TPTP Problem File: SLH0404^1.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_00343_011802__6687074_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1424 ( 586 unt; 146 typ; 0 def)
% Number of atoms : 3654 (1157 equ; 0 cnn)
% Maximal formula atoms : 21 ( 2 avg)
% Number of connectives : 12156 ( 466 ~; 60 |; 255 &;9748 @)
% ( 0 <=>;1627 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 668 ( 668 >; 0 *; 0 +; 0 <<)
% Number of symbols : 133 ( 131 usr; 13 con; 0-4 aty)
% Number of variables : 3802 ( 413 ^;3304 !; 85 ?;3802 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:12:26.318
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_Mt__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
set_Pr3972831103112126087c_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
set_Pr4658907567593863815c_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
set_se7339729205154126530c_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
list_s8081015415394010888c_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
set_se3485332733965609186c_fm_i: $tType ).
thf(ty_n_t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
list_Epistemic_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
set_Epistemic_fm_i: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Epistemic____Logic__Ofm_Itf__i_J,type,
epistemic_fm_i: $tType ).
thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__i,type,
i: $tType ).
% Explicit typings (131)
thf(sy_c_Epistemic__Logic_OAK_001tf__i,type,
epistemic_AK_i: ( epistemic_fm_i > $o ) > epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAx4_001tf__i,type,
epistemic_Ax4_i: epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAx5_001tf__i,type,
epistemic_Ax5_i: epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAxB_001tf__i,type,
epistemic_AxB_i: epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_OAxT_001tf__i,type,
epistemic_AxT_i: epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Oconsistent_001tf__i,type,
episte2285483198712856234tent_i: ( epistemic_fm_i > $o ) > set_Epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Oeval_001tf__i,type,
epistemic_eval_i: ( list_char > $o ) > ( epistemic_fm_i > $o ) > epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Ofm_OCon_001tf__i,type,
epistemic_Con_i: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_ODis_001tf__i,type,
epistemic_Dis_i: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_OFF_001tf__i,type,
epistemic_FF_i: epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_OImp_001tf__i,type,
epistemic_Imp_i: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_OK_001tf__i,type,
epistemic_K_i: i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_OPro_001tf__i,type,
epistemic_Pro_i: list_char > epistemic_fm_i ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001tf__i_001tf__i,type,
epistemic_rel_fm_i_i: ( i > i > $o ) > epistemic_fm_i > epistemic_fm_i > $o ).
thf(sy_c_Epistemic__Logic_Oimply_001tf__i,type,
epistemic_imply_i: list_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Finite__Set_Ofinite_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
finite3304564979551393739c_fm_i: set_Epistemic_fm_i > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
finite7933139204641697195c_fm_i: set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
finite9087983574947184523c_fm_i: set_se7339729205154126530c_fm_i > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
minus_1543973939109951465c_fm_i: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
minus_1073301138667672009c_fm_i: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
minus_8476896128438495145c_fm_i: set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_OUniq_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
uniq_Epistemic_fm_i: ( epistemic_fm_i > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__Nat__Onat,type,
uniq_nat: ( nat > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
uniq_s5891897140743797767c_fm_i: ( set_Epistemic_fm_i > $o ) > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_M_Eo_J,type,
inf_in8976956063904982253fm_i_o: ( epistemic_fm_i > $o ) > ( epistemic_fm_i > $o ) > epistemic_fm_i > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
inf_in853329969422372621fm_i_o: ( set_Epistemic_fm_i > $o ) > ( set_Epistemic_fm_i > $o ) > set_Epistemic_fm_i > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
inf_in3450601097109690352c_fm_i: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
inf_in161960956874937808c_fm_i: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_M_Eo_J,type,
sup_su2844414564337698183fm_i_o: ( epistemic_fm_i > $o ) > ( epistemic_fm_i > $o ) > epistemic_fm_i > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
sup_su6723579903606884263fm_i_o: ( set_Epistemic_fm_i > $o ) > ( set_Epistemic_fm_i > $o ) > set_Epistemic_fm_i > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
sup_su1936195050962291414c_fm_i: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
sup_su2582925890723967158c_fm_i: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
sup_su2317974602288960150c_fm_i: set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Nat__Onat,type,
lattic7487310264250698137_i_nat: ( epistemic_fm_i > nat ) > set_Epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Nat__Onat,type,
lattic8104409813523524473_i_nat: ( set_Epistemic_fm_i > nat ) > set_se3485332733965609186c_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
lattic8495942116038605215c_fm_i: set_se3485332733965609186c_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
lattic7860428167660714879c_fm_i: set_se7339729205154126530c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
lattic3744900211830597177c_fm_i: set_se3485332733965609186c_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
lattic5920028816461561881c_fm_i: set_se7339729205154126530c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_List_Ocoset_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
coset_Epistemic_fm_i: list_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
coset_1342939534705115381c_fm_i: list_s8081015415394010888c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_List_Ofilter_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
filter7636273843821131039c_fm_i: ( epistemic_fm_i > $o ) > list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Ofilter_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
filter3188398074982218495c_fm_i: ( set_Epistemic_fm_i > $o ) > list_s8081015415394010888c_fm_i > list_s8081015415394010888c_fm_i ).
thf(sy_c_List_Olist_OCons_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
cons_Epistemic_fm_i: epistemic_fm_i > list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
cons_s4962720389763977656c_fm_i: set_Epistemic_fm_i > list_s8081015415394010888c_fm_i > list_s8081015415394010888c_fm_i ).
thf(sy_c_List_Olist_Omap_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
map_Ep2755178516647988292c_fm_i: ( epistemic_fm_i > epistemic_fm_i ) > list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
map_se1449444348732536900c_fm_i: ( set_Epistemic_fm_i > set_Epistemic_fm_i ) > list_s8081015415394010888c_fm_i > list_s8081015415394010888c_fm_i ).
thf(sy_c_List_Olist_Orec__list_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
rec_li5834409330748533462c_fm_i: set_Epistemic_fm_i > ( epistemic_fm_i > list_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ) > list_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_List_Olist_Orec__list_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
rec_list_set_nat_nat: set_nat > ( nat > list_nat > set_nat > set_nat ) > list_nat > set_nat ).
thf(sy_c_List_Olist_Orec__list_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
rec_li386023649751468758c_fm_i: set_se3485332733965609186c_fm_i > ( set_Epistemic_fm_i > list_s8081015415394010888c_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ) > list_s8081015415394010888c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_List_Olist_Oset_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
set_Epistemic_fm_i2: list_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
set_se200842218512397079c_fm_i: list_s8081015415394010888c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_List_OremoveAll_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
remove817376155714978094c_fm_i: epistemic_fm_i > list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
remove6226062834164189966c_fm_i: set_Epistemic_fm_i > list_s8081015415394010888c_fm_i > list_s8081015415394010888c_fm_i ).
thf(sy_c_List_Ounion_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
union_Epistemic_fm_i: list_Epistemic_fm_i > list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
union_4878195246919425020c_fm_i: list_s8081015415394010888c_fm_i > list_s8081015415394010888c_fm_i > list_s8081015415394010888c_fm_i ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
maxima1924290493700099598c_fm_i: ( set_Epistemic_fm_i > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_001t__Nat__Onat,type,
maxima5350474959086842340CS_nat: ( set_nat > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
maxima5540402109999912942c_fm_i: ( set_se3485332733965609186c_fm_i > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_Omaximal_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
maxima3264069618988350929c_fm_i: ( set_Epistemic_fm_i > $o ) > set_Epistemic_fm_i > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_Omaximal_001t__Nat__Onat,type,
maxima3892657229671227617al_nat: ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_Omaximal_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
maxima7741914846562630577c_fm_i: ( set_se3485332733965609186c_fm_i > $o ) > set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS__Lim__Ord_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
maxima8116622286610175645c_fm_i: set_Pr4658907567593863815c_fm_i > ( set_Epistemic_fm_i > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS__Lim__Ord_001t__Nat__Onat,type,
maxima4501515429357695765rd_nat: set_Pr1261947904930325089at_nat > ( set_nat > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS__Lim__Ord_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
maxima9073963094320882813c_fm_i: set_Pr3972831103112126087c_fm_i > ( set_se3485332733965609186c_fm_i > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS__Lim__Ord_OextendS_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
maxima5458213620894205884c_fm_i: ( set_Epistemic_fm_i > $o ) > set_Epistemic_fm_i > epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_M_Eo_J,type,
bot_bo2580527446789968623fm_i_o: epistemic_fm_i > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
bot_bo6089404950617257231fm_i_o: set_Epistemic_fm_i > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
bot_bo4194595901900360558c_fm_i: set_Epistemic_fm_i ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
bot_bo145720340923748686c_fm_i: set_se3485332733965609186c_fm_i ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
bot_bo4781621276559889198c_fm_i: set_se7339729205154126530c_fm_i ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_M_Eo_J,type,
ord_le190830114487426235fm_i_o: ( epistemic_fm_i > $o ) > ( epistemic_fm_i > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
ord_le5706303257367236315fm_i_o: ( set_Epistemic_fm_i > $o ) > ( set_Epistemic_fm_i > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
ord_le3843937902494030498c_fm_i: set_Epistemic_fm_i > set_Epistemic_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
ord_le5389487502678872194c_fm_i: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
ord_le2772500130360476258c_fm_i: set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i > $o ).
thf(sy_c_Set_OCollect_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
collec4904205187116291597c_fm_i: ( epistemic_fm_i > $o ) > set_Epistemic_fm_i ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
collec3087743281813070829c_fm_i: ( set_Epistemic_fm_i > $o ) > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
collec2761885132556217805c_fm_i: ( set_se3485332733965609186c_fm_i > $o ) > set_se7339729205154126530c_fm_i ).
thf(sy_c_Set_Ofilter_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
filter6053540608743173075c_fm_i: ( epistemic_fm_i > $o ) > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Ofilter_001t__Nat__Onat,type,
filter_nat2: ( nat > $o ) > set_nat > set_nat ).
thf(sy_c_Set_Ofilter_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
filter8450032351001387443c_fm_i: ( set_Epistemic_fm_i > $o ) > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oinsert_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
insert7817948997695205106c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
insert7698009978809854162c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
insert7111703059596746418c_fm_i: set_se3485332733965609186c_fm_i > set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i ).
thf(sy_c_Set_Ois__singleton_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
is_sin4923117429174228502c_fm_i: set_Epistemic_fm_i > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
is_sin2358732406619996150c_fm_i: set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Stalnaker__Logic_OAx__2_001tf__i,type,
stalnaker_Ax_2_i: epistemic_fm_i > $o ).
thf(sy_c_Stalnaker__Logic_Oconjunct_001tf__i,type,
stalnaker_conjunct_i: list_Epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
member6642669606046002379c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member1104366573291651755c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
member1461078328125707403c_fm_i: set_se3485332733965609186c_fm_i > set_se7339729205154126530c_fm_i > $o ).
thf(sy_v_A,type,
a: epistemic_fm_i > $o ).
thf(sy_v_S_H____,type,
s: list_Epistemic_fm_i ).
thf(sy_v_U,type,
u: set_Epistemic_fm_i ).
thf(sy_v_V,type,
v: set_Epistemic_fm_i ).
thf(sy_v_W,type,
w: set_Epistemic_fm_i ).
thf(sy_v_i,type,
i2: i ).
% Relevant facts (1277)
thf(fact_0_assms_I9_J,axiom,
( member1104366573291651755c_fm_i @ u
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ v ) ) ) ) ) ).
% assms(9)
thf(fact_1__C_K_C_I3_J,axiom,
finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ s ) ).
% "*"(3)
thf(fact_2_set__filter,axiom,
! [P2: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ ( filter3188398074982218495c_fm_i @ P2 @ Xs ) )
= ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
& ( P2 @ X ) ) ) ) ).
% set_filter
thf(fact_3_set__filter,axiom,
! [P2: nat > $o,Xs: list_nat] :
( ( set_nat2 @ ( filter_nat @ P2 @ Xs ) )
= ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( P2 @ X ) ) ) ) ).
% set_filter
thf(fact_4_set__filter,axiom,
! [P2: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( filter7636273843821131039c_fm_i @ P2 @ Xs ) )
= ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( P2 @ X ) ) ) ) ).
% set_filter
thf(fact_5_filter__True,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X2 ) )
=> ( ( filter_nat @ P2 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_6_filter__True,axiom,
! [Xs: list_s8081015415394010888c_fm_i,P2: set_Epistemic_fm_i > $o] :
( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( P2 @ X2 ) )
=> ( ( filter3188398074982218495c_fm_i @ P2 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_7_filter__True,axiom,
! [Xs: list_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( P2 @ X2 ) )
=> ( ( filter7636273843821131039c_fm_i @ P2 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_8_assms_I8_J,axiom,
( member1104366573291651755c_fm_i @ w
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ v ) ) ) ) ) ).
% assms(8)
thf(fact_9__C_K_C_I2_J,axiom,
( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ s )
@ ( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ w ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ u ) ) ) ) ).
% "*"(2)
thf(fact_10_filter__filter,axiom,
! [P2: nat > $o,Q: nat > $o,Xs: list_nat] :
( ( filter_nat @ P2 @ ( filter_nat @ Q @ Xs ) )
= ( filter_nat
@ ^ [X: nat] :
( ( Q @ X )
& ( P2 @ X ) )
@ Xs ) ) ).
% filter_filter
thf(fact_11_filter__filter,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] :
( ( filter3188398074982218495c_fm_i @ P2 @ ( filter3188398074982218495c_fm_i @ Q @ Xs ) )
= ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( Q @ X )
& ( P2 @ X ) )
@ Xs ) ) ).
% filter_filter
thf(fact_12_filter__filter,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( filter7636273843821131039c_fm_i @ P2 @ ( filter7636273843821131039c_fm_i @ Q @ Xs ) )
= ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( Q @ X )
& ( P2 @ X ) )
@ Xs ) ) ).
% filter_filter
thf(fact_13_filter__is__subset,axiom,
! [P2: nat > $o,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( filter_nat @ P2 @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% filter_is_subset
thf(fact_14_filter__is__subset,axiom,
! [P2: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ ( filter3188398074982218495c_fm_i @ P2 @ Xs ) ) @ ( set_se200842218512397079c_fm_i @ Xs ) ) ).
% filter_is_subset
thf(fact_15_filter__is__subset,axiom,
! [P2: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ ( filter7636273843821131039c_fm_i @ P2 @ Xs ) ) @ ( set_Epistemic_fm_i2 @ Xs ) ) ).
% filter_is_subset
thf(fact_16_fm_Oinject_I5_J,axiom,
! [X61: i,X62: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
( ( ( epistemic_K_i @ X61 @ X62 )
= ( epistemic_K_i @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_17_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_18_subsetI,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( member1104366573291651755c_fm_i @ X2 @ B ) )
=> ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_19_subsetI,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( member6642669606046002379c_fm_i @ X2 @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_20_subset__antisym,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_21_subset__antisym,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_22_order__refl,axiom,
! [X3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X3 @ X3 ) ).
% order_refl
thf(fact_23_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_24_order__refl,axiom,
! [X3: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ X3 @ X3 ) ).
