ) )
=> ~ ( member_b @ C @ A ) ) ).
% ComplD
thf(fact_1079_uminus__set__def,axiom,
( uminus_uminus_set_b
= ( ^ [A4: set_b] :
( collect_b
@ ( uminus_uminus_b_o
@ ^ [X2: b] : ( member_b @ X2 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1080_Compl__eq,axiom,
( uminus_uminus_set_b
= ( ^ [A4: set_b] :
( collect_b
@ ^ [X2: b] :
~ ( member_b @ X2 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_1081_imageE,axiom,
! [B2: b,F2: b > b,A: set_b] :
( ( member_b @ B2 @ ( image_b_b @ F2 @ A ) )
=> ~ ! [X: b] :
( ( B2
= ( F2 @ X ) )
=> ~ ( member_b @ X @ A ) ) ) ).
% imageE
thf(fact_1082_Compr__image__eq,axiom,
! [F2: b > b,A: set_b,P4: b > $o] :
( ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ ( image_b_b @ F2 @ A ) )
& ( P4 @ X2 ) ) )
= ( image_b_b @ F2
@ ( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A )
& ( P4 @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1083_rev__image__eqI,axiom,
! [X3: b,A: set_b,B2: b,F2: b > b] :
( ( member_b @ X3 @ A )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member_b @ B2 @ ( image_b_b @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_1084_imageI,axiom,
! [X3: b,A: set_b,F2: b > b] :
( ( member_b @ X3 @ A )
=> ( member_b @ ( F2 @ X3 ) @ ( image_b_b @ F2 @ A ) ) ) ).
% imageI
thf(fact_1085_image__subsetI,axiom,
! [A: set_b,F2: b > b,B: set_b] :
( ! [X: b] :
( ( member_b @ X @ A )
=> ( member_b @ ( F2 @ X ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1086_rangeI,axiom,
! [F2: $o > b,X3: $o] : ( member_b @ ( F2 @ X3 ) @ ( image_o_b @ F2 @ top_top_set_o ) ) ).
% rangeI
thf(fact_1087_range__eqI,axiom,
! [B2: b,F2: $o > b,X3: $o] :
( ( B2
= ( F2 @ X3 ) )
=> ( member_b @ B2 @ ( image_o_b @ F2 @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_1088_rangeE,axiom,
! [B2: b,F2: $o > b] :
( ( member_b @ B2 @ ( image_o_b @ F2 @ top_top_set_o ) )
=> ~ ! [X: $o] :
( B2
!= ( F2 @ X ) ) ) ).
% rangeE
thf(fact_1089_range__subsetD,axiom,
! [F2: $o > b,B: set_b,I2: $o] :
( ( ord_less_eq_set_b @ ( image_o_b @ F2 @ top_top_set_o ) @ B )
=> ( member_b @ ( F2 @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_1090_image__Int__subset,axiom,
! [F2: b > b,A: set_b,B: set_b] : ( ord_less_eq_set_b @ ( image_b_b @ F2 @ ( inf_inf_set_b @ A @ B ) ) @ ( inf_inf_set_b @ ( image_b_b @ F2 @ A ) @ ( image_b_b @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1091_image__constant__conv,axiom,
! [A: set_o,C: $o] :
( ( ( A = bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X2: $o] : C
@ A )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X2: $o] : C
@ A )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_1092_image__constant,axiom,
! [X3: b,A: set_b,C: $o] :
( ( member_b @ X3 @ A )
=> ( ( image_b_o
@ ^ [X2: b] : C
@ A )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_1093_range__eq__singletonD,axiom,
! [F2: $o > $o,A2: $o,X3: $o] :
( ( ( image_o_o @ F2 @ top_top_set_o )
= ( insert_o @ A2 @ bot_bot_set_o ) )
=> ( ( F2 @ X3 )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_1094_empty__bind,axiom,
! [F2: $o > set_o] :
( ( bind_o_o @ bot_bot_set_o @ F2 )
= bot_bot_set_o ) ).
% empty_bind
thf(fact_1095_insert__partition,axiom,
! [X3: set_b,F4: set_set_b] :
( ~ ( member_set_b @ X3 @ F4 )
=> ( ! [X: set_b] :
( ( member_set_b @ X @ ( insert_set_b @ X3 @ F4 ) )
=> ! [Xa: set_b] :
( ( member_set_b @ Xa @ ( insert_set_b @ X3 @ F4 ) )
=> ( ( X != Xa )
=> ( ( inf_inf_set_b @ X @ Xa )
= bot_bot_set_b ) ) ) )
=> ( ( inf_inf_set_b @ X3 @ ( comple2307003614231284044_set_b @ F4 ) )
= bot_bot_set_b ) ) ) ).
% insert_partition
thf(fact_1096_insert__partition,axiom,
! [X3: set_o,F4: set_set_o] :
( ~ ( member_set_o @ X3 @ F4 )
=> ( ! [X: set_o] :
( ( member_set_o @ X @ ( insert_set_o @ X3 @ F4 ) )
=> ! [Xa: set_o] :
( ( member_set_o @ Xa @ ( insert_set_o @ X3 @ F4 ) )
=> ( ( X != Xa )
=> ( ( inf_inf_set_o @ X @ Xa )
= bot_bot_set_o ) ) ) )
=> ( ( inf_inf_set_o @ X3 @ ( comple90263536869209701_set_o @ F4 ) )
= bot_bot_set_o ) ) ) ).
% insert_partition
thf(fact_1097_bind__const,axiom,
! [A: set_o,B: set_o] :
( ( ( A = bot_bot_set_o )
=> ( ( bind_o_o @ A
@ ^ [Uu2: $o] : B )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_o )
=> ( ( bind_o_o @ A
@ ^ [Uu2: $o] : B )
= B ) ) ) ).
% bind_const
thf(fact_1098_Sup__inter__less__eq,axiom,
! [A: set_set_b,B: set_set_b] : ( ord_less_eq_set_b @ ( comple2307003614231284044_set_b @ ( inf_inf_set_set_b @ A @ B ) ) @ ( inf_inf_set_b @ ( comple2307003614231284044_set_b @ A ) @ ( comple2307003614231284044_set_b @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_1099_Int__Union,axiom,
! [A: set_b,B: set_set_b] :
( ( inf_inf_set_b @ A @ ( comple2307003614231284044_set_b @ B ) )
= ( comple2307003614231284044_set_b @ ( image_set_b_set_b @ ( inf_inf_set_b @ A ) @ B ) ) ) ).
% Int_Union
thf(fact_1100_Int__Union2,axiom,
! [B: set_set_b,A: set_b] :
( ( inf_inf_set_b @ ( comple2307003614231284044_set_b @ B ) @ A )
= ( comple2307003614231284044_set_b
@ ( image_set_b_set_b
@ ^ [C3: set_b] : ( inf_inf_set_b @ C3 @ A )
@ B ) ) ) ).
% Int_Union2
thf(fact_1101_Union__disjoint,axiom,
! [C2: set_set_b,A: set_b] :
( ( ( inf_inf_set_b @ ( comple2307003614231284044_set_b @ C2 ) @ A )
= bot_bot_set_b )
= ( ! [X2: set_b] :
( ( member_set_b @ X2 @ C2 )
=> ( ( inf_inf_set_b @ X2 @ A )
= bot_bot_set_b ) ) ) ) ).
% Union_disjoint
thf(fact_1102_Union__disjoint,axiom,
! [C2: set_set_o,A: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ C2 ) @ A )
= bot_bot_set_o )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ C2 )
=> ( ( inf_inf_set_o @ X2 @ A )
= bot_bot_set_o ) ) ) ) ).
% Union_disjoint
thf(fact_1103_Union__Int__subset,axiom,
! [A: set_set_b,B: set_set_b] : ( ord_less_eq_set_b @ ( comple2307003614231284044_set_b @ ( inf_inf_set_set_b @ A @ B ) ) @ ( inf_inf_set_b @ ( comple2307003614231284044_set_b @ A ) @ ( comple2307003614231284044_set_b @ B ) ) ) ).
% Union_Int_subset
thf(fact_1104_Sup__inf,axiom,
! [B: set_set_b,A2: set_b] :
( ( inf_inf_set_b @ ( comple2307003614231284044_set_b @ B ) @ A2 )
= ( comple2307003614231284044_set_b
@ ( image_set_b_set_b
@ ^ [B3: set_b] : ( inf_inf_set_b @ B3 @ A2 )
@ B ) ) ) ).