% order_refl
thf(fact_25_dual__order_Orefl,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_26_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_27_dual__order_Orefl,axiom,
! [A2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_28__092_060open_062set_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060union_062_Aset_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_A_061_Aset_AS_H_092_060close_062,axiom,
( ( sup_su1936195050962291414c_fm_i
@ ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
= ( set_Epistemic_fm_i2 @ s ) ) ).
% \<open>set (filter (\<lambda>p. p \<in> known U i) S') \<union> set (filter (\<lambda>p. p \<in> known W i) S') = set S'\<close>
thf(fact_29_UnCI,axiom,
! [C: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ~ ( member1104366573291651755c_fm_i @ C @ B )
=> ( member1104366573291651755c_fm_i @ C @ A ) )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ) ).
% UnCI
thf(fact_30_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_31_UnCI,axiom,
! [C: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ A ) )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% UnCI
thf(fact_32_Un__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
| ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% Un_iff
thf(fact_33_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_34_Un__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
| ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% Un_iff
thf(fact_35_List_Ofinite__set,axiom,
! [Xs: list_s8081015415394010888c_fm_i] : ( finite7933139204641697195c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) ) ).
% List.finite_set
thf(fact_36_List_Ofinite__set,axiom,
! [Xs: list_Epistemic_fm_i] : ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) ) ).
% List.finite_set
thf(fact_37_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_38_Un__subset__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C2 )
= ( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
& ( ord_le3843937902494030498c_fm_i @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_39_Un__subset__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) @ C2 )
= ( ( ord_le5389487502678872194c_fm_i @ A @ C2 )
& ( ord_le5389487502678872194c_fm_i @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_40_UnE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) )
=> ( ~ ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% UnE
thf(fact_41_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_42_UnE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
=> ( ~ ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% UnE
thf(fact_43_UnI1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ) ).
% UnI1
thf(fact_44_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_45_UnI1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% UnI1
thf(fact_46_UnI2,axiom,
! [C: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ B )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ) ).
% UnI2
thf(fact_47_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_48_UnI2,axiom,
! [C: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% UnI2
thf(fact_49_Un__def,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A3 )
| ( member1104366573291651755c_fm_i @ X @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_50_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
| ( member_nat @ X @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_51_Un__def,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A3 )
| ( member6642669606046002379c_fm_i @ X @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_52_bex__Un,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( ? [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
& ( P2 @ X ) ) )
= ( ? [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P2 @ X ) )
| ? [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B )
& ( P2 @ X ) ) ) ) ).
% bex_Un
thf(fact_53_ball__Un,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
=> ( P2 @ X ) ) )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ( P2 @ X ) )
& ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B )
=> ( P2 @ X ) ) ) ) ).
% ball_Un
thf(fact_54_Un__assoc,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C2 )
= ( sup_su1936195050962291414c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_55_Un__absorb,axiom,
! [A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ A )
= A ) ).
% Un_absorb
thf(fact_56_Un__commute,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_57_Un__left__absorb,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
= ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_left_absorb
thf(fact_58_Collect__disj__eq,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P2 @ X )
| ( Q @ X ) ) )
= ( sup_su2582925890723967158c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_59_Collect__disj__eq,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_60_Collect__disj__eq,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P2 @ X )
| ( Q @ X ) ) )
= ( sup_su1936195050962291414c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_61_Un__left__commute,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) )
= ( sup_su1936195050962291414c_fm_i @ B @ ( sup_su1936195050962291414c_fm_i @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_62_subset__Un__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_63_subset__Un__eq,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_64_subset__UnE,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
=> ~ ! [A4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A4 @ A )
=> ! [B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B3 @ B )
=> ( C2
!= ( sup_su1936195050962291414c_fm_i @ A4 @ B3 ) ) ) ) ) ).
% subset_UnE
thf(fact_65_subset__UnE,axiom,
! [C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C2 @ ( sup_su2582925890723967158c_fm_i @ A @ B ) )
=> ~ ! [A4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A4 @ A )
=> ! [B3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B3 @ B )
=> ( C2
!= ( sup_su2582925890723967158c_fm_i @ A4 @ B3 ) ) ) ) ) ).
% subset_UnE
thf(fact_66_Un__absorb2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_67_Un__absorb2,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( sup_su2582925890723967158c_fm_i @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_68_Un__absorb1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_69_Un__absorb1,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( sup_su2582925890723967158c_fm_i @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_70_Un__upper2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_upper2
thf(fact_71_Un__upper2,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ B @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ).
% Un_upper2
thf(fact_72_Un__upper1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_upper1
thf(fact_73_Un__upper1,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ A @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ).
% Un_upper1
thf(fact_74_Un__least,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_75_Un__least,axiom,
! [A: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ C2 )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ C2 )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_76_Un__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_77_Un__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,D: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ C2 )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ D )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) @ ( sup_su2582925890723967158c_fm_i @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_78_finite__list,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ? [Xs2: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_79_finite__list,axiom,
! [A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ? [Xs2: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_80_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_81_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_82_less__eq__set__def,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ord_le190830114487426235fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A3 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_83_less__eq__set__def,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ord_le5706303257367236315fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A3 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_84_order__antisym__conv,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X3 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_85_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_86_order__antisym__conv,axiom,
! [Y: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ Y @ X3 )
=> ( ( ord_le5389487502678872194c_fm_i @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_87_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_88_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_89_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,F: set_Epistemic_fm_i > nat,C: nat] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_90_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_91_ord__le__eq__subst,axiom,
! [A2: nat,B4: nat,F: nat > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_92_ord__le__eq__subst,axiom,
! [A2: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_93_ord__le__eq__subst,axiom,
! [A2: nat,B4: nat,F: nat > set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_94_ord__le__eq__subst,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,F: set_se3485332733965609186c_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_95_ord__le__eq__subst,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,F: set_se3485332733965609186c_fm_i > nat,C: nat] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_96_ord__le__eq__subst,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,F: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_97_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2
= ( F @ B4 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B4 @ C )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_98_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Epistemic_fm_i > nat,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2
= ( F @ B4 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B4 @ C )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_99_ord__eq__le__subst,axiom,
! [A2: set_se3485332733965609186c_fm_i,F: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2
= ( F @ B4 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B4 @ C )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_100_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_i,F: nat > set_Epistemic_fm_i,B4: nat,C: nat] :
( ( A2
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_101_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B4: nat,C: nat] :
( ( A2
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_102_ord__eq__le__subst,axiom,
! [A2: set_se3485332733965609186c_fm_i,F: nat > set_se3485332733965609186c_fm_i,B4: nat,C: nat] :
( ( A2
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_103_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_i,F: set_se3485332733965609186c_fm_i > set_Epistemic_fm_i,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( A2
= ( F @ B4 ) )
=> ( ( ord_le5389487502678872194c_fm_i @ B4 @ C )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_104_ord__eq__le__subst,axiom,
! [A2: nat,F: set_se3485332733965609186c_fm_i > nat,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( A2
= ( F @ B4 ) )
=> ( ( ord_le5389487502678872194c_fm_i @ B4 @ C )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_105_ord__eq__le__subst,axiom,
! [A2: set_se3485332733965609186c_fm_i,F: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( A2
= ( F @ B4 ) )
=> ( ( ord_le5389487502678872194c_fm_i @ B4 @ C )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_106_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_107_mem__Collect__eq,axiom,
! [A2: epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ A2 @ ( collec4904205187116291597c_fm_i @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_108_mem__Collect__eq,axiom,
! [A2: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ A2 @ ( collec3087743281813070829c_fm_i @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_109_mem__Collect__eq,axiom,
! [A2: nat,P2: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_110_Collect__mem__eq,axiom,
! [A: set_Epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_111_Collect__mem__eq,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_112_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_113_Collect__cong,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ! [X2: epistemic_fm_i] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collec4904205187116291597c_fm_i @ P2 )
= ( collec4904205187116291597c_fm_i @ Q ) ) ) ).
% Collect_cong
thf(fact_114_Collect__cong,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ! [X2: set_Epistemic_fm_i] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collec3087743281813070829c_fm_i @ P2 )
= ( collec3087743281813070829c_fm_i @ Q ) ) ) ).
% Collect_cong
thf(fact_115_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_116_order__eq__refl,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( X3 = Y )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_117_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_118_order__eq__refl,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( X3 = Y )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_119_order__subst2,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F @ B4 ) @ C )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_120_order__subst2,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,F: set_Epistemic_fm_i > nat,C: nat] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_121_order__subst2,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( ( ord_le5389487502678872194c_fm_i @ ( F @ B4 ) @ C )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_122_order__subst2,axiom,
! [A2: nat,B4: nat,F: nat > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_123_order__subst2,axiom,
! [A2: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_124_order__subst2,axiom,
! [A2: nat,B4: nat,F: nat > set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_le5389487502678872194c_fm_i @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_125_order__subst2,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,F: set_se3485332733965609186c_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F @ B4 ) @ C )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_126_order__subst2,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,F: set_se3485332733965609186c_fm_i > nat,C: nat] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_127_order__subst2,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,F: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( ( ord_le5389487502678872194c_fm_i @ ( F @ B4 ) @ C )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_128_order__subst1,axiom,
! [A2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ B4 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B4 @ C )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_129_order__subst1,axiom,
! [A2: set_Epistemic_fm_i,F: nat > set_Epistemic_fm_i,B4: nat,C: nat] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_130_order__subst1,axiom,
! [A2: set_Epistemic_fm_i,F: set_se3485332733965609186c_fm_i > set_Epistemic_fm_i,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ B4 ) )
=> ( ( ord_le5389487502678872194c_fm_i @ B4 @ C )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_131_order__subst1,axiom,
! [A2: nat,F: set_Epistemic_fm_i > nat,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_less_eq_nat @ A2 @ ( F @ B4 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B4 @ C )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_132_order__subst1,axiom,
! [A2: nat,F: nat > nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_133_order__subst1,axiom,
! [A2: nat,F: set_se3485332733965609186c_fm_i > nat,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_less_eq_nat @ A2 @ ( F @ B4 ) )
=> ( ( ord_le5389487502678872194c_fm_i @ B4 @ C )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_134_order__subst1,axiom,
! [A2: set_se3485332733965609186c_fm_i,F: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ ( F @ B4 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B4 @ C )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_135_order__subst1,axiom,
! [A2: set_se3485332733965609186c_fm_i,F: nat > set_se3485332733965609186c_fm_i,B4: nat,C: nat] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_136_order__subst1,axiom,
! [A2: set_se3485332733965609186c_fm_i,F: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ ( F @ B4 ) )
=> ( ( ord_le5389487502678872194c_fm_i @ B4 @ C )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_137_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y3 = Z ) )
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A5 @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_138_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_139_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( Y3 = Z ) )
= ( ^ [A5: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A5 @ B5 )
& ( ord_le5389487502678872194c_fm_i @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_140_antisym,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( ( ord_le3843937902494030498c_fm_i @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% antisym
thf(fact_141_antisym,axiom,
! [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% antisym
thf(fact_142_antisym,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( ( ord_le5389487502678872194c_fm_i @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% antisym
thf(fact_143_dual__order_Otrans,axiom,
! [B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B4 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ B4 )
=> ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_144_dual__order_Otrans,axiom,
! [B4: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B4 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_145_dual__order_Otrans,axiom,
! [B4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B4 @ A2 )
=> ( ( ord_le5389487502678872194c_fm_i @ C @ B4 )
=> ( ord_le5389487502678872194c_fm_i @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_146_dual__order_Oantisym,axiom,
! [B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B4 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( A2 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_147_dual__order_Oantisym,axiom,
! [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B4 )
=> ( A2 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_148_dual__order_Oantisym,axiom,
! [B4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B4 @ A2 )
=> ( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( A2 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_149_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y3 = Z ) )
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B5 @ A5 )
& ( ord_le3843937902494030498c_fm_i @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_150_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_151_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( Y3 = Z ) )
= ( ^ [A5: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B5 @ A5 )
& ( ord_le5389487502678872194c_fm_i @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_152_linorder__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B4: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P2 @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P2 @ B6 @ A6 )
=> ( P2 @ A6 @ B6 ) )
=> ( P2 @ A2 @ B4 ) ) ) ).
% linorder_wlog
thf(fact_153_order__trans,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
=> ( ( ord_le3843937902494030498c_fm_i @ Y @ Z2 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_154_order__trans,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_155_order__trans,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ Y )
=> ( ( ord_le5389487502678872194c_fm_i @ Y @ Z2 )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_156_order_Otrans,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( ( ord_le3843937902494030498c_fm_i @ B4 @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% order.trans
thf(fact_157_order_Otrans,axiom,
! [A2: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_158_order_Otrans,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( ( ord_le5389487502678872194c_fm_i @ B4 @ C )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ C ) ) ) ).
% order.trans
thf(fact_159_order__antisym,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
=> ( ( ord_le3843937902494030498c_fm_i @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_160_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_161_order__antisym,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ Y )
=> ( ( ord_le5389487502678872194c_fm_i @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_162_ord__le__eq__trans,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( ( B4 = C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_163_ord__le__eq__trans,axiom,
! [A2: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_164_ord__le__eq__trans,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( ( B4 = C )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_165_ord__eq__le__trans,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2 = B4 )
=> ( ( ord_le3843937902494030498c_fm_i @ B4 @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_166_ord__eq__le__trans,axiom,
! [A2: nat,B4: nat,C: nat] :
( ( A2 = B4 )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_167_ord__eq__le__trans,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( A2 = B4 )
=> ( ( ord_le5389487502678872194c_fm_i @ B4 @ C )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_168_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y3 = Z ) )
= ( ^ [X: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y4 )
& ( ord_le3843937902494030498c_fm_i @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_169_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_170_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( Y3 = Z ) )
= ( ^ [X: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X @ Y4 )
& ( ord_le5389487502678872194c_fm_i @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_171_le__cases3,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_172_nle__le,axiom,
! [A2: nat,B4: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B4 ) )
= ( ( ord_less_eq_nat @ B4 @ A2 )
& ( B4 != A2 ) ) ) ).
% nle_le
thf(fact_173_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_174_Collect__mono__iff,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) @ ( collec4904205187116291597c_fm_i @ Q ) )
= ( ! [X: epistemic_fm_i] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_175_Collect__mono__iff,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) @ ( collec3087743281813070829c_fm_i @ Q ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_176_set__eq__subset,axiom,
( ( ^ [Y3: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y3 = Z ) )
= ( ^ [A3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B2 )
& ( ord_le3843937902494030498c_fm_i @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_177_set__eq__subset,axiom,
( ( ^ [Y3: set_se3485332733965609186c_fm_i,Z: set_se3485332733965609186c_fm_i] : ( Y3 = Z ) )
= ( ^ [A3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A3 @ B2 )
& ( ord_le5389487502678872194c_fm_i @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_178_subset__trans,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_179_subset__trans,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ C2 )
=> ( ord_le5389487502678872194c_fm_i @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_180_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_181_Collect__mono,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ! [X2: epistemic_fm_i] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ).
% Collect_mono
thf(fact_182_Collect__mono,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ! [X2: set_Epistemic_fm_i] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ).
% Collect_mono
thf(fact_183_subset__refl,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ A ) ).
% subset_refl
thf(fact_184_subset__refl,axiom,
! [A: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ A @ A ) ).