% Sup_inf
thf(fact_1105_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_b,A2: set_b] :
( ( ( inf_inf_set_b @ ( comple2307003614231284044_set_b @ B ) @ A2 )
= bot_bot_set_b )
= ( ! [X2: set_b] :
( ( member_set_b @ X2 @ B )
=> ( ( inf_inf_set_b @ X2 @ A2 )
= bot_bot_set_b ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1106_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_o,A2: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ B ) @ A2 )
= bot_bot_set_o )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ B )
=> ( ( inf_inf_set_o @ X2 @ A2 )
= bot_bot_set_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1107_inf__Sup,axiom,
! [A2: set_b,B: set_set_b] :
( ( inf_inf_set_b @ A2 @ ( comple2307003614231284044_set_b @ B ) )
= ( comple2307003614231284044_set_b @ ( image_set_b_set_b @ ( inf_inf_set_b @ A2 ) @ B ) ) ) ).
% inf_Sup
thf(fact_1108_INT__simps_I1_J,axiom,
! [C2: set_o,A: $o > set_b,B: set_b] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple6135023382983342438_set_b
@ ( image_o_set_b
@ ^ [X2: $o] : ( inf_inf_set_b @ ( A @ X2 ) @ B )
@ C2 ) )
= top_top_set_b ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple6135023382983342438_set_b
@ ( image_o_set_b
@ ^ [X2: $o] : ( inf_inf_set_b @ ( A @ X2 ) @ B )
@ C2 ) )
= ( inf_inf_set_b @ ( comple6135023382983342438_set_b @ ( image_o_set_b @ A @ C2 ) ) @ B ) ) ) ) ).
% INT_simps(1)
thf(fact_1109_INT__simps_I1_J,axiom,
! [C2: set_o,A: $o > set_o,B: set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X2: $o] : ( inf_inf_set_o @ ( A @ X2 ) @ B )
@ C2 ) )
= top_top_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X2: $o] : ( inf_inf_set_o @ ( A @ X2 ) @ B )
@ C2 ) )
= ( inf_inf_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A @ C2 ) ) @ B ) ) ) ) ).
% INT_simps(1)
thf(fact_1110_INT__simps_I2_J,axiom,
! [C2: set_o,A: set_b,B: $o > set_b] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple6135023382983342438_set_b
@ ( image_o_set_b
@ ^ [X2: $o] : ( inf_inf_set_b @ A @ ( B @ X2 ) )
@ C2 ) )
= top_top_set_b ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple6135023382983342438_set_b
@ ( image_o_set_b
@ ^ [X2: $o] : ( inf_inf_set_b @ A @ ( B @ X2 ) )
@ C2 ) )
= ( inf_inf_set_b @ A @ ( comple6135023382983342438_set_b @ ( image_o_set_b @ B @ C2 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_1111_INT__simps_I2_J,axiom,
! [C2: set_o,A: set_o,B: $o > set_o] :
( ( ( C2 = bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X2: $o] : ( inf_inf_set_o @ A @ ( B @ X2 ) )
@ C2 ) )
= top_top_set_o ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X2: $o] : ( inf_inf_set_o @ A @ ( B @ X2 ) )
@ C2 ) )
= ( inf_inf_set_o @ A @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ C2 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_1112_Inf__insert,axiom,
! [A2: $o,A: set_o] :
( ( complete_Inf_Inf_o @ ( insert_o @ A2 @ A ) )
= ( inf_inf_o @ A2 @ ( complete_Inf_Inf_o @ A ) ) ) ).
% Inf_insert
thf(fact_1113_Inf__insert,axiom,
! [A2: set_b,A: set_set_b] :
( ( comple6135023382983342438_set_b @ ( insert_set_b @ A2 @ A ) )
= ( inf_inf_set_b @ A2 @ ( comple6135023382983342438_set_b @ A ) ) ) ).
% Inf_insert
thf(fact_1114_INT__insert,axiom,
! [B: $o > set_b,A2: $o,A: set_o] :
( ( comple6135023382983342438_set_b @ ( image_o_set_b @ B @ ( insert_o @ A2 @ A ) ) )
= ( inf_inf_set_b @ ( B @ A2 ) @ ( comple6135023382983342438_set_b @ ( image_o_set_b @ B @ A ) ) ) ) ).
% INT_insert
thf(fact_1115_INF__UNIV__bool__expand,axiom,
! [A: $o > set_b] :
( ( comple6135023382983342438_set_b @ ( image_o_set_b @ A @ top_top_set_o ) )
= ( inf_inf_set_b @ ( A @ $true ) @ ( A @ $false ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_1116_INT__bool__eq,axiom,
! [A: $o > set_b] :
( ( comple6135023382983342438_set_b @ ( image_o_set_b @ A @ top_top_set_o ) )
= ( inf_inf_set_b @ ( A @ $true ) @ ( A @ $false ) ) ) ).
% INT_bool_eq
thf(fact_1117_Inter__Un__distrib,axiom,
! [A: set_set_b,B: set_set_b] :
( ( comple6135023382983342438_set_b @ ( sup_sup_set_set_b @ A @ B ) )
= ( inf_inf_set_b @ ( comple6135023382983342438_set_b @ A ) @ ( comple6135023382983342438_set_b @ B ) ) ) ).
% Inter_Un_distrib
thf(fact_1118_Inf__union__distrib,axiom,
! [A: set_set_b,B: set_set_b] :
( ( comple6135023382983342438_set_b @ ( sup_sup_set_set_b @ A @ B ) )
= ( inf_inf_set_b @ ( comple6135023382983342438_set_b @ A ) @ ( comple6135023382983342438_set_b @ B ) ) ) ).
% Inf_union_distrib
thf(fact_1119_INF__absorb,axiom,
! [K4: b,I4: set_b,A: b > set_b] :
( ( member_b @ K4 @ I4 )
=> ( ( inf_inf_set_b @ ( A @ K4 ) @ ( comple6135023382983342438_set_b @ ( image_b_set_b @ A @ I4 ) ) )
= ( comple6135023382983342438_set_b @ ( image_b_set_b @ A @ I4 ) ) ) ) ).
% INF_absorb
thf(fact_1120_Inter__insert,axiom,
! [A2: set_b,B: set_set_b] :
( ( comple6135023382983342438_set_b @ ( insert_set_b @ A2 @ B ) )
= ( inf_inf_set_b @ A2 @ ( comple6135023382983342438_set_b @ B ) ) ) ).
% Inter_insert
thf(fact_1121_INT__absorb,axiom,
! [K4: b,I4: set_b,A: b > set_b] :
( ( member_b @ K4 @ I4 )
=> ( ( inf_inf_set_b @ ( A @ K4 ) @ ( comple6135023382983342438_set_b @ ( image_b_set_b @ A @ I4 ) ) )
= ( comple6135023382983342438_set_b @ ( image_b_set_b @ A @ I4 ) ) ) ) ).
% INT_absorb
thf(fact_1122_Int__Inter__eq_I1_J,axiom,
! [B6: set_set_b,A: set_b] :
( ( ( B6 = bot_bot_set_set_b )
=> ( ( inf_inf_set_b @ A @ ( comple6135023382983342438_set_b @ B6 ) )
= A ) )
& ( ( B6 != bot_bot_set_set_b )
=> ( ( inf_inf_set_b @ A @ ( comple6135023382983342438_set_b @ B6 ) )
= ( comple6135023382983342438_set_b @ ( image_set_b_set_b @ ( inf_inf_set_b @ A ) @ B6 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_1123_Int__Inter__eq_I2_J,axiom,
! [B6: set_set_b,A: set_b] :
( ( ( B6 = bot_bot_set_set_b )
=> ( ( inf_inf_set_b @ ( comple6135023382983342438_set_b @ B6 ) @ A )
= A ) )
& ( ( B6 != bot_bot_set_set_b )
=> ( ( inf_inf_set_b @ ( comple6135023382983342438_set_b @ B6 ) @ A )
= ( comple6135023382983342438_set_b
@ ( image_set_b_set_b
@ ^ [B4: set_b] : ( inf_inf_set_b @ B4 @ A )
@ B6 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_1124_finite__Inf__in,axiom,
! [A: set_set_b] :
( ( finite_finite_set_b @ A )
=> ( ( A != bot_bot_set_set_b )
=> ( ! [X: set_b,Y3: set_b] :
( ( member_set_b @ X @ A )
=> ( ( member_set_b @ Y3 @ A )
=> ( member_set_b @ ( inf_inf_set_b @ X @ Y3 ) @ A ) ) )
=> ( member_set_b @ ( comple6135023382983342438_set_b @ A ) @ A ) ) ) ) ).