% subset_refl
thf(fact_185_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_186_subset__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
! [T: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ T @ A3 )
=> ( member6642669606046002379c_fm_i @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_187_subset__iff,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
! [T: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ T @ A3 )
=> ( member1104366573291651755c_fm_i @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_188_equalityD2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A = B )
=> ( ord_le3843937902494030498c_fm_i @ B @ A ) ) ).
% equalityD2
thf(fact_189_equalityD2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( A = B )
=> ( ord_le5389487502678872194c_fm_i @ B @ A ) ) ).
% equalityD2
thf(fact_190_equalityD1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A = B )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% equalityD1
thf(fact_191_equalityD1,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( A = B )
=> ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ).
% equalityD1
thf(fact_192_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_193_subset__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A3 )
=> ( member6642669606046002379c_fm_i @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_194_subset__eq,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A3 )
=> ( member1104366573291651755c_fm_i @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_195_equalityE,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A = B )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ~ ( ord_le3843937902494030498c_fm_i @ B @ A ) ) ) ).
% equalityE
thf(fact_196_equalityE,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( A = B )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ~ ( ord_le5389487502678872194c_fm_i @ B @ A ) ) ) ).
% equalityE
thf(fact_197_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_198_subsetD,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_199_subsetD,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_200_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_201_in__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X3: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( member6642669606046002379c_fm_i @ X3 @ B ) ) ) ).
% in_mono
thf(fact_202_in__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( member1104366573291651755c_fm_i @ X3 @ B ) ) ) ).
% in_mono
thf(fact_203_Collect__subset,axiom,
! [A: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_204_Collect__subset,axiom,
! [A: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_205_Collect__subset,axiom,
! [A: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o] :
( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P2 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_206_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_207_subset__code_I1_J,axiom,
! [Xs: list_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ B )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_208_subset__code_I1_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ B )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( member1104366573291651755c_fm_i @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_209_filter__id__conv,axiom,
! [P2: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( ( filter7636273843821131039c_fm_i @ P2 @ Xs )
= Xs )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( P2 @ X ) ) ) ) ).
% filter_id_conv
thf(fact_210_filter__id__conv,axiom,
! [P2: nat > $o,Xs: list_nat] :
( ( ( filter_nat @ P2 @ Xs )
= Xs )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P2 @ X ) ) ) ) ).
% filter_id_conv
thf(fact_211_filter__id__conv,axiom,
! [P2: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] :
( ( ( filter3188398074982218495c_fm_i @ P2 @ Xs )
= Xs )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( P2 @ X ) ) ) ) ).
% filter_id_conv
thf(fact_212_filter__cong,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( Xs = Ys )
=> ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Ys ) )
=> ( ( P2 @ X2 )
= ( Q @ X2 ) ) )
=> ( ( filter7636273843821131039c_fm_i @ P2 @ Xs )
= ( filter7636273843821131039c_fm_i @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_213_filter__cong,axiom,
! [Xs: list_nat,Ys: list_nat,P2: nat > $o,Q: nat > $o] :
( ( Xs = Ys )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( P2 @ X2 )
= ( Q @ X2 ) ) )
=> ( ( filter_nat @ P2 @ Xs )
= ( filter_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_214_filter__cong,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( Xs = Ys )
=> ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Ys ) )
=> ( ( P2 @ X2 )
= ( Q @ X2 ) ) )
=> ( ( filter3188398074982218495c_fm_i @ P2 @ Xs )
= ( filter3188398074982218495c_fm_i @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_215_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_216_finite__Collect__subsets,axiom,
! [A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_217_finite__Collect__subsets,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( finite9087983574947184523c_fm_i
@ ( collec2761885132556217805c_fm_i
@ ^ [B2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_218_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_219_finite__Un,axiom,
! [F2: set_Epistemic_fm_i,G: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ F2 @ G ) )
= ( ( finite3304564979551393739c_fm_i @ F2 )
& ( finite3304564979551393739c_fm_i @ G ) ) ) ).
% finite_Un
thf(fact_220_sup_Obounded__iff,axiom,
! [B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B4 @ C ) @ A2 )
= ( ( ord_le3843937902494030498c_fm_i @ B4 @ A2 )
& ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_221_sup_Obounded__iff,axiom,
! [B4: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B4 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_222_sup_Obounded__iff,axiom,
! [B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ B4 @ C ) @ A2 )
= ( ( ord_le5389487502678872194c_fm_i @ B4 @ A2 )
& ( ord_le5389487502678872194c_fm_i @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_223_le__sup__iff,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ Z2 )
= ( ( ord_le3843937902494030498c_fm_i @ X3 @ Z2 )
& ( ord_le3843937902494030498c_fm_i @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_224_le__sup__iff,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X3 @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_225_le__sup__iff,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) @ Z2 )
= ( ( ord_le5389487502678872194c_fm_i @ X3 @ Z2 )
& ( ord_le5389487502678872194c_fm_i @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_226_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_227_set__union,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ ( union_4878195246919425020c_fm_i @ Xs @ Ys ) )
= ( sup_su2582925890723967158c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) ) ) ).
% set_union
thf(fact_228_set__union,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( union_Epistemic_fm_i @ Xs @ Ys ) )
= ( sup_su1936195050962291414c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) ) ) ).
% set_union
thf(fact_229_finite__Collect__conjI,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) )
| ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ Q ) ) )
=> ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_230_finite__Collect__conjI,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) )
| ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ Q ) ) )
=> ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_231_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_232_finite__Collect__disjI,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) )
& ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_233_finite__Collect__disjI,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) )
& ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_234_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_235_sup_Oidem,axiom,
! [A2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_236_sup__idem,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_237_sup_Oleft__idem,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) )
= ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) ) ).
% sup.left_idem
thf(fact_238_sup__left__idem,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) )
= ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_239_sup_Oright__idem,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) @ B4 )
= ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) ) ).
% sup.right_idem
thf(fact_240_sup__set__def,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ( sup_su6723579903606884263fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A3 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_241_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_242_sup__set__def,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( sup_su2844414564337698183fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A3 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_243_sup__left__commute,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z2 ) )
= ( sup_su1936195050962291414c_fm_i @ Y @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_244_sup_Oleft__commute,axiom,
! [B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ B4 @ ( sup_su1936195050962291414c_fm_i @ A2 @ C ) )
= ( sup_su1936195050962291414c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B4 @ C ) ) ) ).
% sup.left_commute
thf(fact_245_sup__commute,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ Y4 @ X ) ) ) ).
% sup_commute
thf(fact_246_sup_Ocommute,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_247_sup__assoc,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ Z2 )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_248_sup_Oassoc,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) @ C )
= ( sup_su1936195050962291414c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B4 @ C ) ) ) ).
% sup.assoc
thf(fact_249_inf__sup__aci_I5_J,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ Y4 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_250_inf__sup__aci_I6_J,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ Z2 )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_251_inf__sup__aci_I7_J,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z2 ) )
= ( sup_su1936195050962291414c_fm_i @ Y @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_252_inf__sup__aci_I8_J,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) )
= ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_253_pigeonhole__infinite__rel,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i,R: set_Epistemic_fm_i > epistemic_fm_i > $o] :
( ~ ( finite7933139204641697195c_fm_i @ A )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ? [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ B )
& ~ ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [A5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_254_pigeonhole__infinite__rel,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_nat,R: set_Epistemic_fm_i > nat > $o] :
( ~ ( finite7933139204641697195c_fm_i @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [A5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_255_pigeonhole__infinite__rel,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,R: epistemic_fm_i > epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ A )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ? [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ B )
& ~ ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [A5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_256_pigeonhole__infinite__rel,axiom,
! [A: set_Epistemic_fm_i,B: set_nat,R: epistemic_fm_i > nat > $o] :
( ~ ( finite3304564979551393739c_fm_i @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [A5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_257_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_Epistemic_fm_i,R: nat > epistemic_fm_i > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_258_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_259_not__finite__existsD,axiom,
! [P2: set_Epistemic_fm_i > $o] :
( ~ ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) )
=> ? [X_1: set_Epistemic_fm_i] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_260_not__finite__existsD,axiom,
! [P2: epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) )
=> ? [X_1: epistemic_fm_i] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_261_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_262_finite__has__maximal2,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
& ( ord_le3843937902494030498c_fm_i @ A2 @ X2 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_263_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_264_finite__has__maximal2,axiom,
! [A: set_se7339729205154126530c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ A2 @ A )
=> ? [X2: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X2 @ A )
& ( ord_le5389487502678872194c_fm_i @ A2 @ X2 )
& ! [Xa: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ Xa @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_265_finite__has__minimal2,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
& ( ord_le3843937902494030498c_fm_i @ X2 @ A2 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_266_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_267_finite__has__minimal2,axiom,
! [A: set_se7339729205154126530c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ A2 @ A )
=> ? [X2: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X2 @ A )
& ( ord_le5389487502678872194c_fm_i @ X2 @ A2 )
& ! [Xa: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ Xa @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_268_inf__sup__ord_I4_J,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_269_inf__sup__ord_I4_J,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_270_inf__sup__ord_I4_J,axiom,
! [Y: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ Y @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_271_inf__sup__ord_I3_J,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_272_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_273_inf__sup__ord_I3_J,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_274_le__supE,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) @ X3 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A2 @ X3 )
=> ~ ( ord_le3843937902494030498c_fm_i @ B4 @ X3 ) ) ) ).
% le_supE
thf(fact_275_le__supE,axiom,
! [A2: nat,B4: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B4 ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_nat @ B4 @ X3 ) ) ) ).
% le_supE
thf(fact_276_le__supE,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A2 @ B4 ) @ X3 )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ A2 @ X3 )
=> ~ ( ord_le5389487502678872194c_fm_i @ B4 @ X3 ) ) ) ).
% le_supE
thf(fact_277_le__supI,axiom,
! [A2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ X3 )
=> ( ( ord_le3843937902494030498c_fm_i @ B4 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) @ X3 ) ) ) ).
% le_supI
thf(fact_278_le__supI,axiom,
! [A2: nat,X3: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ( ord_less_eq_nat @ B4 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B4 ) @ X3 ) ) ) ).
% le_supI
thf(fact_279_le__supI,axiom,
! [A2: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ X3 )
=> ( ( ord_le5389487502678872194c_fm_i @ B4 @ X3 )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A2 @ B4 ) @ X3 ) ) ) ).
% le_supI
thf(fact_280_sup__ge1,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_281_sup__ge1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_282_sup__ge1,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_283_sup__ge2,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_284_sup__ge2,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_285_sup__ge2,axiom,
! [Y: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ Y @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_286_le__supI1,axiom,
! [X3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) ) ) ).
% le_supI1
thf(fact_287_le__supI1,axiom,
! [X3: nat,A2: nat,B4: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B4 ) ) ) ).
% le_supI1
thf(fact_288_le__supI1,axiom,
! [X3: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ A2 )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ A2 @ B4 ) ) ) ).
% le_supI1
thf(fact_289_le__supI2,axiom,
! [X3: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ B4 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) ) ) ).
% le_supI2
thf(fact_290_le__supI2,axiom,
! [X3: nat,B4: nat,A2: nat] :
( ( ord_less_eq_nat @ X3 @ B4 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B4 ) ) ) ).
% le_supI2
thf(fact_291_le__supI2,axiom,
! [X3: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ B4 )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ A2 @ B4 ) ) ) ).
% le_supI2
thf(fact_292_sup_Omono,axiom,
! [C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ D2 @ B4 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ C @ D2 ) @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) ) ) ) ).
% sup.mono
thf(fact_293_sup_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B4: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B4 ) ) ) ) ).
% sup.mono
thf(fact_294_sup_Omono,axiom,
! [C: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,D2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C @ A2 )
=> ( ( ord_le5389487502678872194c_fm_i @ D2 @ B4 )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ C @ D2 ) @ ( sup_su2582925890723967158c_fm_i @ A2 @ B4 ) ) ) ) ).
% sup.mono
thf(fact_295_sup__mono,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B4 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) @ ( sup_su1936195050962291414c_fm_i @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_296_sup__mono,axiom,
! [A2: nat,C: nat,B4: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B4 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B4 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_297_sup__mono,axiom,
! [A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,D2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ C )
=> ( ( ord_le5389487502678872194c_fm_i @ B4 @ D2 )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A2 @ B4 ) @ ( sup_su2582925890723967158c_fm_i @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_298_sup__least,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X3 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z2 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_299_sup__least,axiom,
! [Y: nat,X3: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_300_sup__least,axiom,
! [Y: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ Y @ X3 )
=> ( ( ord_le5389487502678872194c_fm_i @ Z2 @ X3 )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_301_le__iff__sup,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_302_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( sup_sup_nat @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_303_le__iff__sup,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [X: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_304_sup_OorderE,axiom,
! [B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B4 @ A2 )
=> ( A2
= ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) ) ) ).
% sup.orderE
thf(fact_305_sup_OorderE,axiom,
! [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B4 ) ) ) ).
% sup.orderE
thf(fact_306_sup_OorderE,axiom,
! [B4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B4 @ A2 )
=> ( A2
= ( sup_su2582925890723967158c_fm_i @ A2 @ B4 ) ) ) ).
% sup.orderE
thf(fact_307_sup_OorderI,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( A2
= ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) )
=> ( ord_le3843937902494030498c_fm_i @ B4 @ A2 ) ) ).
% sup.orderI
thf(fact_308_sup_OorderI,axiom,
! [A2: nat,B4: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B4 ) )
=> ( ord_less_eq_nat @ B4 @ A2 ) ) ).
% sup.orderI
thf(fact_309_sup_OorderI,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( A2
= ( sup_su2582925890723967158c_fm_i @ A2 @ B4 ) )
=> ( ord_le5389487502678872194c_fm_i @ B4 @ A2 ) ) ).
% sup.orderI
thf(fact_310_sup__unique,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y2 @ X2 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z3 @ X2 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_311_sup__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_312_sup__unique,axiom,
! [F: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ Y2 @ X2 )
=> ( ( ord_le5389487502678872194c_fm_i @ Z3 @ X2 )
=> ( ord_le5389487502678872194c_fm_i @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su2582925890723967158c_fm_i @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_313_sup_Oabsorb1,axiom,
! [B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B4 @ A2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B4 )
= A2 ) ) ).
% sup.absorb1
thf(fact_314_sup_Oabsorb1,axiom,
! [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B4 )
= A2 ) ) ).
% sup.absorb1
thf(fact_315_sup_Oabsorb1,axiom,
! [B4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B4 @ A2 )
=> ( ( sup_su2582925890723967158c_fm_i @ A2 @ B4 )
= A2 ) ) ).
% sup.absorb1
thf(fact_316_sup_Oabsorb2,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_317_sup_Oabsorb2,axiom,
! [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( sup_sup_nat @ A2 @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_318_sup_Oabsorb2,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( ( sup_su2582925890723967158c_fm_i @ A2 @ B4 )
= B4 ) ) ).