% finite_Inf_in
thf(fact_1125_finite__Inf__in,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X: $o,Y3: $o] :
( ( member_o @ X @ A )
=> ( ( member_o @ Y3 @ A )
=> ( member_o @ ( inf_inf_o @ X @ Y3 ) @ A ) ) )
=> ( member_o @ ( complete_Inf_Inf_o @ A ) @ A ) ) ) ) ).
% finite_Inf_in
thf(fact_1126_INF__inf__const2,axiom,
! [I4: set_o,F2: $o > set_b,X3: set_b] :
( ( I4 != bot_bot_set_o )
=> ( ( comple6135023382983342438_set_b
@ ( image_o_set_b
@ ^ [I: $o] : ( inf_inf_set_b @ ( F2 @ I ) @ X3 )
@ I4 ) )
= ( inf_inf_set_b @ ( comple6135023382983342438_set_b @ ( image_o_set_b @ F2 @ I4 ) ) @ X3 ) ) ) ).
% INF_inf_const2
thf(fact_1127_INF__inf__const1,axiom,
! [I4: set_o,X3: set_b,F2: $o > set_b] :
( ( I4 != bot_bot_set_o )
=> ( ( comple6135023382983342438_set_b
@ ( image_o_set_b
@ ^ [I: $o] : ( inf_inf_set_b @ X3 @ ( F2 @ I ) )
@ I4 ) )
= ( inf_inf_set_b @ X3 @ ( comple6135023382983342438_set_b @ ( image_o_set_b @ F2 @ I4 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_1128_INF__insert,axiom,
! [F2: $o > set_b,A2: $o,A: set_o] :
( ( comple6135023382983342438_set_b @ ( image_o_set_b @ F2 @ ( insert_o @ A2 @ A ) ) )
= ( inf_inf_set_b @ ( F2 @ A2 ) @ ( comple6135023382983342438_set_b @ ( image_o_set_b @ F2 @ A ) ) ) ) ).
% INF_insert
thf(fact_1129_INT__extend__simps_I1_J,axiom,
! [C2: set_o,A: $o > set_b,B: set_b] :
( ( ( C2 = bot_bot_set_o )
=> ( ( inf_inf_set_b @ ( comple6135023382983342438_set_b @ ( image_o_set_b @ A @ C2 ) ) @ B )
= B ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( inf_inf_set_b @ ( comple6135023382983342438_set_b @ ( image_o_set_b @ A @ C2 ) ) @ B )
= ( comple6135023382983342438_set_b
@ ( image_o_set_b
@ ^ [X2: $o] : ( inf_inf_set_b @ ( A @ X2 ) @ B )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_1130_INT__extend__simps_I2_J,axiom,
! [C2: set_o,A: set_b,B: $o > set_b] :
( ( ( C2 = bot_bot_set_o )
=> ( ( inf_inf_set_b @ A @ ( comple6135023382983342438_set_b @ ( image_o_set_b @ B @ C2 ) ) )
= A ) )
& ( ( C2 != bot_bot_set_o )
=> ( ( inf_inf_set_b @ A @ ( comple6135023382983342438_set_b @ ( image_o_set_b @ B @ C2 ) ) )
= ( comple6135023382983342438_set_b
@ ( image_o_set_b
@ ^ [X2: $o] : ( inf_inf_set_b @ A @ ( B @ X2 ) )
@ C2 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_1131_inf__img__fin__dom_H,axiom,
! [F2: b > $o,A: set_b] :
( ( finite_finite_o @ ( image_b_o @ F2 @ A ) )
=> ( ~ ( finite_finite_b @ A )
=> ? [X: $o] :
( ( member_o @ X @ ( image_b_o @ F2 @ A ) )
& ~ ( finite_finite_b @ ( inf_inf_set_b @ ( vimage_b_o @ F2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_1132_vimageI,axiom,
! [F2: b > b,A2: b,B2: b,B: set_b] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_b @ B2 @ B )
=> ( member_b @ A2 @ ( vimage_b_b @ F2 @ B ) ) ) ) ).
% vimageI
thf(fact_1133_vimage__eq,axiom,
! [A2: b,F2: b > b,B: set_b] :
( ( member_b @ A2 @ ( vimage_b_b @ F2 @ B ) )
= ( member_b @ ( F2 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_1134_vimage__UNIV,axiom,
! [F2: $o > $o] :
( ( vimage_o_o @ F2 @ top_top_set_o )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_1135_vimage__empty,axiom,
! [F2: $o > $o] :
( ( vimage_o_o @ F2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_1136_vimage__Int,axiom,
! [F2: b > b,A: set_b,B: set_b] :
( ( vimage_b_b @ F2 @ ( inf_inf_set_b @ A @ B ) )
= ( inf_inf_set_b @ ( vimage_b_b @ F2 @ A ) @ ( vimage_b_b @ F2 @ B ) ) ) ).
% vimage_Int
thf(fact_1137_vimage__const,axiom,
! [C: b,A: set_b] :
( ( ( member_b @ C @ A )
=> ( ( vimage_o_b
@ ^ [X2: $o] : C
@ A )
= top_top_set_o ) )
& ( ~ ( member_b @ C @ A )
=> ( ( vimage_o_b
@ ^ [X2: $o] : C
@ A )
= bot_bot_set_o ) ) ) ).
% vimage_const
thf(fact_1138_image__vimage__eq,axiom,
! [F2: $o > b,A: set_b] :
( ( image_o_b @ F2 @ ( vimage_o_b @ F2 @ A ) )
= ( inf_inf_set_b @ A @ ( image_o_b @ F2 @ top_top_set_o ) ) ) ).
% image_vimage_eq
thf(fact_1139_vimage__if,axiom,
! [C: b,A: set_b,D: b,B: set_b] :
( ( ( member_b @ C @ A )
=> ( ( ( member_b @ D @ A )
=> ( ( vimage_b_b
@ ^ [X2: b] : ( if_b @ ( member_b @ X2 @ B ) @ C @ D )
@ A )
= top_top_set_b ) )
& ( ~ ( member_b @ D @ A )
=> ( ( vimage_b_b
@ ^ [X2: b] : ( if_b @ ( member_b @ X2 @ B ) @ C @ D )
@ A )
= B ) ) ) )
& ( ~ ( member_b @ C @ A )
=> ( ( ( member_b @ D @ A )
=> ( ( vimage_b_b
@ ^ [X2: b] : ( if_b @ ( member_b @ X2 @ B ) @ C @ D )
@ A )
= ( uminus_uminus_set_b @ B ) ) )
& ( ~ ( member_b @ D @ A )
=> ( ( vimage_b_b
@ ^ [X2: b] : ( if_b @ ( member_b @ X2 @ B ) @ C @ D )
@ A )
= bot_bot_set_b ) ) ) ) ) ).
% vimage_if
thf(fact_1140_vimage__if,axiom,
! [C: b,A: set_b,D: b,B: set_o] :
( ( ( member_b @ C @ A )
=> ( ( ( member_b @ D @ A )
=> ( ( vimage_o_b
@ ^ [X2: $o] : ( if_b @ ( member_o @ X2 @ B ) @ C @ D )
@ A )
= top_top_set_o ) )
& ( ~ ( member_b @ D @ A )
=> ( ( vimage_o_b
@ ^ [X2: $o] : ( if_b @ ( member_o @ X2 @ B ) @ C @ D )
@ A )
= B ) ) ) )
& ( ~ ( member_b @ C @ A )
=> ( ( ( member_b @ D @ A )
=> ( ( vimage_o_b
@ ^ [X2: $o] : ( if_b @ ( member_o @ X2 @ B ) @ C @ D )
@ A )
= ( uminus_uminus_set_o @ B ) ) )
& ( ~ ( member_b @ D @ A )
=> ( ( vimage_o_b
@ ^ [X2: $o] : ( if_b @ ( member_o @ X2 @ B ) @ C @ D )
@ A )
= bot_bot_set_o ) ) ) ) ) ).