% sup.absorb2
thf(fact_319_sup__absorb1,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X3 )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_320_sup__absorb1,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_321_sup__absorb1,axiom,
! [Y: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ Y @ X3 )
=> ( ( sup_su2582925890723967158c_fm_i @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_322_sup__absorb2,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_323_sup__absorb2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( sup_sup_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_324_sup__absorb2,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ Y )
=> ( ( sup_su2582925890723967158c_fm_i @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_325_sup_OboundedE,axiom,
! [B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B4 @ C ) @ A2 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ B4 @ A2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_326_sup_OboundedE,axiom,
! [B4: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B4 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_327_sup_OboundedE,axiom,
! [B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ B4 @ C ) @ A2 )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ B4 @ A2 )
=> ~ ( ord_le5389487502678872194c_fm_i @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_328_sup_OboundedI,axiom,
! [B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B4 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B4 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_329_sup_OboundedI,axiom,
! [B4: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B4 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_330_sup_OboundedI,axiom,
! [B4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B4 @ A2 )
=> ( ( ord_le5389487502678872194c_fm_i @ C @ A2 )
=> ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ B4 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_331_sup_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B5: set_Epistemic_fm_i,A5: set_Epistemic_fm_i] :
( A5
= ( sup_su1936195050962291414c_fm_i @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_332_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_333_sup_Oorder__iff,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [B5: set_se3485332733965609186c_fm_i,A5: set_se3485332733965609186c_fm_i] :
( A5
= ( sup_su2582925890723967158c_fm_i @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_334_sup_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) ) ).
% sup.cobounded1
thf(fact_335_sup_Ocobounded1,axiom,
! [A2: nat,B4: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B4 ) ) ).
% sup.cobounded1
thf(fact_336_sup_Ocobounded1,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ A2 @ ( sup_su2582925890723967158c_fm_i @ A2 @ B4 ) ) ).
% sup.cobounded1
thf(fact_337_sup_Ocobounded2,axiom,
! [B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B4 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) ) ).
% sup.cobounded2
thf(fact_338_sup_Ocobounded2,axiom,
! [B4: nat,A2: nat] : ( ord_less_eq_nat @ B4 @ ( sup_sup_nat @ A2 @ B4 ) ) ).
% sup.cobounded2
thf(fact_339_sup_Ocobounded2,axiom,
! [B4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ B4 @ ( sup_su2582925890723967158c_fm_i @ A2 @ B4 ) ) ).
% sup.cobounded2
thf(fact_340_sup_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B5: set_Epistemic_fm_i,A5: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_341_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_342_sup_Oabsorb__iff1,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [B5: set_se3485332733965609186c_fm_i,A5: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_343_sup_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_344_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_345_sup_Oabsorb__iff2,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A5: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_346_sup_OcoboundedI1,axiom,
! [C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_347_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B4: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_348_sup_OcoboundedI1,axiom,
! [C: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C @ A2 )
=> ( ord_le5389487502678872194c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A2 @ B4 ) ) ) ).
% sup.coboundedI1
thf(fact_349_sup_OcoboundedI2,axiom,
! [C: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ B4 )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_350_sup_OcoboundedI2,axiom,
! [C: nat,B4: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B4 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_351_sup_OcoboundedI2,axiom,
! [C: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C @ B4 )
=> ( ord_le5389487502678872194c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A2 @ B4 ) ) ) ).
% sup.coboundedI2
thf(fact_352_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_353_finite__subset,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% finite_subset
thf(fact_354_finite__subset,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( finite7933139204641697195c_fm_i @ A ) ) ) ).
% finite_subset
thf(fact_355_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_356_infinite__super,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S @ T2 )
=> ( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ T2 ) ) ) ).
% infinite_super
thf(fact_357_infinite__super,axiom,
! [S: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ S @ T2 )
=> ( ~ ( finite7933139204641697195c_fm_i @ S )
=> ~ ( finite7933139204641697195c_fm_i @ T2 ) ) ) ).
% infinite_super
thf(fact_358_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_359_rev__finite__subset,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% rev_finite_subset
thf(fact_360_rev__finite__subset,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ B )
=> ( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( finite7933139204641697195c_fm_i @ A ) ) ) ).
% rev_finite_subset
thf(fact_361_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_362_infinite__Un,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( ~ ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) ) )
= ( ~ ( finite3304564979551393739c_fm_i @ S )
| ~ ( finite3304564979551393739c_fm_i @ T2 ) ) ) ).
% infinite_Un
thf(fact_363_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_364_Un__infinite,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_365_finite__UnI,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_366_finite__UnI,axiom,
! [F2: set_Epistemic_fm_i,G: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( finite3304564979551393739c_fm_i @ G )
=> ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_367_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_368_pred__subset__eq,axiom,
! [R: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ R )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ S ) )
= ( ord_le3843937902494030498c_fm_i @ R @ S ) ) ).
% pred_subset_eq
thf(fact_369_pred__subset__eq,axiom,
! [R: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( ord_le5706303257367236315fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ R )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ S ) )
= ( ord_le5389487502678872194c_fm_i @ R @ S ) ) ).
% pred_subset_eq
thf(fact_370__092_060open_062_092_060not_062_Aconsistent_AA_A_Iknown_AW_Ai_A_092_060union_062_Aknown_AU_Ai_J_092_060close_062,axiom,
~ ( episte2285483198712856234tent_i @ a
@ ( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ w ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ u ) ) ) ) ).
% \<open>\<not> consistent A (known W i \<union> known U i)\<close>
thf(fact_371_sup__Un__eq,axiom,
! [R: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( sup_su6723579903606884263fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ R )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ S ) )
= ( ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ ( sup_su2582925890723967158c_fm_i @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_372_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_373_sup__Un__eq,axiom,
! [R: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( sup_su2844414564337698183fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ R )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ S ) )
= ( ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_374_conj__subset__def,axiom,
! [A: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P2 ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_375_conj__subset__def,axiom,
! [A: set_Epistemic_fm_i,P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ A
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le3843937902494030498c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P2 ) )
& ( ord_le3843937902494030498c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_376_conj__subset__def,axiom,
! [A: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ A
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le5389487502678872194c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P2 ) )
& ( ord_le5389487502678872194c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_377_prop__restrict,axiom,
! [X3: nat,Z4: set_nat,X4: set_nat,P2: nat > $o] :
( ( member_nat @ X3 @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X4 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_378_prop__restrict,axiom,
! [X3: epistemic_fm_i,Z4: set_Epistemic_fm_i,X4: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ X3 @ Z4 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z4
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ X4 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_379_prop__restrict,axiom,
! [X3: set_Epistemic_fm_i,Z4: set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ X3 @ Z4 )
=> ( ( ord_le5389487502678872194c_fm_i @ Z4
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ X4 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_380_Collect__restrict,axiom,
! [X4: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X4 )
& ( P2 @ X ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_381_Collect__restrict,axiom,
! [X4: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ X4 )
& ( P2 @ X ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_382_Collect__restrict,axiom,
! [X4: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o] :
( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ X4 )
& ( P2 @ X ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_383_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_384_subset__CollectI,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i,Q: epistemic_fm_i > $o,P2: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B )
& ( Q @ X ) ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_385_subset__CollectI,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,Q: set_Epistemic_fm_i > $o,P2: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ B )
& ( Q @ X ) ) )
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_386_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_387_subset__Collect__iff,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ B
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_388_subset__Collect__iff,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ B
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P2 @ X ) ) ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ B )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_389_union__coset__filter,axiom,
! [Xs: list_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( coset_nat @ Xs ) @ A )
= ( coset_nat
@ ( filter_nat
@ ^ [X: nat] :
~ ( member_nat @ X @ A )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_390_union__coset__filter,axiom,
! [Xs: list_s8081015415394010888c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( coset_1342939534705115381c_fm_i @ Xs ) @ A )
= ( coset_1342939534705115381c_fm_i
@ ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X @ A )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_391_union__coset__filter,axiom,
! [Xs: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( coset_Epistemic_fm_i @ Xs ) @ A )
= ( coset_Epistemic_fm_i
@ ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X @ A )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_392_assms_I1_J,axiom,
! [P3: epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ P3 )
=> ( a @ P3 ) ) ).
% assms(1)
thf(fact_393_assms_I4_J,axiom,
episte2285483198712856234tent_i @ a @ w ).
% assms(4)
thf(fact_394_assms_I6_J,axiom,
episte2285483198712856234tent_i @ a @ u ).
% assms(6)
thf(fact_395_assms_I2_J,axiom,
episte2285483198712856234tent_i @ a @ v ).
% assms(2)
thf(fact_396_assms_I5_J,axiom,
maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ w ).
% assms(5)
thf(fact_397_assms_I7_J,axiom,
maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ u ).
% assms(7)
thf(fact_398_assms_I3_J,axiom,
maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ v ).
% assms(3)
thf(fact_399_consistent__hereditary,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,S2: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( ( ord_le3843937902494030498c_fm_i @ S2 @ S )
=> ( episte2285483198712856234tent_i @ A @ S2 ) ) ) ).
% consistent_hereditary
thf(fact_400_inconsistent__finite,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ S )
=> ? [S3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S3 @ S )
& ( finite3304564979551393739c_fm_i @ S3 )
& ~ ( episte2285483198712856234tent_i @ A @ S3 ) ) ) ).
% inconsistent_finite
thf(fact_401_subset__code_I2_J,axiom,
! [A: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A @ ( coset_nat @ Ys ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_402_subset__code_I2_J,axiom,
! [A: set_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( coset_Epistemic_fm_i @ Ys ) )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Ys ) )
=> ~ ( member6642669606046002379c_fm_i @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_403_subset__code_I2_J,axiom,
! [A: set_se3485332733965609186c_fm_i,Ys: list_s8081015415394010888c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( coset_1342939534705115381c_fm_i @ Ys ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Ys ) )
=> ~ ( member1104366573291651755c_fm_i @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_404__092_060open_062_092_060nexists_062X_O_Aconsistent_AA_AX_A_092_060and_062_Amaximal_AA_AX_A_092_060and_062_AX_A_092_060in_062_A_123Wa_O_Aknown_AW_Ai_A_092_060subseteq_062_AWa_125_A_092_060inter_062_A_123W_O_Aknown_AU_Ai_A_092_060subseteq_062_AW_125_092_060close_062,axiom,
~ ? [X5: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ a @ X5 )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ X5 )
& ( member1104366573291651755c_fm_i @ X5
@ ( inf_in161960956874937808c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ w ) ) ) )
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ u ) ) ) ) ) ) ) ).
% \<open>\<nexists>X. consistent A X \<and> maximal A X \<and> X \<in> {Wa. known W i \<subseteq> Wa} \<inter> {W. known U i \<subseteq> W}\<close>
thf(fact_405__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062S_H_O_A_092_060lbrakk_062A_A_092_060turnstile_062_AS_H_A_092_060_094bold_062_092_060leadsto_062_A_092_060_094bold_062_092_060bottom_062_059_Aset_AS_H_A_092_060subseteq_062_Aknown_AW_Ai_A_092_060union_062_Aknown_AU_Ai_059_Afinite_A_Iset_AS_H_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [S3: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ a @ ( epistemic_imply_i @ S3 @ epistemic_FF_i ) )
=> ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ S3 )
@ ( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ w ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ u ) ) ) )
=> ~ ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ S3 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>S'. \<lbrakk>A \<turnstile> S' \<^bold>\<leadsto> \<^bold>\<bottom>; set S' \<subseteq> known W i \<union> known U i; finite (set S')\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_406__092_060open_062A_A_092_060turnstile_062_Afilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_A_092_060_094bold_062_092_060leadsto_062_Afilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_A_092_060_094bold_062_092_060leadsto_062_A_092_060_094bold_062_092_060bottom_062_092_060close_062,axiom,
( epistemic_AK_i @ a
@ ( epistemic_imply_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s )
@ ( epistemic_imply_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s )
@ epistemic_FF_i ) ) ) ).
% \<open>A \<turnstile> filter (\<lambda>p. p \<in> known U i) S' \<^bold>\<leadsto> filter (\<lambda>p. p \<in> known W i) S' \<^bold>\<leadsto> \<^bold>\<bottom>\<close>
thf(fact_407_filter__set,axiom,
! [P2: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( filter6053540608743173075c_fm_i @ P2 @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( set_Epistemic_fm_i2 @ ( filter7636273843821131039c_fm_i @ P2 @ Xs ) ) ) ).
% filter_set
thf(fact_408_filter__set,axiom,
! [P2: nat > $o,Xs: list_nat] :
( ( filter_nat2 @ P2 @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( filter_nat @ P2 @ Xs ) ) ) ).
% filter_set
thf(fact_409_filter__set,axiom,
! [P2: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] :
( ( filter8450032351001387443c_fm_i @ P2 @ ( set_se200842218512397079c_fm_i @ Xs ) )
= ( set_se200842218512397079c_fm_i @ ( filter3188398074982218495c_fm_i @ P2 @ Xs ) ) ) ).
% filter_set
thf(fact_410__C_K_C_I1_J,axiom,
epistemic_AK_i @ a @ ( epistemic_imply_i @ s @ epistemic_FF_i ) ).
% "*"(1)
thf(fact_411_inf_Oidem,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_412_inf__idem,axiom,
! [X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_413_inf_Oleft__idem,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) )
= ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) ) ).
% inf.left_idem
thf(fact_414_inf__left__idem,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) )
= ( inf_in161960956874937808c_fm_i @ X3 @ Y ) ) ).
% inf_left_idem
thf(fact_415_inf_Oright__idem,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) @ B4 )
= ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) ) ).
% inf.right_idem
thf(fact_416_inf__right__idem,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ Y )
= ( inf_in161960956874937808c_fm_i @ X3 @ Y ) ) ).
% inf_right_idem
thf(fact_417_IntI,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% IntI
thf(fact_418_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_419_IntI,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( ( member1104366573291651755c_fm_i @ C @ B )
=> ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% IntI
thf(fact_420_Int__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
& ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% Int_iff
thf(fact_421_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_422_Int__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
& ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% Int_iff
thf(fact_423_member__filter,axiom,
! [X3: epistemic_fm_i,P2: epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( filter6053540608743173075c_fm_i @ P2 @ A ) )
= ( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) ).
% member_filter
thf(fact_424_member__filter,axiom,
! [X3: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( filter8450032351001387443c_fm_i @ P2 @ A ) )
= ( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) ).
% member_filter
thf(fact_425_member__filter,axiom,
! [X3: nat,P2: nat > $o,A: set_nat] :
( ( member_nat @ X3 @ ( filter_nat2 @ P2 @ A ) )
= ( ( member_nat @ X3 @ A )
& ( P2 @ X3 ) ) ) ).