% vimage_if
thf(fact_1141_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1142_vimage__singleton__eq,axiom,
! [A2: b,F2: b > $o,B2: $o] :
( ( member_b @ A2 @ ( vimage_b_o @ F2 @ ( insert_o @ B2 @ bot_bot_set_o ) ) )
= ( ( F2 @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_1143_vimageD,axiom,
! [A2: b,F2: b > b,A: set_b] :
( ( member_b @ A2 @ ( vimage_b_b @ F2 @ A ) )
=> ( member_b @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1144_vimageE,axiom,
! [A2: b,F2: b > b,B: set_b] :
( ( member_b @ A2 @ ( vimage_b_b @ F2 @ B ) )
=> ( member_b @ ( F2 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1145_vimageI2,axiom,
! [F2: b > b,A2: b,A: set_b] :
( ( member_b @ ( F2 @ A2 ) @ A )
=> ( member_b @ A2 @ ( vimage_b_b @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_1146_finite__finite__vimage__IntI,axiom,
! [F4: set_b,H2: b > b,A: set_b] :
( ( finite_finite_b @ F4 )
=> ( ! [Y3: b] :
( ( member_b @ Y3 @ F4 )
=> ( finite_finite_b @ ( inf_inf_set_b @ ( vimage_b_b @ H2 @ ( insert_b @ Y3 @ bot_bot_set_b ) ) @ A ) ) )
=> ( finite_finite_b @ ( inf_inf_set_b @ ( vimage_b_b @ H2 @ F4 ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1147_finite__finite__vimage__IntI,axiom,
! [F4: set_o,H2: b > $o,A: set_b] :
( ( finite_finite_o @ F4 )
=> ( ! [Y3: $o] :
( ( member_o @ Y3 @ F4 )
=> ( finite_finite_b @ ( inf_inf_set_b @ ( vimage_b_o @ H2 @ ( insert_o @ Y3 @ bot_bot_set_o ) ) @ A ) ) )
=> ( finite_finite_b @ ( inf_inf_set_b @ ( vimage_b_o @ H2 @ F4 ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1148_inf__img__fin__domE_H,axiom,
! [F2: b > b,A: set_b] :
( ( finite_finite_b @ ( image_b_b @ F2 @ A ) )
=> ( ~ ( finite_finite_b @ A )
=> ~ ! [Y3: b] :
( ( member_b @ Y3 @ ( image_b_b @ F2 @ A ) )
=> ( finite_finite_b @ ( inf_inf_set_b @ ( vimage_b_b @ F2 @ ( insert_b @ Y3 @ bot_bot_set_b ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_1149_inf__img__fin__domE_H,axiom,
! [F2: b > $o,A: set_b] :
( ( finite_finite_o @ ( image_b_o @ F2 @ A ) )
=> ( ~ ( finite_finite_b @ A )
=> ~ ! [Y3: $o] :
( ( member_o @ Y3 @ ( image_b_o @ F2 @ A ) )
=> ( finite_finite_b @ ( inf_inf_set_b @ ( vimage_b_o @ F2 @ ( insert_o @ Y3 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_1150_inj__on__vimage__singleton,axiom,
! [F2: b > $o,A: set_b,A2: $o] :
( ( inj_on_b_o @ F2 @ A )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ ( vimage_b_o @ F2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ A )
@ ( insert_b
@ ( the_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A )
& ( ( F2 @ X2 )
= A2 ) ) )
@ bot_bot_set_b ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1151_inj__on__vimage__singleton,axiom,
! [F2: $o > $o,A: set_o,A2: $o] :
( ( inj_on_o_o @ F2 @ A )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ ( vimage_o_o @ F2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ A )
@ ( insert_o
@ ( the_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( ( F2 @ X2 )
= A2 ) ) )
@ bot_bot_set_o ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1152_Pow__singleton__iff,axiom,
! [X5: set_o,Y5: set_o] :
( ( ( pow_o @ X5 )
= ( insert_set_o @ Y5 @ bot_bot_set_set_o ) )
= ( ( X5 = bot_bot_set_o )
& ( Y5 = bot_bot_set_o ) ) ) ).
% Pow_singleton_iff
thf(fact_1153_Pow__UNIV,axiom,
( ( pow_o @ top_top_set_o )
= top_top_set_set_o ) ).
% Pow_UNIV
thf(fact_1154_Pow__Int__eq,axiom,
! [A: set_b,B: set_b] :
( ( pow_b @ ( inf_inf_set_b @ A @ B ) )
= ( inf_inf_set_set_b @ ( pow_b @ A ) @ ( pow_b @ B ) ) ) ).
% Pow_Int_eq
thf(fact_1155_Pow__empty,axiom,
( ( pow_o @ bot_bot_set_o )
= ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ).
% Pow_empty
thf(fact_1156_Pow__bottom,axiom,
! [B: set_o] : ( member_set_o @ bot_bot_set_o @ ( pow_o @ B ) ) ).
% Pow_bottom
thf(fact_1157_image__Int,axiom,
! [F2: b > b,A: set_b,B: set_b] :
( ( inj_on_b_b @ F2 @ top_top_set_b )
=> ( ( image_b_b @ F2 @ ( inf_inf_set_b @ A @ B ) )
= ( inf_inf_set_b @ ( image_b_b @ F2 @ A ) @ ( image_b_b @ F2 @ B ) ) ) ) ).
% image_Int
thf(fact_1158_image__Int,axiom,
! [F2: $o > b,A: set_o,B: set_o] :
( ( inj_on_o_b @ F2 @ top_top_set_o )
=> ( ( image_o_b @ F2 @ ( inf_inf_set_o @ A @ B ) )
= ( inf_inf_set_b @ ( image_o_b @ F2 @ A ) @ ( image_o_b @ F2 @ B ) ) ) ) ).
% image_Int
thf(fact_1159_inj__on__image__Int,axiom,
! [F2: b > b,C2: set_b,A: set_b,B: set_b] :
( ( inj_on_b_b @ F2 @ C2 )
=> ( ( ord_less_eq_set_b @ A @ C2 )
=> ( ( ord_less_eq_set_b @ B @ C2 )
=> ( ( image_b_b @ F2 @ ( inf_inf_set_b @ A @ B ) )
= ( inf_inf_set_b @ ( image_b_b @ F2 @ A ) @ ( image_b_b @ F2 @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1160_inj__on__disjoint__Un,axiom,
! [F2: b > b,A: set_b,G3: b > b,B: set_b] :
( ( inj_on_b_b @ F2 @ A )
=> ( ( inj_on_b_b @ G3 @ B )
=> ( ( ( inf_inf_set_b @ ( image_b_b @ F2 @ A ) @ ( image_b_b @ G3 @ B ) )
= bot_bot_set_b )
=> ( inj_on_b_b
@ ^ [X2: b] : ( if_b @ ( member_b @ X2 @ A ) @ ( F2 @ X2 ) @ ( G3 @ X2 ) )
@ ( sup_sup_set_b @ A @ B ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_1161_inj__on__disjoint__Un,axiom,
! [F2: b > $o,A: set_b,G3: b > $o,B: set_b] :
( ( inj_on_b_o @ F2 @ A )
=> ( ( inj_on_b_o @ G3 @ B )
=> ( ( ( inf_inf_set_o @ ( image_b_o @ F2 @ A ) @ ( image_b_o @ G3 @ B ) )
= bot_bot_set_o )
=> ( inj_on_b_o
@ ^ [X2: b] :
( ( ( member_b @ X2 @ A )
=> ( F2 @ X2 ) )
& ( ~ ( member_b @ X2 @ A )
=> ( G3 @ X2 ) ) )
@ ( sup_sup_set_b @ A @ B ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_1162_Pow__insert,axiom,
! [A2: $o,A: set_o] :
( ( pow_o @ ( insert_o @ A2 @ A ) )
= ( sup_sup_set_set_o @ ( pow_o @ A ) @ ( image_set_o_set_o @ ( insert_o @ A2 ) @ ( pow_o @ A ) ) ) ) ).
% Pow_insert
thf(fact_1163_Inf__fold__inf,axiom,
! [A: set_set_b] :
( ( finite_finite_set_b @ A )
=> ( ( comple6135023382983342438_set_b @ A )
= ( finite7473193970610274952_set_b @ inf_inf_set_b @ top_top_set_b @ A ) ) ) ).
% Inf_fold_inf
thf(fact_1164_Inf__fold__inf,axiom,
! [A: set_set_o] :
( ( finite_finite_set_o @ A )
=> ( ( comple3063163877087187839_set_o @ A )
= ( finite4337638375924247368_set_o @ inf_inf_set_o @ top_top_set_o @ A ) ) ) ).
% Inf_fold_inf
thf(fact_1165_DiffI,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ A )
=> ( ~ ( member_b @ C @ B )
=> ( member_b @ C @ ( minus_minus_set_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1166_Diff__iff,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
= ( ( member_b @ C @ A )
& ~ ( member_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1167_Diff__cancel,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ A )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_1168_empty__Diff,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_1169_Diff__empty,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ bot_bot_set_o )
= A ) ).