% member_filter
thf(fact_426_le__inf__iff,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y @ Z2 ) )
= ( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
& ( ord_le3843937902494030498c_fm_i @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_427_le__inf__iff,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X3 @ Y )
& ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_428_le__inf__iff,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z2 ) )
= ( ( ord_le5389487502678872194c_fm_i @ X3 @ Y )
& ( ord_le5389487502678872194c_fm_i @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_429_inf_Obounded__iff,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B4 @ C ) )
= ( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
& ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_430_inf_Obounded__iff,axiom,
! [A2: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B4 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B4 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_431_inf_Obounded__iff,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B4 @ C ) )
= ( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
& ( ord_le5389487502678872194c_fm_i @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_432_sup__inf__absorb,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_433_sup__inf__absorb,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_434_inf__sup__absorb,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_435_inf__sup__absorb,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_436_finite__Int,axiom,
! [F2: set_Epistemic_fm_i,G: set_Epistemic_fm_i] :
( ( ( finite3304564979551393739c_fm_i @ F2 )
| ( finite3304564979551393739c_fm_i @ G ) )
=> ( finite3304564979551393739c_fm_i @ ( inf_in3450601097109690352c_fm_i @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_437_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_438_finite__Int,axiom,
! [F2: set_se3485332733965609186c_fm_i,G: set_se3485332733965609186c_fm_i] :
( ( ( finite7933139204641697195c_fm_i @ F2 )
| ( finite7933139204641697195c_fm_i @ G ) )
=> ( finite7933139204641697195c_fm_i @ ( inf_in161960956874937808c_fm_i @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_439_Int__subset__iff,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
= ( ( ord_le3843937902494030498c_fm_i @ C2 @ A )
& ( ord_le3843937902494030498c_fm_i @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_440_Int__subset__iff,axiom,
! [C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= ( ( ord_le5389487502678872194c_fm_i @ C2 @ A )
& ( ord_le5389487502678872194c_fm_i @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_441_Un__Int__eq_I1_J,axiom,
! [S: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_442_Un__Int__eq_I1_J,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_443_Un__Int__eq_I2_J,axiom,
! [S: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_444_Un__Int__eq_I2_J,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_445_Un__Int__eq_I3_J,axiom,
! [S: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ S @ ( sup_su2582925890723967158c_fm_i @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_446_Un__Int__eq_I3_J,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ S @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_447_Un__Int__eq_I4_J,axiom,
! [T2: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ T2 @ ( sup_su2582925890723967158c_fm_i @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_448_Un__Int__eq_I4_J,axiom,
! [T2: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ T2 @ ( sup_su1936195050962291414c_fm_i @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_449_Int__Un__eq_I1_J,axiom,
! [S: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_450_Int__Un__eq_I1_J,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_451_Int__Un__eq_I2_J,axiom,
! [S: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_452_Int__Un__eq_I2_J,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_453_Int__Un__eq_I3_J,axiom,
! [S: set_se3485332733965609186c_fm_i,T2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ S @ ( inf_in161960956874937808c_fm_i @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_454_Int__Un__eq_I3_J,axiom,
! [S: set_Epistemic_fm_i,T2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ S @ ( inf_in3450601097109690352c_fm_i @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_455_Int__Un__eq_I4_J,axiom,
! [T2: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ T2 @ ( inf_in161960956874937808c_fm_i @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_456_Int__Un__eq_I4_J,axiom,
! [T2: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ T2 @ ( inf_in3450601097109690352c_fm_i @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_457_deriv__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( epistemic_AK_i @ A @ P4 )
=> ( member6642669606046002379c_fm_i @ P4 @ V ) ) ) ) ).
% deriv_in_maximal
thf(fact_458_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ Y @ Z2 ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ ( inf_in161960956874937808c_fm_i @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_459_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_460_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z2 ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) @ ( sup_su2582925890723967158c_fm_i @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_461_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y @ Z2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_462_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y @ Z2 ) @ X3 )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ Y @ X3 ) @ ( inf_in161960956874937808c_fm_i @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_463_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y @ Z2 ) @ X3 )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y @ X3 ) @ ( inf_in3450601097109690352c_fm_i @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_464_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ Y @ Z2 ) @ X3 )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y @ X3 ) @ ( sup_su2582925890723967158c_fm_i @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_465_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y @ Z2 ) @ X3 )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y @ X3 ) @ ( sup_su1936195050962291414c_fm_i @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_466_Int__def,axiom,
( inf_in3450601097109690352c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A3 )
& ( member6642669606046002379c_fm_i @ X @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_467_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( member_nat @ X @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_468_Int__def,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A3 )
& ( member1104366573291651755c_fm_i @ X @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_469_Int__Collect,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P2 ) ) )
= ( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) ).
% Int_Collect
thf(fact_470_Int__Collect,axiom,
! [X3: nat,A: set_nat,P2: nat > $o] :
( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P2 ) ) )
= ( ( member_nat @ X3 @ A )
& ( P2 @ X3 ) ) ) ).
% Int_Collect
thf(fact_471_Int__Collect,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P2 ) ) )
= ( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( P2 @ X3 ) ) ) ).
% Int_Collect
thf(fact_472_Collect__conj__eq,axiom,
! [P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P2 @ X )
& ( Q @ X ) ) )
= ( inf_in3450601097109690352c_fm_i @ ( collec4904205187116291597c_fm_i @ P2 ) @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_473_Collect__conj__eq,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_474_Collect__conj__eq,axiom,
! [P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P2 @ X )
& ( Q @ X ) ) )
= ( inf_in161960956874937808c_fm_i @ ( collec3087743281813070829c_fm_i @ P2 ) @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_475_Ax,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i] :
( ( A @ P4 )
=> ( epistemic_AK_i @ A @ P4 ) ) ).
% Ax
thf(fact_476_IntE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C @ A )
=> ~ ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% IntE
thf(fact_477_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_478_IntE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
=> ~ ( ( member1104366573291651755c_fm_i @ C @ A )
=> ~ ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% IntE
thf(fact_479_IntD1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
=> ( member6642669606046002379c_fm_i @ C @ A ) ) ).
% IntD1
thf(fact_480_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_481_IntD1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
=> ( member1104366573291651755c_fm_i @ C @ A ) ) ).
% IntD1
thf(fact_482_IntD2,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ).
% IntD2
thf(fact_483_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_484_IntD2,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ).
% IntD2
thf(fact_485_Int__assoc,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_486_Int__absorb,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ A )
= A ) ).
% Int_absorb
thf(fact_487_Int__commute,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] : ( inf_in161960956874937808c_fm_i @ B2 @ A3 ) ) ) ).
% Int_commute
thf(fact_488_Int__left__absorb,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ).
% Int_left_absorb
thf(fact_489_Int__left__commute,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) )
= ( inf_in161960956874937808c_fm_i @ B @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_490_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_se3485332733965609186c_fm_i,K: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( B
= ( inf_in161960956874937808c_fm_i @ K @ B4 ) )
=> ( ( inf_in161960956874937808c_fm_i @ A2 @ B )
= ( inf_in161960956874937808c_fm_i @ K @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_491_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_se3485332733965609186c_fm_i,K: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( A
= ( inf_in161960956874937808c_fm_i @ K @ A2 ) )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B4 )
= ( inf_in161960956874937808c_fm_i @ K @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_492_inf__sup__aci_I4_J,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) )
= ( inf_in161960956874937808c_fm_i @ X3 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_493_inf__sup__aci_I3_J,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z2 ) )
= ( inf_in161960956874937808c_fm_i @ Y @ ( inf_in161960956874937808c_fm_i @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_494_inf__sup__aci_I2_J,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ Z2 )
= ( inf_in161960956874937808c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_495_inf__sup__aci_I1_J,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [X: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] : ( inf_in161960956874937808c_fm_i @ Y4 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_496_inf_Oassoc,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) @ C )
= ( inf_in161960956874937808c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B4 @ C ) ) ) ).
% inf.assoc
thf(fact_497_inf__assoc,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ Z2 )
= ( inf_in161960956874937808c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_498_inf_Ocommute,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A5: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] : ( inf_in161960956874937808c_fm_i @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_499_inf__commute,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [X: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] : ( inf_in161960956874937808c_fm_i @ Y4 @ X ) ) ) ).
% inf_commute
thf(fact_500_inf_Oleft__commute,axiom,
! [B4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ B4 @ ( inf_in161960956874937808c_fm_i @ A2 @ C ) )
= ( inf_in161960956874937808c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B4 @ C ) ) ) ).
% inf.left_commute
thf(fact_501_inf__left__commute,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z2 ) )
= ( inf_in161960956874937808c_fm_i @ Y @ ( inf_in161960956874937808c_fm_i @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_502_inter__Set__filter,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= ( filter6053540608743173075c_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_503_inter__Set__filter,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= ( filter_nat2
@ ^ [X: nat] : ( member_nat @ X @ A )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_504_inter__Set__filter,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ B )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B )
= ( filter8450032351001387443c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_505_consistent__def,axiom,
( episte2285483198712856234tent_i
= ( ^ [A3: epistemic_fm_i > $o,S4: set_Epistemic_fm_i] :
~ ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ S4 )
& ( epistemic_AK_i @ A3 @ ( epistemic_imply_i @ Qs @ epistemic_FF_i ) ) ) ) ) ).
% consistent_def
thf(fact_506_K__imply__weaken,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,Q3: epistemic_fm_i,Ps2: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q3 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ps ) @ ( set_Epistemic_fm_i2 @ Ps2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps2 @ Q3 ) ) ) ) ).
% K_imply_weaken
thf(fact_507_R2,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,I: i] :
( ( epistemic_AK_i @ A @ P4 )
=> ( epistemic_AK_i @ A @ ( epistemic_K_i @ I @ P4 ) ) ) ).
% R2
thf(fact_508_inf__sup__ord_I2_J,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_509_inf__sup__ord_I2_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_510_inf__sup__ord_I2_J,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_511_inf__sup__ord_I1_J,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_512_inf__sup__ord_I1_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_513_inf__sup__ord_I1_J,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_514_inf__le1,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_515_inf__le1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_516_inf__le1,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_517_inf__le2,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_518_inf__le2,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_519_inf__le2,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_520_le__infE,axiom,
! [X3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ A2 @ B4 ) )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ X3 @ A2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ X3 @ B4 ) ) ) ).
% le_infE
thf(fact_521_le__infE,axiom,
! [X3: nat,A2: nat,B4: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B4 ) )
=> ~ ( ( ord_less_eq_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_nat @ X3 @ B4 ) ) ) ).
% le_infE
thf(fact_522_le__infE,axiom,
! [X3: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ X3 @ A2 )
=> ~ ( ord_le5389487502678872194c_fm_i @ X3 @ B4 ) ) ) ).
% le_infE
thf(fact_523_le__infI,axiom,
! [X3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ B4 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ A2 @ B4 ) ) ) ) ).
% le_infI
thf(fact_524_le__infI,axiom,
! [X3: nat,A2: nat,B4: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ B4 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B4 ) ) ) ) ).
% le_infI
thf(fact_525_le__infI,axiom,
! [X3: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ A2 )
=> ( ( ord_le5389487502678872194c_fm_i @ X3 @ B4 )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) ) ) ) ).
% le_infI
thf(fact_526_inf__mono,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B4 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B4 ) @ ( inf_in3450601097109690352c_fm_i @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_527_inf__mono,axiom,
! [A2: nat,C: nat,B4: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B4 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_528_inf__mono,axiom,
! [A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,D2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ C )
=> ( ( ord_le5389487502678872194c_fm_i @ B4 @ D2 )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) @ ( inf_in161960956874937808c_fm_i @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_529_le__infI1,axiom,
! [A2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B4 ) @ X3 ) ) ).
% le_infI1
thf(fact_530_le__infI1,axiom,
! [A2: nat,X3: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ X3 ) ) ).
% le_infI1
thf(fact_531_le__infI1,axiom,
! [A2: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ X3 )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) @ X3 ) ) ).
% le_infI1
thf(fact_532_le__infI2,axiom,
! [B4: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B4 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B4 ) @ X3 ) ) ).
% le_infI2
thf(fact_533_le__infI2,axiom,
! [B4: nat,X3: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ X3 ) ) ).
% le_infI2
thf(fact_534_le__infI2,axiom,
! [B4: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B4 @ X3 )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) @ X3 ) ) ).
% le_infI2
thf(fact_535_inf_OorderE,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( A2
= ( inf_in3450601097109690352c_fm_i @ A2 @ B4 ) ) ) ).
% inf.orderE
thf(fact_536_inf_OorderE,axiom,
! [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( A2
= ( inf_inf_nat @ A2 @ B4 ) ) ) ).
% inf.orderE
thf(fact_537_inf_OorderE,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( A2
= ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) ) ) ).
% inf.orderE
thf(fact_538_inf_OorderI,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( A2
= ( inf_in3450601097109690352c_fm_i @ A2 @ B4 ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ B4 ) ) ).
% inf.orderI
thf(fact_539_inf_OorderI,axiom,
! [A2: nat,B4: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B4 ) )
=> ( ord_less_eq_nat @ A2 @ B4 ) ) ).
% inf.orderI
thf(fact_540_inf_OorderI,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( A2
= ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ B4 ) ) ).
% inf.orderI
thf(fact_541_inf__unique,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X2 @ Z3 )
=> ( ord_le3843937902494030498c_fm_i @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_542_inf__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_543_inf__unique,axiom,
! [F: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X2 @ Y2 )
=> ( ( ord_le5389487502678872194c_fm_i @ X2 @ Z3 )
=> ( ord_le5389487502678872194c_fm_i @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_in161960956874937808c_fm_i @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_544_le__iff__inf,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_545_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( inf_inf_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_546_le__iff__inf,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [X: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_547_inf_Oabsorb1,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ B4 )
= A2 ) ) ).
% inf.absorb1
thf(fact_548_inf_Oabsorb1,axiom,
! [A2: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( inf_inf_nat @ A2 @ B4 )
= A2 ) ) ).
% inf.absorb1
thf(fact_549_inf_Oabsorb1,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( ( inf_in161960956874937808c_fm_i @ A2 @ B4 )
= A2 ) ) ).
% inf.absorb1
thf(fact_550_inf_Oabsorb2,axiom,
! [B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B4 @ A2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_551_inf_Oabsorb2,axiom,
! [B4: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_552_inf_Oabsorb2,axiom,
! [B4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B4 @ A2 )
=> ( ( inf_in161960956874937808c_fm_i @ A2 @ B4 )
= B4 ) ) ).
% inf.absorb2
thf(fact_553_inf__absorb1,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_554_inf__absorb1,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( inf_inf_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_555_inf__absorb1,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ Y )
=> ( ( inf_in161960956874937808c_fm_i @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_556_inf__absorb2,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y @ X3 )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_557_inf__absorb2,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_558_inf__absorb2,axiom,
! [Y: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ Y @ X3 )
=> ( ( inf_in161960956874937808c_fm_i @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_559_inf_OboundedE,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B4 @ C ) )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ~ ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_560_inf_OboundedE,axiom,
! [A2: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B4 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B4 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_561_inf_OboundedE,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B4 @ C ) )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ~ ( ord_le5389487502678872194c_fm_i @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_562_inf_OboundedI,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B4 )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B4 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_563_inf_OboundedI,axiom,
! [A2: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B4 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B4 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_564_inf_OboundedI,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B4 )
=> ( ( ord_le5389487502678872194c_fm_i @ A2 @ C )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B4 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_565_inf__greatest,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Z2 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_566_inf__greatest,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_567_inf__greatest,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ Y )
=> ( ( ord_le5389487502678872194c_fm_i @ X3 @ Z2 )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_568_inf_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( A5
= ( inf_in3450601097109690352c_fm_i @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_569_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( A5
= ( inf_inf_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_570_inf_Oorder__iff,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A5: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] :
( A5
= ( inf_in161960956874937808c_fm_i @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_571_inf_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B4 ) @ A2 ) ).
% inf.cobounded1
thf(fact_572_inf_Ocobounded1,axiom,
! [A2: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ A2 ) ).
% inf.cobounded1
thf(fact_573_inf_Ocobounded1,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) @ A2 ) ).
% inf.cobounded1
thf(fact_574_inf_Ocobounded2,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_575_inf_Ocobounded2,axiom,
! [A2: nat,B4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_576_inf_Ocobounded2,axiom,
! [A2: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) @ B4 ) ).
% inf.cobounded2
thf(fact_577_inf_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A5: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_578_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_579_inf_Oabsorb__iff1,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A5: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_580_inf_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B5: set_Epistemic_fm_i,A5: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_581_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_582_inf_Oabsorb__iff2,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [B5: set_se3485332733965609186c_fm_i,A5: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_583_inf_OcoboundedI1,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B4 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_584_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_585_inf_OcoboundedI1,axiom,
! [A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,B4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ C )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_586_inf_OcoboundedI2,axiom,
! [B4: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B4 @ C )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B4 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_587_inf_OcoboundedI2,axiom,
! [B4: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B4 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_588_inf_OcoboundedI2,axiom,
! [B4: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B4 @ C )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B4 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_589_sup__inf__distrib2,axiom,
! [Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ Y @ Z2 ) @ X3 )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y @ X3 ) @ ( sup_su2582925890723967158c_fm_i @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_590_sup__inf__distrib2,axiom,
! [Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y @ Z2 ) @ X3 )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y @ X3 ) @ ( sup_su1936195050962291414c_fm_i @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_591_sup__inf__distrib1,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z2 ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) @ ( sup_su2582925890723967158c_fm_i @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_592_sup__inf__distrib1,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y @ Z2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_593_inf__sup__distrib2,axiom,
! [Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y @ Z2 ) @ X3 )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ Y @ X3 ) @ ( inf_in161960956874937808c_fm_i @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_594_inf__sup__distrib2,axiom,
! [Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i,X3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y @ Z2 ) @ X3 )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y @ X3 ) @ ( inf_in3450601097109690352c_fm_i @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_595_inf__sup__distrib1,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ Y @ Z2 ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ ( inf_in161960956874937808c_fm_i @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_596_inf__sup__distrib1,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_597_distrib__imp2,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X2 @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X2 @ Y2 ) @ ( sup_su2582925890723967158c_fm_i @ X2 @ Z3 ) ) )
=> ( ( inf_in161960956874937808c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ Y @ Z2 ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ ( inf_in161960956874937808c_fm_i @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_598_distrib__imp2,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X2 @ ( inf_in3450601097109690352c_fm_i @ Y2 @ Z3 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X2 @ Y2 ) @ ( sup_su1936195050962291414c_fm_i @ X2 @ Z3 ) ) )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_599_distrib__imp1,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] :
( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X2 @ ( sup_su2582925890723967158c_fm_i @ Y2 @ Z3 ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X2 @ Y2 ) @ ( inf_in161960956874937808c_fm_i @ X2 @ Z3 ) ) )
=> ( ( sup_su2582925890723967158c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z2 ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) @ ( sup_su2582925890723967158c_fm_i @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_600_distrib__imp1,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X2 @ ( sup_su1936195050962291414c_fm_i @ Y2 @ Z3 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X2 @ Y2 ) @ ( inf_in3450601097109690352c_fm_i @ X2 @ Z3 ) ) )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y @ Z2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_601_Int__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_602_Int__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,D: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ C2 )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ D )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( inf_in161960956874937808c_fm_i @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_603_Int__lower1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_604_Int__lower1,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_605_Int__lower2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_606_Int__lower2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_607_Int__absorb1,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_608_Int__absorb1,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_609_Int__absorb2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_610_Int__absorb2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_611_Int__greatest,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ C2 @ B )
=> ( ord_le3843937902494030498c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_612_Int__greatest,axiom,
! [C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C2 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ C2 @ B )
=> ( ord_le5389487502678872194c_fm_i @ C2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_613_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_614_Int__Collect__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,P2: epistemic_fm_i > $o,Q: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P2 ) ) @ ( inf_in3450601097109690352c_fm_i @ B @ ( collec4904205187116291597c_fm_i @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_615_Int__Collect__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,P2: set_Epistemic_fm_i > $o,Q: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P2 ) ) @ ( inf_in161960956874937808c_fm_i @ B @ ( collec3087743281813070829c_fm_i @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_616_fm_Odistinct_I9_J,axiom,
! [X61: i,X62: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_617_Set_Ofilter__def,axiom,
( filter6053540608743173075c_fm_i
= ( ^ [P5: epistemic_fm_i > $o,A3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [A5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A5 @ A3 )
& ( P5 @ A5 ) ) ) ) ) ).
% Set.filter_def
thf(fact_618_Set_Ofilter__def,axiom,
( filter8450032351001387443c_fm_i
= ( ^ [P5: set_Epistemic_fm_i > $o,A3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [A5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A5 @ A3 )
& ( P5 @ A5 ) ) ) ) ) ).
% Set.filter_def
thf(fact_619_Set_Ofilter__def,axiom,
( filter_nat2
= ( ^ [P5: nat > $o,A3: set_nat] :
( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A3 )
& ( P5 @ A5 ) ) ) ) ) ).
% Set.filter_def
thf(fact_620_Un__Int__distrib2,axiom,
! [B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) @ A )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ B @ A ) @ ( sup_su2582925890723967158c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_621_Un__Int__distrib2,axiom,
! [B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) @ A )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B @ A ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_622_Int__Un__distrib2,axiom,
! [B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) @ A )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ B @ A ) @ ( inf_in161960956874937808c_fm_i @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_623_Int__Un__distrib2,axiom,
! [B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) @ A )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ B @ A ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_624_Un__Int__distrib,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) @ ( sup_su2582925890723967158c_fm_i @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_625_Un__Int__distrib,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ ( sup_su1936195050962291414c_fm_i @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_626_Int__Un__distrib,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_627_Int__Un__distrib,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_628_Un__Int__crazy,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) @ ( inf_in161960956874937808c_fm_i @ C2 @ A ) )
= ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) ) @ ( sup_su2582925890723967158c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_629_Un__Int__crazy,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ A ) )
= ( inf_in3450601097109690352c_fm_i @ ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_630_ax__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( A @ P4 )
=> ( member6642669606046002379c_fm_i @ P4 @ V ) ) ) ) ).
% ax_in_maximal
thf(fact_631_distrib__sup__le,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y @ Z2 ) ) @ ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_632_distrib__sup__le,axiom,
! [X3: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y ) @ ( sup_sup_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_633_distrib__sup__le,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y @ Z2 ) ) @ ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y ) @ ( sup_su2582925890723967158c_fm_i @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_634_distrib__inf__le,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z2 ) ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_635_distrib__inf__le,axiom,
! [X3: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y ) @ ( inf_inf_nat @ X3 @ Z2 ) ) @ ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_636_distrib__inf__le,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i,Z2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y ) @ ( inf_in161960956874937808c_fm_i @ X3 @ Z2 ) ) @ ( inf_in161960956874937808c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_637_Un__Int__assoc__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) )
= ( ord_le3843937902494030498c_fm_i @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_638_Un__Int__assoc__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ A @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) ) )
= ( ord_le5389487502678872194c_fm_i @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_639_finite__filter,axiom,
! [S: set_Epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ S )
=> ( finite3304564979551393739c_fm_i @ ( filter6053540608743173075c_fm_i @ P2 @ S ) ) ) ).
% finite_filter
thf(fact_640_finite__filter,axiom,
! [S: set_nat,P2: nat > $o] :
( ( finite_finite_nat @ S )
=> ( finite_finite_nat @ ( filter_nat2 @ P2 @ S ) ) ) ).
% finite_filter
thf(fact_641_inter__set__filter,axiom,
! [A: set_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_642_inter__set__filter,axiom,
! [A: set_nat,Xs: list_nat] :
( ( inf_inf_set_nat @ A @ ( set_nat2 @ Xs ) )
= ( set_nat2
@ ( filter_nat
@ ^ [X: nat] : ( member_nat @ X @ A )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_643_inter__set__filter,axiom,
! [A: set_se3485332733965609186c_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( set_se200842218512397079c_fm_i @ Xs ) )
= ( set_se200842218512397079c_fm_i
@ ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_644_maximal__extension,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ~ ! [W: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ V @ W )
=> ( ( episte2285483198712856234tent_i @ A @ W )
=> ~ ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) ) ) ).
% maximal_extension
thf(fact_645_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Epistemic_fm_i,K: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( B
= ( sup_su1936195050962291414c_fm_i @ K @ B4 ) )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B )
= ( sup_su1936195050962291414c_fm_i @ K @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_646_boolean__algebra__cancel_Osup1,axiom,
! [A: set_Epistemic_fm_i,K: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( A
= ( sup_su1936195050962291414c_fm_i @ K @ A2 ) )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B4 )
= ( sup_su1936195050962291414c_fm_i @ K @ ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_647__092_060open_062A_A_092_060turnstile_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060_094bold_062_092_060longrightarrow_062_Afilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_A_092_060_094bold_062_092_060leadsto_062_A_092_060_094bold_062_092_060bottom_062_092_060close_062,axiom,
( epistemic_AK_i @ a
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( epistemic_imply_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s )
@ epistemic_FF_i ) ) ) ).
% \<open>A \<turnstile> conjunct (filter (\<lambda>p. p \<in> known U i) S') \<^bold>\<longrightarrow> filter (\<lambda>p. p \<in> known W i) S' \<^bold>\<leadsto> \<^bold>\<bottom>\<close>
thf(fact_648__092_060open_062A_A_092_060turnstile_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_092_060close_062,axiom,
( epistemic_AK_i @ a
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ epistemic_FF_i ) ) ) ).
% \<open>A \<turnstile> conjunct (filter (\<lambda>p. p \<in> known U i) S') \<^bold>\<longrightarrow> \<^bold>\<not> conjunct (filter (\<lambda>p. p \<in> known W i) S')\<close>
thf(fact_649__092_060open_062A_A_092_060turnstile_062_Afilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_A_092_060_094bold_062_092_060leadsto_062_A_092_060_094bold_062_092_060bottom_062_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_092_060close_062,axiom,
( epistemic_AK_i @ a
@ ( epistemic_Imp_i
@ ( epistemic_imply_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s )
@ epistemic_FF_i )
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ epistemic_FF_i ) ) ) ).
% \<open>A \<turnstile> filter (\<lambda>p. p \<in> known W i) S' \<^bold>\<leadsto> \<^bold>\<bottom> \<^bold>\<longrightarrow> \<^bold>\<not> conjunct (filter (\<lambda>p. p \<in> known W i) S')\<close>
thf(fact_650_mcs__conjunction__mult,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,S: list_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ S ) @ V )
& ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ S ) ) )
=> ( member6642669606046002379c_fm_i @ ( stalnaker_conjunct_i @ S ) @ V ) ) ) ) ).
% mcs_conjunction_mult
thf(fact_651_o1,axiom,
( epistemic_AK_i @ a
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ epistemic_FF_i ) ) ).
% o1
thf(fact_652_AxB__symmetric_H,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,W2: set_Epistemic_fm_i,I: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ W2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W2 )
=> ( ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ V
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ W2 ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_653_Ax5__Euclidean,axiom,
! [A: epistemic_fm_i > $o,U: set_Epistemic_fm_i,V: set_Epistemic_fm_i,W2: set_Epistemic_fm_i,I: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ U )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ U )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ W2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W2 )
=> ( ( member1104366573291651755c_fm_i @ V
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ U ) ) ) ) )
=> ( ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ U ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_654_Ax4__transitive,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,W2: set_Epistemic_fm_i,I: i,U: set_Epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) ) )
=> ( ( member1104366573291651755c_fm_i @ U
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ W2 ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ U
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_655_fm_Oinject_I4_J,axiom,
! [X51: epistemic_fm_i,X52: epistemic_fm_i,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
( ( ( epistemic_Imp_i @ X51 @ X52 )
= ( epistemic_Imp_i @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_656_fm_Oinject_I3_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
( ( ( epistemic_Con_i @ X41 @ X42 )
= ( epistemic_Con_i @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_657_K__multi__imply,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i,B4: epistemic_fm_i,C: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ ( epistemic_Imp_i @ B4 @ C ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ A2 @ B4 ) @ C ) ) ) ).
% K_multi_imply
thf(fact_658_K__imply__multi,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i,B4: epistemic_fm_i,C: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ B4 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ C ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ ( epistemic_Con_i @ B4 @ C ) ) ) ) ) ).
% K_imply_multi
thf(fact_659_fm_Odistinct_I25_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i,X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( epistemic_Con_i @ X41 @ X42 )
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_660_Ax4_Ointros,axiom,
! [I: i,P4: epistemic_fm_i] : ( epistemic_Ax4_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ ( epistemic_K_i @ I @ P4 ) ) ) ) ).
% Ax4.intros
thf(fact_661_Ax4_Osimps,axiom,
( epistemic_Ax4_i
= ( ^ [A5: epistemic_fm_i] :
? [I2: i,P: epistemic_fm_i] :
( A5
= ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_662_Ax4_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_Ax4_i @ A2 )
=> ~ ! [I3: i,P6: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P6 ) @ ( epistemic_K_i @ I3 @ ( epistemic_K_i @ I3 @ P6 ) ) ) ) ) ).
% Ax4.cases
thf(fact_663_K__conj__imply__factor,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i,Q3: epistemic_fm_i,R2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ Q3 ) ) @ R2 ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Con_i @ P4 @ Q3 ) ) @ R2 ) ) ) ).
% K_conj_imply_factor
thf(fact_664_K__conjunction__out,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i,Q3: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ Q3 ) ) @ ( epistemic_K_i @ I @ ( epistemic_Con_i @ P4 @ Q3 ) ) ) ) ).
% K_conjunction_out
thf(fact_665_K__conjunction__in,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i,Q3: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Con_i @ P4 @ Q3 ) ) @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ Q3 ) ) ) ) ).
% K_conjunction_in
thf(fact_666_A2,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i,Q3: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ Q3 ) ) ) @ ( epistemic_K_i @ I @ Q3 ) ) ) ).
% A2
thf(fact_667_fm_Odistinct_I29_J,axiom,
! [X51: epistemic_fm_i,X52: epistemic_fm_i,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Imp_i @ X51 @ X52 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_668_fm_Odistinct_I7_J,axiom,
! [X51: epistemic_fm_i,X52: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_669_K__imp__trans,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Q3: epistemic_fm_i,R2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ Q3 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ Q3 @ R2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ R2 ) ) ) ) ).
% K_imp_trans
thf(fact_670_K__imp__trans_H,axiom,
! [A: epistemic_fm_i > $o,Q3: epistemic_fm_i,R2: epistemic_fm_i,P4: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ Q3 @ R2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P4 @ Q3 ) @ ( epistemic_Imp_i @ P4 @ R2 ) ) ) ) ).
% K_imp_trans'
thf(fact_671_K__trans,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Q3: epistemic_fm_i,R2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P4 @ Q3 ) @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ Q3 @ R2 ) @ ( epistemic_Imp_i @ P4 @ R2 ) ) ) ) ).
% K_trans
thf(fact_672_R1,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ P4 )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ Q3 ) )
=> ( epistemic_AK_i @ A @ Q3 ) ) ) ).