% Diff_empty
thf(fact_1170_insert__Diff1,axiom,
! [X3: $o,B: set_o,A: set_o] :
( ( member_o @ X3 @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X3 @ A ) @ B )
= ( minus_minus_set_o @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1171_insert__Diff1,axiom,
! [X3: b,B: set_b,A: set_b] :
( ( member_b @ X3 @ B )
=> ( ( minus_minus_set_b @ ( insert_b @ X3 @ A ) @ B )
= ( minus_minus_set_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1172_Diff__insert0,axiom,
! [X3: $o,A: set_o,B: set_o] :
( ~ ( member_o @ X3 @ A )
=> ( ( minus_minus_set_o @ A @ ( insert_o @ X3 @ B ) )
= ( minus_minus_set_o @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1173_Diff__insert0,axiom,
! [X3: b,A: set_b,B: set_b] :
( ~ ( member_b @ X3 @ A )
=> ( ( minus_minus_set_b @ A @ ( insert_b @ X3 @ B ) )
= ( minus_minus_set_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1174_Diff__UNIV,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ top_top_set_o )
= bot_bot_set_o ) ).
% Diff_UNIV
thf(fact_1175_Diff__eq__empty__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( minus_minus_set_o @ A @ B )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1176_insert__Diff__single,axiom,
! [A2: $o,A: set_o] :
( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( insert_o @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_1177_Diff__disjoint,axiom,
! [A: set_b,B: set_b] :
( ( inf_inf_set_b @ A @ ( minus_minus_set_b @ B @ A ) )
= bot_bot_set_b ) ).
% Diff_disjoint
thf(fact_1178_Diff__disjoint,axiom,
! [A: set_o,B: set_o] :
( ( inf_inf_set_o @ A @ ( minus_minus_set_o @ B @ A ) )
= bot_bot_set_o ) ).
% Diff_disjoint
thf(fact_1179_Diff__Compl,axiom,
! [A: set_b,B: set_b] :
( ( minus_minus_set_b @ A @ ( uminus_uminus_set_b @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ).
% Diff_Compl
thf(fact_1180_Diff__insert__absorb,axiom,
! [X3: b,A: set_b] :
( ~ ( member_b @ X3 @ A )
=> ( ( minus_minus_set_b @ ( insert_b @ X3 @ A ) @ ( insert_b @ X3 @ bot_bot_set_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1181_Diff__insert__absorb,axiom,
! [X3: $o,A: set_o] :
( ~ ( member_o @ X3 @ A )
=> ( ( minus_minus_set_o @ ( insert_o @ X3 @ A ) @ ( insert_o @ X3 @ bot_bot_set_o ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1182_Diff__insert2,axiom,
! [A: set_o,A2: $o,B: set_o] :
( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ B ) ) ).
% Diff_insert2
thf(fact_1183_insert__Diff,axiom,
! [A2: b,A: set_b] :
( ( member_b @ A2 @ A )
=> ( ( insert_b @ A2 @ ( minus_minus_set_b @ A @ ( insert_b @ A2 @ bot_bot_set_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1184_insert__Diff,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1185_Diff__insert,axiom,
! [A: set_o,A2: $o,B: set_o] :
( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A @ B ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_1186_diff__shunt__var,axiom,
! [X3: set_o,Y2: set_o] :
( ( ( minus_minus_set_o @ X3 @ Y2 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X3 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1187_Int__Diff__disjoint,axiom,
! [A: set_b,B: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B ) @ ( minus_minus_set_b @ A @ B ) )
= bot_bot_set_b ) ).
% Int_Diff_disjoint
thf(fact_1188_Int__Diff__disjoint,axiom,
! [A: set_o,B: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A @ B ) @ ( minus_minus_set_o @ A @ B ) )
= bot_bot_set_o ) ).
% Int_Diff_disjoint
thf(fact_1189_Diff__triv,axiom,
! [A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
=> ( ( minus_minus_set_b @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1190_Diff__triv,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
=> ( ( minus_minus_set_o @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1191_insert__Diff__if,axiom,
! [X3: $o,B: set_o,A: set_o] :
( ( ( member_o @ X3 @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X3 @ A ) @ B )
= ( minus_minus_set_o @ A @ B ) ) )
& ( ~ ( member_o @ X3 @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X3 @ A ) @ B )
= ( insert_o @ X3 @ ( minus_minus_set_o @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1192_insert__Diff__if,axiom,
! [X3: b,B: set_b,A: set_b] :
( ( ( member_b @ X3 @ B )
=> ( ( minus_minus_set_b @ ( insert_b @ X3 @ A ) @ B )
= ( minus_minus_set_b @ A @ B ) ) )
& ( ~ ( member_b @ X3 @ B )
=> ( ( minus_minus_set_b @ ( insert_b @ X3 @ A ) @ B )
= ( insert_b @ X3 @ ( minus_minus_set_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1193_subset__Diff__insert,axiom,
! [A: set_o,B: set_o,X3: $o,C2: set_o] :
( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B @ ( insert_o @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B @ C2 ) )
& ~ ( member_o @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1194_subset__Diff__insert,axiom,
! [A: set_b,B: set_b,X3: b,C2: set_b] :
( ( ord_less_eq_set_b @ A @ ( minus_minus_set_b @ B @ ( insert_b @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_b @ A @ ( minus_minus_set_b @ B @ C2 ) )
& ~ ( member_b @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1195_DiffE,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
=> ~ ( ( member_b @ C @ A )
=> ( member_b @ C @ B ) ) ) ).
% DiffE
thf(fact_1196_DiffD1,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
=> ( member_b @ C @ A ) ) ).
% DiffD1
thf(fact_1197_DiffD2,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A @ B ) )
=> ~ ( member_b @ C @ B ) ) ).
% DiffD2
thf(fact_1198_set__diff__eq,axiom,
( minus_minus_set_b
= ( ^ [A4: set_b,B4: set_b] :
( collect_b
@ ^ [X2: b] :
( ( member_b @ X2 @ A4 )
& ~ ( member_b @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1199_diff__eq,axiom,
( minus_minus_set_b
= ( ^ [X2: set_b,Y: set_b] : ( inf_inf_set_b @ X2 @ ( uminus_uminus_set_b @ Y ) ) ) ) ).
% diff_eq
thf(fact_1200_Diff__eq,axiom,
( minus_minus_set_b
= ( ^ [A4: set_b,B4: set_b] : ( inf_inf_set_b @ A4 @ ( uminus_uminus_set_b @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_1201_Int__Diff,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( minus_minus_set_b @ ( inf_inf_set_b @ A @ B ) @ C2 )
= ( inf_inf_set_b @ A @ ( minus_minus_set_b @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_1202_Diff__Int2,axiom,
! [A: set_b,C2: set_b,B: set_b] :
( ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ ( inf_inf_set_b @ B @ C2 ) )
= ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_1203_Diff__Diff__Int,axiom,
! [A: set_b,B: set_b] :
( ( minus_minus_set_b @ A @ ( minus_minus_set_b @ A @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_1204_Diff__Int__distrib,axiom,
! [C2: set_b,A: set_b,B: set_b] :
( ( inf_inf_set_b @ C2 @ ( minus_minus_set_b @ A @ B ) )
= ( minus_minus_set_b @ ( inf_inf_set_b @ C2 @ A ) @ ( inf_inf_set_b @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_1205_Diff__Int__distrib2,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( inf_inf_set_b @ ( minus_minus_set_b @ A @ B ) @ C2 )
= ( minus_minus_set_b @ ( inf_inf_set_b @ A @ C2 ) @ ( inf_inf_set_b @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1206_Diff__Un,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( minus_minus_set_b @ A @ ( sup_sup_set_b @ B @ C2 ) )
= ( inf_inf_set_b @ ( minus_minus_set_b @ A @ B ) @ ( minus_minus_set_b @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1207_Diff__Int,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( minus_minus_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) )
= ( sup_sup_set_b @ ( minus_minus_set_b @ A @ B ) @ ( minus_minus_set_b @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1208_Int__Diff__Un,axiom,
! [A: set_b,B: set_b] :
( ( sup_sup_set_b @ ( inf_inf_set_b @ A @ B ) @ ( minus_minus_set_b @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1209_Un__Diff__Int,axiom,
! [A: set_b,B: set_b] :
( ( sup_sup_set_b @ ( minus_minus_set_b @ A @ B ) @ ( inf_inf_set_b @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1210_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_o
= ( minus_minus_set_o @ top_top_set_o ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1211_subset__insert__iff,axiom,
! [A: set_b,X3: b,B: set_b] :
( ( ord_less_eq_set_b @ A @ ( insert_b @ X3 @ B ) )
= ( ( ( member_b @ X3 @ A )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A @ ( insert_b @ X3 @ bot_bot_set_b ) ) @ B ) )
& ( ~ ( member_b @ X3 @ A )
=> ( ord_less_eq_set_b @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1212_subset__insert__iff,axiom,
! [A: set_o,X3: $o,B: set_o] :
( ( ord_less_eq_set_o @ A @ ( insert_o @ X3 @ B ) )
= ( ( ( member_o @ X3 @ A )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B ) )
& ( ~ ( member_o @ X3 @ A )
=> ( ord_less_eq_set_o @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1213_Diff__single__insert,axiom,
! [A: set_o,X3: $o,B: set_o] :
( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B )
=> ( ord_less_eq_set_o @ A @ ( insert_o @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1214_inf__Inf__fold__inf,axiom,
! [A: set_set_b,B: set_b] :
( ( finite_finite_set_b @ A )
=> ( ( inf_inf_set_b @ ( comple6135023382983342438_set_b @ A ) @ B )
= ( finite7473193970610274952_set_b @ inf_inf_set_b @ B @ A ) ) ) ).