% R1
thf(fact_673_Ax5_Ointros,axiom,
! [I: i,P4: epistemic_fm_i] : ( epistemic_Ax5_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% Ax5.intros
thf(fact_674_Ax5_Osimps,axiom,
( epistemic_Ax5_i
= ( ^ [A5: epistemic_fm_i] :
? [I2: i,P: epistemic_fm_i] :
( A5
= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_675_Ax5_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_Ax5_i @ A2 )
=> ~ ! [I3: i,P6: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P6 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P6 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% Ax5.cases
thf(fact_676_KB4__5,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% KB4_5
thf(fact_677_AxB_Ointros,axiom,
! [P4: epistemic_fm_i,I: i] : ( epistemic_AxB_i @ ( epistemic_Imp_i @ P4 @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% AxB.intros
thf(fact_678_AxB_Osimps,axiom,
( epistemic_AxB_i
= ( ^ [A5: epistemic_fm_i] :
? [P: epistemic_fm_i,I2: i] :
( A5
= ( epistemic_Imp_i @ P @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% AxB.simps
thf(fact_679_AxB_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_AxB_i @ A2 )
=> ~ ! [P6: epistemic_fm_i,I3: i] :
( A2
!= ( epistemic_Imp_i @ P6 @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P6 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% AxB.cases
thf(fact_680_fm_Odistinct_I27_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Con_i @ X41 @ X42 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_681_fm_Odistinct_I5_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_682_K__conjunct__imply,axiom,
! [A: epistemic_fm_i > $o,G: list_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G ) @ P4 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ P4 ) ) ) ).
% K_conjunct_imply
thf(fact_683_K__imply__conjunct,axiom,
! [A: epistemic_fm_i > $o,G: list_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ P4 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G ) @ P4 ) ) ) ).
% K_imply_conjunct
thf(fact_684_inf__set__def,axiom,
( inf_in3450601097109690352c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( inf_in8976956063904982253fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A3 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_685_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_686_inf__set__def,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ( inf_in853329969422372621fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A3 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_687_inf__Int__eq,axiom,
! [R: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( inf_in8976956063904982253fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ R )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ S ) )
= ( ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ ( inf_in3450601097109690352c_fm_i @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_688_inf__Int__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( inf_inf_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( inf_inf_set_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_689_inf__Int__eq,axiom,
! [R: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( inf_in853329969422372621fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ R )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ S ) )
= ( ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ ( inf_in161960956874937808c_fm_i @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_690_K__thm,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i,Q3: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ Q3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_Con_i @ P4 @ Q3 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ).
% K_thm
thf(fact_691_K4__L,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% K4_L
thf(fact_692_K5__L,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ P4 ) ) ) ) ).
% K5_L
thf(fact_693_K__map,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Q3: epistemic_fm_i,I: i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ Q3 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ Q3 ) ) ) ) ).
% K_map
thf(fact_694_K__A2_H,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i,Q3: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ Q3 ) ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ Q3 ) ) ) ) ).
% K_A2'
thf(fact_695_K__right__mp,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,P4: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ P4 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ ( epistemic_Imp_i @ P4 @ Q3 ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q3 ) ) ) ) ).
% K_right_mp
thf(fact_696_consequent__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ P4 @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P4 @ Q3 ) @ V )
=> ( member6642669606046002379c_fm_i @ Q3 @ V ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_697_mcs__conjunction,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( ( member6642669606046002379c_fm_i @ P4 @ V )
& ( member6642669606046002379c_fm_i @ Q3 @ V ) )
=> ( member6642669606046002379c_fm_i @ ( epistemic_Con_i @ P4 @ Q3 ) @ V ) ) ) ) ).
% mcs_conjunction
thf(fact_698_K__L__dual,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ P4 ) ) ) ).
% K_L_dual
thf(fact_699_K__LK,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ epistemic_FF_i ) ) ) ).
% K_LK
thf(fact_700_exactly__one__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ P4 @ V )
= ( ~ ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ V ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_701_Ax__2_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ A2 )
=> ~ ! [I3: i,P6: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P6 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P6 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% Ax_2.cases
thf(fact_702_Ax__2_Osimps,axiom,
( stalnaker_Ax_2_i
= ( ^ [A5: epistemic_fm_i] :
? [I2: i,P: epistemic_fm_i] :
( A5
= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% Ax_2.simps
thf(fact_703_Ax__2_Ointros,axiom,
! [I: i,P4: epistemic_fm_i] : ( stalnaker_Ax_2_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% Ax_2.intros
thf(fact_704_dual__reach,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,I: i,P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ V )
=> ? [W: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ W
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) ) )
& ( member6642669606046002379c_fm_i @ P4 @ W ) ) ) ) ) ).
% dual_reach
thf(fact_705_reach__dualK,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,W2: set_Epistemic_fm_i,I: i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ W2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W2 )
=> ( ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) ) )
=> ! [P3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ P3 @ W2 )
=> ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ V ) ) ) ) ) ) ) ).
% reach_dualK
thf(fact_706__092_060open_062tautology_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_A_092_060_094bold_062_092_060leadsto_062_A_092_060_094bold_062_092_060bottom_062_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_J_092_060close_062,axiom,
! [G2: list_char > $o,H: epistemic_fm_i > $o] :
( epistemic_eval_i @ G2 @ H
@ ( epistemic_Imp_i
@ ( epistemic_imply_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s )
@ epistemic_FF_i )
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ epistemic_FF_i ) ) ) ).
% \<open>tautology (filter (\<lambda>p. p \<in> known W i) S' \<^bold>\<leadsto> \<^bold>\<bottom> \<^bold>\<longrightarrow> \<^bold>\<not> conjunct (filter (\<lambda>p. p \<in> known W i) S'))\<close>
thf(fact_707_S5__S5_H__assms,axiom,
! [G: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_AxB_i @ P )
| ( epistemic_Ax4_i @ P ) )
@ ( epistemic_imply_i @ Qs @ P4 ) ) ) )
= ( ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_Ax5_i @ P ) )
@ ( epistemic_imply_i @ Qs @ P4 ) ) ) ) ) ).
% S5_S5'_assms
thf(fact_708_S5_H__B,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,I: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% S5'_B
thf(fact_709_AxT__reflexive,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,I: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( member1104366573291651755c_fm_i @ V
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_710_S5_H__4,axiom,
! [A: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ ( epistemic_K_i @ I @ P4 ) ) ) ) ) ) ).
% S5'_4
thf(fact_711_T__L,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,I: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% T_L
thf(fact_712_eval_Osimps_I5_J,axiom,
! [G3: list_char > $o,H2: epistemic_fm_i > $o,P4: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( epistemic_eval_i @ G3 @ H2 @ ( epistemic_Imp_i @ P4 @ Q3 ) )
= ( ( epistemic_eval_i @ G3 @ H2 @ P4 )
=> ( epistemic_eval_i @ G3 @ H2 @ Q3 ) ) ) ).
% eval.simps(5)
thf(fact_713_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H2: epistemic_fm_i > $o,I: i,P4: epistemic_fm_i] :
( ( epistemic_eval_i @ Ux @ H2 @ ( epistemic_K_i @ I @ P4 ) )
= ( H2 @ ( epistemic_K_i @ I @ P4 ) ) ) ).
% eval.simps(6)
thf(fact_714_eval_Osimps_I1_J,axiom,
! [Uu: list_char > $o,Uv: epistemic_fm_i > $o] :
~ ( epistemic_eval_i @ Uu @ Uv @ epistemic_FF_i ) ).
% eval.simps(1)
thf(fact_715_A1,axiom,
! [P4: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [G4: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G4 @ H3 @ P4 )
=> ( epistemic_AK_i @ A @ P4 ) ) ).
% A1
thf(fact_716_eval_Osimps_I4_J,axiom,
! [G3: list_char > $o,H2: epistemic_fm_i > $o,P4: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( epistemic_eval_i @ G3 @ H2 @ ( epistemic_Con_i @ P4 @ Q3 ) )
= ( ( epistemic_eval_i @ G3 @ H2 @ P4 )
& ( epistemic_eval_i @ G3 @ H2 @ Q3 ) ) ) ).
% eval.simps(4)
thf(fact_717_AxT_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_AxT_i @ A2 )
=> ~ ! [I3: i,P6: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P6 ) @ P6 ) ) ) ).
% AxT.cases
thf(fact_718_AxT_Osimps,axiom,
( epistemic_AxT_i
= ( ^ [A5: epistemic_fm_i] :
? [I2: i,P: epistemic_fm_i] :
( A5
= ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P ) @ P ) ) ) ) ).
% AxT.simps
thf(fact_719_AxT_Ointros,axiom,
! [I: i,P4: epistemic_fm_i] : ( epistemic_AxT_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ P4 ) ) ).
% AxT.intros
thf(fact_720_duality__taut,axiom,
! [I: i,P4: epistemic_fm_i,Q3: epistemic_fm_i,G2: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G2 @ H @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ Q3 @ epistemic_FF_i ) ) ) @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ Q3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P4 ) @ epistemic_FF_i ) ) ) ) ).
% duality_taut
thf(fact_721_conjunct__imply,axiom,
! [G: list_Epistemic_fm_i,P4: epistemic_fm_i,G2: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G2 @ H @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G ) @ P4 ) @ ( epistemic_imply_i @ G @ P4 ) ) ) ).
% conjunct_imply
thf(fact_722_imply__conjunct,axiom,
! [G: list_Epistemic_fm_i,P4: epistemic_fm_i,G2: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G2 @ H @ ( epistemic_Imp_i @ ( epistemic_imply_i @ G @ P4 ) @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G ) @ P4 ) ) ) ).
% imply_conjunct
thf(fact_723_AK_Ocases,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ A2 )
=> ( ~ ! [G2: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G2 @ H @ A2 )
=> ( ! [I3: i,P6: epistemic_fm_i,Q4: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P6 ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P6 @ Q4 ) ) ) @ ( epistemic_K_i @ I3 @ Q4 ) ) )
=> ( ~ ( A @ A2 )
=> ( ! [P6: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ P6 )
=> ~ ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P6 @ A2 ) ) )
=> ~ ! [P6: epistemic_fm_i] :
( ? [I3: i] :
( A2
= ( epistemic_K_i @ I3 @ P6 ) )
=> ~ ( epistemic_AK_i @ A @ P6 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_724_AK_Osimps,axiom,
( epistemic_AK_i
= ( ^ [A3: epistemic_fm_i > $o,A5: epistemic_fm_i] :
( ? [P: epistemic_fm_i] :
( ( A5 = P )
& ! [G5: list_char > $o,H4: epistemic_fm_i > $o] : ( epistemic_eval_i @ G5 @ H4 @ P ) )
| ? [I2: i,P: epistemic_fm_i,Q2: epistemic_fm_i] :
( A5
= ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I2 @ P ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P @ Q2 ) ) ) @ ( epistemic_K_i @ I2 @ Q2 ) ) )
| ? [P: epistemic_fm_i] :
( ( A5 = P )
& ( A3 @ P ) )
| ? [P: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( A5 = Q2 )
& ( epistemic_AK_i @ A3 @ P )
& ( epistemic_AK_i @ A3 @ ( epistemic_Imp_i @ P @ Q2 ) ) )
| ? [P: epistemic_fm_i,I2: i] :
( ( A5
= ( epistemic_K_i @ I2 @ P ) )
& ( epistemic_AK_i @ A3 @ P ) ) ) ) ) ).
% AK.simps
thf(fact_725_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_i,Qs2: list_Epistemic_fm_i,R2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ps ) @ ( set_Epistemic_fm_i2 @ Qs2 ) )
=> ! [G2: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G2 @ H @ ( epistemic_Imp_i @ ( epistemic_imply_i @ Ps @ R2 ) @ ( epistemic_imply_i @ Qs2 @ R2 ) ) ) ) ).
% tautology_imply_superset
thf(fact_726_S5__S5_H,axiom,
! [P4: epistemic_fm_i] :
( ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_AxB_i @ P )
| ( epistemic_Ax4_i @ P ) )
@ P4 )
=> ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_Ax5_i @ P ) )
@ P4 ) ) ).
% S5_S5'
thf(fact_727_S5_H__S5,axiom,
! [P4: epistemic_fm_i] :
( ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_Ax5_i @ P ) )
@ P4 )
=> ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_AxB_i @ P )
| ( epistemic_Ax4_i @ P ) )
@ P4 ) ) ).
% S5'_S5
thf(fact_728_K__Boole,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,G: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ G ) @ epistemic_FF_i ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ P4 ) ) ) ).
% K_Boole
thf(fact_729_K__conjunction__in__mult,axiom,
! [A: epistemic_fm_i > $o,I: i,G: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( stalnaker_conjunct_i @ G ) ) @ ( stalnaker_conjunct_i @ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ I ) @ G ) ) ) ) ).
% K_conjunction_in_mult
thf(fact_730_K__conjunction__out__mult,axiom,
! [A: epistemic_fm_i > $o,I: i,G: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ I ) @ G ) ) @ ( epistemic_K_i @ I @ ( stalnaker_conjunct_i @ G ) ) ) ) ).
% K_conjunction_out_mult
thf(fact_731_MCS_Omaximal_Ocong,axiom,
maxima3264069618988350929c_fm_i = maxima3264069618988350929c_fm_i ).
% MCS.maximal.cong
thf(fact_732_inconsistent__imply,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,G: list_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ ( set_Epistemic_fm_i2 @ G ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ P4 ) ) ) ).
% inconsistent_imply
thf(fact_733_empty__Collect__eq,axiom,
! [P2: epistemic_fm_i > $o] :
( ( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ P2 ) )
= ( ! [X: epistemic_fm_i] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_734_empty__Collect__eq,axiom,
! [P2: set_Epistemic_fm_i > $o] :
( ( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ P2 ) )
= ( ! [X: set_Epistemic_fm_i] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_735_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X: nat] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_736_Collect__empty__eq,axiom,
! [P2: epistemic_fm_i > $o] :
( ( ( collec4904205187116291597c_fm_i @ P2 )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X: epistemic_fm_i] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_737_Collect__empty__eq,axiom,
! [P2: set_Epistemic_fm_i > $o] :
( ( ( collec3087743281813070829c_fm_i @ P2 )
= bot_bo145720340923748686c_fm_i )
= ( ! [X: set_Epistemic_fm_i] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_738_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_739_all__not__in__conv,axiom,
! [A: set_Epistemic_fm_i] :
( ( ! [X: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X @ A ) )
= ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% all_not_in_conv
thf(fact_740_all__not__in__conv,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ! [X: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X @ A ) )
= ( A = bot_bo145720340923748686c_fm_i ) ) ).
% all_not_in_conv
thf(fact_741_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_742_empty__iff,axiom,
! [C: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ C @ bot_bo4194595901900360558c_fm_i ) ).
% empty_iff
thf(fact_743_empty__iff,axiom,
! [C: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ C @ bot_bo145720340923748686c_fm_i ) ).