% inf_Inf_fold_inf
thf(fact_1215_Compl__insert,axiom,
! [X3: $o,A: set_o] :
( ( uminus_uminus_set_o @ ( insert_o @ X3 @ A ) )
= ( minus_minus_set_o @ ( uminus_uminus_set_o @ A ) @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% Compl_insert
thf(fact_1216_in__image__insert__iff,axiom,
! [B: set_set_b,X3: b,A: set_b] :
( ! [C4: set_b] :
( ( member_set_b @ C4 @ B )
=> ~ ( member_b @ X3 @ C4 ) )
=> ( ( member_set_b @ A @ ( image_set_b_set_b @ ( insert_b @ X3 ) @ B ) )
= ( ( member_b @ X3 @ A )
& ( member_set_b @ ( minus_minus_set_b @ A @ ( insert_b @ X3 @ bot_bot_set_b ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1217_in__image__insert__iff,axiom,
! [B: set_set_o,X3: $o,A: set_o] :
( ! [C4: set_o] :
( ( member_set_o @ C4 @ B )
=> ~ ( member_o @ X3 @ C4 ) )
=> ( ( member_set_o @ A @ ( image_set_o_set_o @ ( insert_o @ X3 ) @ B ) )
= ( ( member_o @ X3 @ A )
& ( member_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1218_remove__def,axiom,
( remove_o
= ( ^ [X2: $o,A4: set_o] : ( minus_minus_set_o @ A4 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% remove_def
thf(fact_1219_member__remove,axiom,
! [X3: b,Y2: b,A: set_b] :
( ( member_b @ X3 @ ( remove_b @ Y2 @ A ) )
= ( ( member_b @ X3 @ A )
& ( X3 != Y2 ) ) ) ).
% member_remove
thf(fact_1220_minus__set__def,axiom,
( minus_minus_set_b
= ( ^ [A4: set_b,B4: set_b] :
( collect_b
@ ( minus_minus_b_o
@ ^ [X2: b] : ( member_b @ X2 @ A4 )
@ ^ [X2: b] : ( member_b @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1221_pairwise__alt,axiom,
( pairwise_o
= ( ^ [R4: $o > $o > $o,S2: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ S2 )
=> ! [Y: $o] :
( ( member_o @ Y @ ( minus_minus_set_o @ S2 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
=> ( R4 @ X2 @ Y ) ) ) ) ) ).
% pairwise_alt
thf(fact_1222_pairwiseI,axiom,
! [S: set_b,R3: b > b > $o] :
( ! [X: b,Y3: b] :
( ( member_b @ X @ S )
=> ( ( member_b @ Y3 @ S )
=> ( ( X != Y3 )
=> ( R3 @ X @ Y3 ) ) ) )
=> ( pairwise_b @ R3 @ S ) ) ).
% pairwiseI
thf(fact_1223_pairwiseD,axiom,
! [R3: b > b > $o,S: set_b,X3: b,Y2: b] :
( ( pairwise_b @ R3 @ S )
=> ( ( member_b @ X3 @ S )
=> ( ( member_b @ Y2 @ S )
=> ( ( X3 != Y2 )
=> ( R3 @ X3 @ Y2 ) ) ) ) ) ).
% pairwiseD
thf(fact_1224_pairwise__insert,axiom,
! [R: $o > $o > $o,X3: $o,S4: set_o] :
( ( pairwise_o @ R @ ( insert_o @ X3 @ S4 ) )
= ( ! [Y: $o] :
( ( ( member_o @ Y @ S4 )
& ( Y != X3 ) )
=> ( ( R @ X3 @ Y )
& ( R @ Y @ X3 ) ) )
& ( pairwise_o @ R @ S4 ) ) ) ).
% pairwise_insert
thf(fact_1225_pairwise__insert,axiom,
! [R: b > b > $o,X3: b,S4: set_b] :
( ( pairwise_b @ R @ ( insert_b @ X3 @ S4 ) )
= ( ! [Y: b] :
( ( ( member_b @ Y @ S4 )
& ( Y != X3 ) )
=> ( ( R @ X3 @ Y )
& ( R @ Y @ X3 ) ) )
& ( pairwise_b @ R @ S4 ) ) ) ).
% pairwise_insert
thf(fact_1226_pairwise__empty,axiom,
! [P4: $o > $o > $o] : ( pairwise_o @ P4 @ bot_bot_set_o ) ).
% pairwise_empty
thf(fact_1227_Inf__set__fold,axiom,
! [Xs: list_set_b] :
( ( comple6135023382983342438_set_b @ ( set_set_b2 @ Xs ) )
= ( fold_set_b_set_b @ inf_inf_set_b @ Xs @ top_top_set_b ) ) ).
% Inf_set_fold
thf(fact_1228_Inf__set__fold,axiom,
! [Xs: list_set_o] :
( ( comple3063163877087187839_set_o @ ( set_set_o2 @ Xs ) )
= ( fold_set_o_set_o @ inf_inf_set_o @ Xs @ top_top_set_o ) ) ).
% Inf_set_fold
thf(fact_1229_pairwise__singleton,axiom,
! [P4: $o > $o > $o,A: $o] : ( pairwise_o @ P4 @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% pairwise_singleton
thf(fact_1230_Inf__fin_Oset__eq__fold,axiom,
! [X3: set_b,Xs: list_set_b] :
( ( lattic8209813469468118012_set_b @ ( set_set_b2 @ ( cons_set_b @ X3 @ Xs ) ) )
= ( fold_set_b_set_b @ inf_inf_set_b @ Xs @ X3 ) ) ).
% Inf_fin.set_eq_fold
thf(fact_1231_inter__coset__fold,axiom,
! [A: set_b,Xs: list_b] :
( ( inf_inf_set_b @ A @ ( coset_b @ Xs ) )
= ( fold_b_set_b @ remove_b @ Xs @ A ) ) ).
% inter_coset_fold
thf(fact_1232_Inf__fin_Oinsert,axiom,
! [A: set_set_b,X3: set_b] :
( ( finite_finite_set_b @ A )
=> ( ( A != bot_bot_set_set_b )
=> ( ( lattic8209813469468118012_set_b @ ( insert_set_b @ X3 @ A ) )
= ( inf_inf_set_b @ X3 @ ( lattic8209813469468118012_set_b @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1233_Inf__fin_Oinsert,axiom,
! [A: set_o,X3: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X3 @ A ) )
= ( inf_inf_o @ X3 @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1234_Inf__fin_Oinsert__remove,axiom,
! [A: set_set_b,X3: set_b] :
( ( finite_finite_set_b @ A )
=> ( ( ( ( minus_5807331545291222566_set_b @ A @ ( insert_set_b @ X3 @ bot_bot_set_set_b ) )
= bot_bot_set_set_b )
=> ( ( lattic8209813469468118012_set_b @ ( insert_set_b @ X3 @ A ) )
= X3 ) )
& ( ( ( minus_5807331545291222566_set_b @ A @ ( insert_set_b @ X3 @ bot_bot_set_set_b ) )
!= bot_bot_set_set_b )
=> ( ( lattic8209813469468118012_set_b @ ( insert_set_b @ X3 @ A ) )
= ( inf_inf_set_b @ X3 @ ( lattic8209813469468118012_set_b @ ( minus_5807331545291222566_set_b @ A @ ( insert_set_b @ X3 @ bot_bot_set_set_b ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1235_Inf__fin_Oinsert__remove,axiom,
! [A: set_o,X3: $o] :
( ( finite_finite_o @ A )
=> ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X3 @ A ) )
= ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X3 @ bot_bot_set_o ) )
= bot_bot_set_o )
=> X3 )
& ( ( ( minus_minus_set_o @ A @ ( insert_o @ X3 @ bot_bot_set_o ) )
!= bot_bot_set_o )
=> ( inf_inf_o @ X3 @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1236_Inf__fin_Oin__idem,axiom,
! [A: set_set_b,X3: set_b] :
( ( finite_finite_set_b @ A )
=> ( ( member_set_b @ X3 @ A )
=> ( ( inf_inf_set_b @ X3 @ ( lattic8209813469468118012_set_b @ A ) )
= ( lattic8209813469468118012_set_b @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1237_Inf__fin_Ohom__commute,axiom,
! [H2: set_b > set_b,N: set_set_b] :
( ! [X: set_b,Y3: set_b] :
( ( H2 @ ( inf_inf_set_b @ X @ Y3 ) )
= ( inf_inf_set_b @ ( H2 @ X ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite_set_b @ N )
=> ( ( N != bot_bot_set_set_b )
=> ( ( H2 @ ( lattic8209813469468118012_set_b @ N ) )
= ( lattic8209813469468118012_set_b @ ( image_set_b_set_b @ H2 @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1238_Inf__fin_Ohom__commute,axiom,
! [H2: $o > $o,N: set_o] :
( ! [X: $o,Y3: $o] :
( ( H2 @ ( inf_inf_o @ X @ Y3 ) )
= ( inf_inf_o @ ( H2 @ X ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite_o @ N )
=> ( ( N != bot_bot_set_o )
=> ( ( H2 @ ( lattic4107685809792843317_fin_o @ N ) )
= ( lattic4107685809792843317_fin_o @ ( image_o_o @ H2 @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1239_Inf__fin_Osubset,axiom,
! [A: set_set_b,B: set_set_b] :
( ( finite_finite_set_b @ A )
=> ( ( B != bot_bot_set_set_b )
=> ( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( ( inf_inf_set_b @ ( lattic8209813469468118012_set_b @ B ) @ ( lattic8209813469468118012_set_b @ A ) )
= ( lattic8209813469468118012_set_b @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1240_Inf__fin_Osubset,axiom,
! [A: set_o,B: set_o] :
( ( finite_finite_o @ A )
=> ( ( B != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ B @ A )
=> ( ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ B ) @ ( lattic4107685809792843317_fin_o @ A ) )
= ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1241_Inf__fin_Oclosed,axiom,
! [A: set_set_b] :
( ( finite_finite_set_b @ A )
=> ( ( A != bot_bot_set_set_b )
=> ( ! [X: set_b,Y3: set_b] : ( member_set_b @ ( inf_inf_set_b @ X @ Y3 ) @ ( insert_set_b @ X @ ( insert_set_b @ Y3 @ bot_bot_set_set_b ) ) )
=> ( member_set_b @ ( lattic8209813469468118012_set_b @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_1242_Inf__fin_Oclosed,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X: $o,Y3: $o] : ( member_o @ ( inf_inf_o @ X @ Y3 ) @ ( insert_o @ X @ ( insert_o @ Y3 @ bot_bot_set_o ) ) )
=> ( member_o @ ( lattic4107685809792843317_fin_o @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_1243_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_set_b,X3: set_b] :
( ( finite_finite_set_b @ A )
=> ( ~ ( member_set_b @ X3 @ A )
=> ( ( A != bot_bot_set_set_b )
=> ( ( lattic8209813469468118012_set_b @ ( insert_set_b @ X3 @ A ) )
= ( inf_inf_set_b @ X3 @ ( lattic8209813469468118012_set_b @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1244_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_o,X3: $o] :
( ( finite_finite_o @ A )
=> ( ~ ( member_o @ X3 @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X3 @ A ) )
= ( inf_inf_o @ X3 @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1245_Inf__fin_Ounion,axiom,
! [A: set_set_b,B: set_set_b] :
( ( finite_finite_set_b @ A )
=> ( ( A != bot_bot_set_set_b )
=> ( ( finite_finite_set_b @ B )
=> ( ( B != bot_bot_set_set_b )
=> ( ( lattic8209813469468118012_set_b @ ( sup_sup_set_set_b @ A @ B ) )
= ( inf_inf_set_b @ ( lattic8209813469468118012_set_b @ A ) @ ( lattic8209813469468118012_set_b @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1246_Inf__fin_Ounion,axiom,
! [A: set_o,B: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( finite_finite_o @ B )
=> ( ( B != bot_bot_set_o )
=> ( ( lattic4107685809792843317_fin_o @ ( sup_sup_set_o @ A @ B ) )
= ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ A ) @ ( lattic4107685809792843317_fin_o @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1247_Inf__fin_Oeq__fold,axiom,
! [A: set_o,X3: $o] :
( ( finite_finite_o @ A )
=> ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X3 @ A ) )
= ( finite_fold_o_o @ inf_inf_o @ X3 @ A ) ) ) ).
% Inf_fin.eq_fold
thf(fact_1248_Inf__fin_Oeq__fold,axiom,
! [A: set_set_b,X3: set_b] :
( ( finite_finite_set_b @ A )
=> ( ( lattic8209813469468118012_set_b @ ( insert_set_b @ X3 @ A ) )
= ( finite7473193970610274952_set_b @ inf_inf_set_b @ X3 @ A ) ) ) ).
% Inf_fin.eq_fold
thf(fact_1249_Inf__fin_Oremove,axiom,
! [A: set_set_b,X3: set_b] :
( ( finite_finite_set_b @ A )
=> ( ( member_set_b @ X3 @ A )
=> ( ( ( ( minus_5807331545291222566_set_b @ A @ ( insert_set_b @ X3 @ bot_bot_set_set_b ) )
= bot_bot_set_set_b )
=> ( ( lattic8209813469468118012_set_b @ A )
= X3 ) )
& ( ( ( minus_5807331545291222566_set_b @ A @ ( insert_set_b @ X3 @ bot_bot_set_set_b ) )
!= bot_bot_set_set_b )
=> ( ( lattic8209813469468118012_set_b @ A )
= ( inf_inf_set_b @ X3 @ ( lattic8209813469468118012_set_b @ ( minus_5807331545291222566_set_b @ A @ ( insert_set_b @ X3 @ bot_bot_set_set_b ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1250_Inf__fin_Oremove,axiom,
! [A: set_o,X3: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X3 @ A )
=> ( ( lattic4107685809792843317_fin_o @ A )
= ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X3 @ bot_bot_set_o ) )
= bot_bot_set_o )
=> X3 )
& ( ( ( minus_minus_set_o @ A @ ( insert_o @ X3 @ bot_bot_set_o ) )
!= bot_bot_set_o )
=> ( inf_inf_o @ X3 @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1251_psubset__insert__iff,axiom,
! [A: set_b,X3: b,B: set_b] :
( ( ord_less_set_b @ A @ ( insert_b @ X3 @ B ) )
= ( ( ( member_b @ X3 @ B )
=> ( ord_less_set_b @ A @ B ) )
& ( ~ ( member_b @ X3 @ B )
=> ( ( ( member_b @ X3 @ A )
=> ( ord_less_set_b @ ( minus_minus_set_b @ A @ ( insert_b @ X3 @ bot_bot_set_b ) ) @ B ) )
& ( ~ ( member_b @ X3 @ A )
=> ( ord_less_eq_set_b @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1252_psubset__insert__iff,axiom,
! [A: set_o,X3: $o,B: set_o] :
( ( ord_less_set_o @ A @ ( insert_o @ X3 @ B ) )
= ( ( ( member_o @ X3 @ B )
=> ( ord_less_set_o @ A @ B ) )
& ( ~ ( member_o @ X3 @ B )
=> ( ( ( member_o @ X3 @ A )
=> ( ord_less_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B ) )
& ( ~ ( member_o @ X3 @ A )
=> ( ord_less_eq_set_o @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1253_not__psubset__empty,axiom,
! [A: set_o] :
~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).
% not_psubset_empty
thf(fact_1254_psubsetD,axiom,
! [A: set_b,B: set_b,C: b] :
( ( ord_less_set_b @ A @ B )
=> ( ( member_b @ C @ A )
=> ( member_b @ C @ B ) ) ) ).
% psubsetD
thf(fact_1255_less__infI1,axiom,
! [A2: set_b,X3: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ X3 )
=> ( ord_less_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X3 ) ) ).
% less_infI1
thf(fact_1256_less__infI2,axiom,
! [B2: set_b,X3: set_b,A2: set_b] :
( ( ord_less_set_b @ B2 @ X3 )
=> ( ord_less_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X3 ) ) ).
% less_infI2
thf(fact_1257_inf_Oabsorb3,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_1258_inf_Oabsorb4,axiom,
! [B2: set_b,A2: set_b] :
( ( ord_less_set_b @ B2 @ A2 )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_1259_inf_Ostrict__boundedE,axiom,
! [A2: set_b,B2: set_b,C: set_b] :
( ( ord_less_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C ) )
=> ~ ( ( ord_less_set_b @ A2 @ B2 )
=> ~ ( ord_less_set_b @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1260_inf_Ostrict__order__iff,axiom,
( ord_less_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( A3
= ( inf_inf_set_b @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1261_inf_Ostrict__coboundedI1,axiom,
! [A2: set_b,C: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ C )
=> ( ord_less_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1262_inf_Ostrict__coboundedI2,axiom,
! [B2: set_b,C: set_b,A2: set_b] :
( ( ord_less_set_b @ B2 @ C )
=> ( ord_less_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1263_psubset__imp__ex__mem,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_set_b @ A @ B )
=> ? [B7: b] : ( member_b @ B7 @ ( minus_minus_set_b @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1264_card__Diff__subset__Int,axiom,
! [A: set_b,B: set_b] :
( ( finite_finite_b @ ( inf_inf_set_b @ A @ B ) )
=> ( ( finite_card_b @ ( minus_minus_set_b @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_b @ A ) @ ( finite_card_b @ ( inf_inf_set_b @ A @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1265_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila2554085542299052326_set_o @ sup_sup_set_o @ bot_bot_set_o
@ ^ [X2: set_o,Y: set_o] : ( ord_less_eq_set_o @ Y @ X2 )
@ ^ [X2: set_o,Y: set_o] : ( ord_less_set_o @ Y @ X2 ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_1266_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila2496817879753468813_set_b @ inf_inf_set_b @ top_top_set_b @ ord_less_eq_set_b @ ord_less_set_b ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1267_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila2554085542299052326_set_o @ inf_inf_set_o @ top_top_set_o @ ord_less_eq_set_o @ ord_less_set_o ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1268_less__set__def,axiom,
( ord_less_set_b
= ( ^ [A4: set_b,B4: set_b] :
( ord_less_b_o
@ ^ [X2: b] : ( member_b @ X2 @ A4 )
@ ^ [X2: b] : ( member_b @ X2 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_1269_card__partition,axiom,
! [C2: set_set_b,K4: nat] :
( ( finite_finite_set_b @ C2 )
=> ( ( finite_finite_b @ ( comple2307003614231284044_set_b @ C2 ) )
=> ( ! [C5: set_b] :
( ( member_set_b @ C5 @ C2 )
=> ( ( finite_card_b @ C5 )
= K4 ) )
=> ( ! [C1: set_b,C22: set_b] :
( ( member_set_b @ C1 @ C2 )
=> ( ( member_set_b @ C22 @ C2 )
=> ( ( C1 != C22 )
=> ( ( inf_inf_set_b @ C1 @ C22 )
= bot_bot_set_b ) ) ) )
=> ( ( times_times_nat @ K4 @ ( finite_card_set_b @ C2 ) )
= ( finite_card_b @ ( comple2307003614231284044_set_b @ C2 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_1270_card__partition,axiom,
! [C2: set_set_o,K4: nat] :
( ( finite_finite_set_o @ C2 )
=> ( ( finite_finite_o @ ( comple90263536869209701_set_o @ C2 ) )
=> ( ! [C5: set_o] :
( ( member_set_o @ C5 @ C2 )
=> ( ( finite_card_o @ C5 )
= K4 ) )
=> ( ! [C1: set_o,C22: set_o] :
( ( member_set_o @ C1 @ C2 )
=> ( ( member_set_o @ C22 @ C2 )
=> ( ( C1 != C22 )
=> ( ( inf_inf_set_o @ C1 @ C22 )
= bot_bot_set_o ) ) ) )
=> ( ( times_times_nat @ K4 @ ( finite_card_set_o @ C2 ) )
= ( finite_card_o @ ( comple90263536869209701_set_o @ C2 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_1271_card__Un__disjoint,axiom,
! [A: set_b,B: set_b] :
( ( finite_finite_b @ A )
=> ( ( finite_finite_b @ B )
=> ( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
=> ( ( finite_card_b @ ( sup_sup_set_b @ A @ B ) )
= ( plus_plus_nat @ ( finite_card_b @ A ) @ ( finite_card_b @ B ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1272_card__Un__disjoint,axiom,
! [A: set_o,B: set_o] :
( ( finite_finite_o @ A )
=> ( ( finite_finite_o @ B )
=> ( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
=> ( ( finite_card_o @ ( sup_sup_set_o @ A @ B ) )
= ( plus_plus_nat @ ( finite_card_o @ A ) @ ( finite_card_o @ B ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1273_card__Un__Int,axiom,
! [A: set_b,B: set_b] :
( ( finite_finite_b @ A )
=> ( ( finite_finite_b @ B )
=> ( ( plus_plus_nat @ ( finite_card_b @ A ) @ ( finite_card_b @ B ) )
= ( plus_plus_nat @ ( finite_card_b @ ( sup_sup_set_b @ A @ B ) ) @ ( finite_card_b @ ( inf_inf_set_b @ A @ B ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_1274_dvd__partition,axiom,
! [C2: set_set_b,K4: nat] :
( ( finite_finite_b @ ( comple2307003614231284044_set_b @ C2 ) )
=> ( ! [X: set_b] :
( ( member_set_b @ X @ C2 )
=> ( dvd_dvd_nat @ K4 @ ( finite_card_b @ X ) ) )
=> ( ! [X: set_b] :
( ( member_set_b @ X @ C2 )
=> ! [Xa: set_b] :
( ( member_set_b @ Xa @ C2 )
=> ( ( X != Xa )
=> ( ( inf_inf_set_b @ X @ Xa )
= bot_bot_set_b ) ) ) )
=> ( dvd_dvd_nat @ K4 @ ( finite_card_b @ ( comple2307003614231284044_set_b @ C2 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_1275_dvd__partition,axiom,
! [C2: set_set_o,K4: nat] :
( ( finite_finite_o @ ( comple90263536869209701_set_o @ C2 ) )
=> ( ! [X: set_o] :
( ( member_set_o @ X @ C2 )
=> ( dvd_dvd_nat @ K4 @ ( finite_card_o @ X ) ) )
=> ( ! [X: set_o] :
( ( member_set_o @ X @ C2 )
=> ! [Xa: set_o] :
( ( member_set_o @ Xa @ C2 )
=> ( ( X != Xa )
=> ( ( inf_inf_set_o @ X @ Xa )
= bot_bot_set_o ) ) ) )
=> ( dvd_dvd_nat @ K4 @ ( finite_card_o @ ( comple90263536869209701_set_o @ C2 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_1276_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic8986249274379242937_set_b @ inf_inf_set_b @ ord_less_eq_set_b @ ord_less_set_b ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1277_inf_Osemilattice__order__axioms,axiom,
semila4706084625072599247_set_b @ inf_inf_set_b @ ord_less_eq_set_b @ ord_less_set_b ).
% inf.semilattice_order_axioms
thf(fact_1278_sum__Un__nat,axiom,
! [A: set_b,B: set_b,F2: b > nat] :
( ( finite_finite_b @ A )
=> ( ( finite_finite_b @ B )
=> ( ( groups7570001007293516437_b_nat @ F2 @ ( sup_sup_set_b @ A @ B ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( groups7570001007293516437_b_nat @ F2 @ A ) @ ( groups7570001007293516437_b_nat @ F2 @ B ) ) @ ( groups7570001007293516437_b_nat @ F2 @ ( inf_inf_set_b @ A @ B ) ) ) ) ) ) ).
% sum_Un_nat
% Helper facts (3)
thf(help_If_3_1_If_001tf__b_T,axiom,
! [P4: $o] :
( ( P4 = $true )
| ( P4 = $false ) ) ).
thf(help_If_2_1_If_001tf__b_T,axiom,
! [X3: b,Y2: b] :
( ( if_b @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__b_T,axiom,
! [X3: b,Y2: b] :
( ( if_b @ $true @ X3 @ Y2 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X: b] :
( ( member_b @ X @ ( inf_inf_set_b @ ( episte1782384855165018035t_unit @ m ) @ ( episte1071096959727133607t_unit @ m @ i @ w ) ) )
=> ? [Xa3: b] :
( ( member_b @ Xa3 @ ( inf_inf_set_b @ ( episte1782384855165018035t_unit @ m ) @ ( episte1071096959727133607t_unit @ m @ i @ X ) ) )
& ( episte295617885132580261cs_a_b @ m @ Xa3 @ p ) ) ) ).
%------------------------------------------------------------------------------