% empty_iff
thf(fact_744_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_745_insert__iff,axiom,
! [A2: epistemic_fm_i,B4: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B4 @ A ) )
= ( ( A2 = B4 )
| ( member6642669606046002379c_fm_i @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_746_insert__iff,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B4 @ A ) )
= ( ( A2 = B4 )
| ( member1104366573291651755c_fm_i @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_747_insert__iff,axiom,
! [A2: nat,B4: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B4 @ A ) )
= ( ( A2 = B4 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_748_insertCI,axiom,
! [A2: epistemic_fm_i,B: set_Epistemic_fm_i,B4: epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ A2 @ B )
=> ( A2 = B4 ) )
=> ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B4 @ B ) ) ) ).
% insertCI
thf(fact_749_insertCI,axiom,
! [A2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,B4: set_Epistemic_fm_i] :
( ( ~ ( member1104366573291651755c_fm_i @ A2 @ B )
=> ( A2 = B4 ) )
=> ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B4 @ B ) ) ) ).
% insertCI
thf(fact_750_insertCI,axiom,
! [A2: nat,B: set_nat,B4: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B4 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B4 @ B ) ) ) ).
% insertCI
thf(fact_751_empty__subsetI,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A ) ).
% empty_subsetI
thf(fact_752_empty__subsetI,axiom,
! [A: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ bot_bo145720340923748686c_fm_i @ A ) ).
% empty_subsetI
thf(fact_753_subset__empty,axiom,
! [A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% subset_empty
thf(fact_754_subset__empty,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ bot_bo145720340923748686c_fm_i )
= ( A = bot_bo145720340923748686c_fm_i ) ) ).
% subset_empty
thf(fact_755_inf__bot__right,axiom,
! [X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i )
= bot_bo145720340923748686c_fm_i ) ).
% inf_bot_right
thf(fact_756_inf__bot__left,axiom,
! [X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ bot_bo145720340923748686c_fm_i @ X3 )
= bot_bo145720340923748686c_fm_i ) ).
% inf_bot_left
thf(fact_757_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ bot_bo145720340923748686c_fm_i @ X3 )
= bot_bo145720340923748686c_fm_i ) ).
% boolean_algebra.conj_zero_left
thf(fact_758_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i )
= bot_bo145720340923748686c_fm_i ) ).
% boolean_algebra.conj_zero_right
thf(fact_759_sup__bot__left,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_760_sup__bot__right,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i )
= X3 ) ).
% sup_bot_right
thf(fact_761_bot__eq__sup__iff,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( sup_su1936195050962291414c_fm_i @ X3 @ Y ) )
= ( ( X3 = bot_bo4194595901900360558c_fm_i )
& ( Y = bot_bo4194595901900360558c_fm_i ) ) ) ).
% bot_eq_sup_iff
thf(fact_762_sup__eq__bot__iff,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ X3 @ Y )
= bot_bo4194595901900360558c_fm_i )
= ( ( X3 = bot_bo4194595901900360558c_fm_i )
& ( Y = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_eq_bot_iff
thf(fact_763_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A2 @ B4 )
= bot_bo4194595901900360558c_fm_i )
= ( ( A2 = bot_bo4194595901900360558c_fm_i )
& ( B4 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_764_sup__bot_Oleft__neutral,axiom,
! [A2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_765_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( sup_su1936195050962291414c_fm_i @ A2 @ B4 ) )
= ( ( A2 = bot_bo4194595901900360558c_fm_i )
& ( B4 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_766_sup__bot_Oright__neutral,axiom,
! [A2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_767_singletonI,axiom,
! [A2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singletonI
thf(fact_768_singletonI,axiom,
! [A2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singletonI
thf(fact_769_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_770_finite__insert,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) )
= ( finite3304564979551393739c_fm_i @ A ) ) ).
% finite_insert
thf(fact_771_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_772_insert__subset,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_773_insert__subset,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ B )
= ( ( member6642669606046002379c_fm_i @ X3 @ B )
& ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ) ).
% insert_subset
thf(fact_774_insert__subset,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ B )
= ( ( member1104366573291651755c_fm_i @ X3 @ B )
& ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ) ).
% insert_subset
thf(fact_775_Un__empty,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B = bot_bo4194595901900360558c_fm_i ) ) ) ).
% Un_empty
thf(fact_776_Int__insert__right__if1,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_777_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_778_Int__insert__right__if1,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_779_Int__insert__right__if0,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_780_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_781_Int__insert__right__if0,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_782_insert__inter__insert,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ A ) @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_783_Int__insert__left__if1,axiom,
! [A2: epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_784_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_785_Int__insert__left__if1,axiom,
! [A2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B ) @ C2 )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_786_Int__insert__left__if0,axiom,
! [A2: epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_787_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_788_Int__insert__left__if0,axiom,
! [A2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_789_Un__insert__left,axiom,
! [A2: epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_790_Un__insert__right,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_791_singleton__conv2,axiom,
! [A2: epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ( ^ [Y3: epistemic_fm_i,Z: epistemic_fm_i] : ( Y3 = Z )
@ A2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singleton_conv2
thf(fact_792_singleton__conv2,axiom,
! [A2: set_Epistemic_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ( ^ [Y3: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y3 = Z )
@ A2 ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singleton_conv2
thf(fact_793_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_794_singleton__conv,axiom,
! [A2: epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : ( X = A2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singleton_conv
thf(fact_795_singleton__conv,axiom,
! [A2: set_Epistemic_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( X = A2 ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singleton_conv
thf(fact_796_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_797_singleton__insert__inj__eq_H,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B4: epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= ( insert7817948997695205106c_fm_i @ B4 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( A2 = B4 )
& ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_798_singleton__insert__inj__eq_H,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B4: set_Epistemic_fm_i] :
( ( ( insert7698009978809854162c_fm_i @ A2 @ A )
= ( insert7698009978809854162c_fm_i @ B4 @ bot_bo145720340923748686c_fm_i ) )
= ( ( A2 = B4 )
& ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ B4 @ bot_bo145720340923748686c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_799_singleton__insert__inj__eq,axiom,
! [B4: epistemic_fm_i,A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ B4 @ bot_bo4194595901900360558c_fm_i )
= ( insert7817948997695205106c_fm_i @ A2 @ A ) )
= ( ( A2 = B4 )
& ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_800_singleton__insert__inj__eq,axiom,
! [B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ( insert7698009978809854162c_fm_i @ B4 @ bot_bo145720340923748686c_fm_i )
= ( insert7698009978809854162c_fm_i @ A2 @ A ) )
= ( ( A2 = B4 )
& ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ B4 @ bot_bo145720340923748686c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_801_list_Osimps_I15_J,axiom,
! [X21: epistemic_fm_i,X22: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X22 ) )
= ( insert7817948997695205106c_fm_i @ X21 @ ( set_Epistemic_fm_i2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_802_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_803_list_Osimps_I15_J,axiom,
! [X21: set_Epistemic_fm_i,X22: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X21 @ X22 ) )
= ( insert7698009978809854162c_fm_i @ X21 @ ( set_se200842218512397079c_fm_i @ X22 ) ) ) ).
% list.simps(15)
thf(fact_804_disjoint__insert_I2_J,axiom,
! [A: set_Epistemic_fm_i,B4: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B4 @ B ) ) )
= ( ~ ( member6642669606046002379c_fm_i @ B4 @ A )
& ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_805_disjoint__insert_I2_J,axiom,
! [A: set_nat,B4: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B4 @ B ) ) )
= ( ~ ( member_nat @ B4 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_806_disjoint__insert_I2_J,axiom,
! [A: set_se3485332733965609186c_fm_i,B4: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ B4 @ B ) ) )
= ( ~ ( member1104366573291651755c_fm_i @ B4 @ A )
& ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_807_disjoint__insert_I1_J,axiom,
! [B: set_Epistemic_fm_i,A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A2 @ A ) )
= bot_bo4194595901900360558c_fm_i )
= ( ~ ( member6642669606046002379c_fm_i @ A2 @ B )
& ( ( inf_in3450601097109690352c_fm_i @ B @ A )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% disjoint_insert(1)
thf(fact_808_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_809_disjoint__insert_I1_J,axiom,
! [B: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ B @ ( insert7698009978809854162c_fm_i @ A2 @ A ) )
= bot_bo145720340923748686c_fm_i )
= ( ~ ( member1104366573291651755c_fm_i @ A2 @ B )
& ( ( inf_in161960956874937808c_fm_i @ B @ A )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% disjoint_insert(1)
thf(fact_810_insert__disjoint_I2_J,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) @ B ) )
= ( ~ ( member6642669606046002379c_fm_i @ A2 @ B )
& ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_811_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_812_insert__disjoint_I2_J,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ A ) @ B ) )
= ( ~ ( member1104366573291651755c_fm_i @ A2 @ B )
& ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_813_insert__disjoint_I1_J,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ~ ( member6642669606046002379c_fm_i @ A2 @ B )
& ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% insert_disjoint(1)
thf(fact_814_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_815_insert__disjoint_I1_J,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ A ) @ B )
= bot_bo145720340923748686c_fm_i )
= ( ~ ( member1104366573291651755c_fm_i @ A2 @ B )
& ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% insert_disjoint(1)
thf(fact_816_finite__subset__induct_H,axiom,
! [F2: set_nat,A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ A6 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_817_finite__subset__induct_H,axiom,
! [F2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( ord_le3843937902494030498c_fm_i @ F2 @ A )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A6: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ F3 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ A6 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_818_finite__subset__induct_H,axiom,
! [F2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P2: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( ord_le5389487502678872194c_fm_i @ F2 @ A )
=> ( ( P2 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A6: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ F3 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7698009978809854162c_fm_i @ A6 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_819_finite__subset__induct,axiom,
! [F2: set_nat,A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ A6 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_820_finite__subset__induct,axiom,
! [F2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( ord_le3843937902494030498c_fm_i @ F2 @ A )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A6: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ A6 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_821_finite__subset__induct,axiom,
! [F2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P2: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( ord_le5389487502678872194c_fm_i @ F2 @ A )
=> ( ( P2 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A6: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ A6 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7698009978809854162c_fm_i @ A6 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_822_list_Omap__ident__strong,axiom,
! [T3: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ! [Z3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z3 @ ( set_Epistemic_fm_i2 @ T3 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_Ep2755178516647988292c_fm_i @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_823_list_Omap__ident__strong,axiom,
! [T3: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ T3 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_824_list_Omap__ident__strong,axiom,
! [T3: list_s8081015415394010888c_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i] :
( ! [Z3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z3 @ ( set_se200842218512397079c_fm_i @ T3 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_se1449444348732536900c_fm_i @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_825_map__idI,axiom,
! [Xs: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_826_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_827_map__idI,axiom,
! [Xs: list_s8081015415394010888c_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i] :
( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_se1449444348732536900c_fm_i @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_828_bot_Oextremum,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A2 ) ).
% bot.extremum
thf(fact_829_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_830_bot_Oextremum,axiom,
! [A2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ bot_bo145720340923748686c_fm_i @ A2 ) ).
% bot.extremum
thf(fact_831_bot_Oextremum__unique,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i )
= ( A2 = bot_bo4194595901900360558c_fm_i ) ) ).
% bot.extremum_unique
thf(fact_832_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_833_bot_Oextremum__unique,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i )
= ( A2 = bot_bo145720340923748686c_fm_i ) ) ).
% bot.extremum_unique
thf(fact_834_bot_Oextremum__uniqueI,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i )
=> ( A2 = bot_bo4194595901900360558c_fm_i ) ) ).
% bot.extremum_uniqueI
thf(fact_835_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_836_bot_Oextremum__uniqueI,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i )
=> ( A2 = bot_bo145720340923748686c_fm_i ) ) ).
% bot.extremum_uniqueI
thf(fact_837_infinite__finite__induct,axiom,
! [P2: set_se3485332733965609186c_fm_i > $o,A: set_se3485332733965609186c_fm_i] :
( ! [A7: set_se3485332733965609186c_fm_i] :
( ~ ( finite7933139204641697195c_fm_i @ A7 )
=> ( P2 @ A7 ) )
=> ( ( P2 @ bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ~ ( member1104366573291651755c_fm_i @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7698009978809854162c_fm_i @ X2 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_838_infinite__finite__induct,axiom,
! [P2: set_Epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ! [A7: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ A7 )
=> ( P2 @ A7 ) )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X2: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ~ ( member6642669606046002379c_fm_i @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ X2 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_839_infinite__finite__induct,axiom,
! [P2: set_nat > $o,A: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P2 @ A7 ) )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_840_finite__ne__induct,axiom,
! [F2: set_se3485332733965609186c_fm_i,P2: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( F2 != bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i] : ( P2 @ ( insert7698009978809854162c_fm_i @ X2 @ bot_bo145720340923748686c_fm_i ) )
=> ( ! [X2: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( F3 != bot_bo145720340923748686c_fm_i )
=> ( ~ ( member1104366573291651755c_fm_i @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7698009978809854162c_fm_i @ X2 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_841_finite__ne__induct,axiom,
! [F2: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( F2 != bot_bo4194595901900360558c_fm_i )
=> ( ! [X2: epistemic_fm_i] : ( P2 @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) )
=> ( ! [X2: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( F3 != bot_bo4194595901900360558c_fm_i )
=> ( ~ ( member6642669606046002379c_fm_i @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ X2 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_842_finite__ne__induct,axiom,
! [F2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X2 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_843_finite__induct,axiom,
! [F2: set_se3485332733965609186c_fm_i,P2: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( P2 @ bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ~ ( member1104366573291651755c_fm_i @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7698009978809854162c_fm_i @ X2 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_844_finite__induct,axiom,
! [F2: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X2: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ~ ( member6642669606046002379c_fm_i @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ X2 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_845_finite__induct,axiom,
! [F2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_846_finite_Osimps,axiom,
( finite3304564979551393739c_fm_i
= ( ^ [A5: set_Epistemic_fm_i] :
( ( A5 = bot_bo4194595901900360558c_fm_i )
| ? [A3: set_Epistemic_fm_i,B5: epistemic_fm_i] :
( ( A5
= ( insert7817948997695205106c_fm_i @ B5 @ A3 ) )
& ( finite3304564979551393739c_fm_i @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_847_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A3: set_nat,B5: nat] :
( ( A5
= ( insert_nat @ B5 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_848_finite_Ocases,axiom,
! [A2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( A2 != bot_bo4194595901900360558c_fm_i )
=> ~ ! [A7: set_Epistemic_fm_i] :
( ? [A6: epistemic_fm_i] :
( A2
= ( insert7817948997695205106c_fm_i @ A6 @ A7 ) )
=> ~ ( finite3304564979551393739c_fm_i @ A7 ) ) ) ) ).
% finite.cases
thf(fact_849_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A6: nat] :
( A2
= ( insert_nat @ A6 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_850_mk__disjoint__insert,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ? [B7: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ A2 @ B7 ) )
& ~ ( member6642669606046002379c_fm_i @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_851_mk__disjoint__insert,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [B7: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ A2 @ B7 ) )
& ~ ( member1104366573291651755c_fm_i @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_852_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B7: set_nat] :
( ( A
= ( insert_nat @ A2 @ B7 ) )
& ~ ( member_nat @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_853_Collect__conv__if2,axiom,
! [P2: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( ( P2 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( A2 = X )
& ( P2 @ X ) ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( A2 = X )
& ( P2 @ X ) ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_conv_if2
thf(fact_854_Collect__conv__if2,axiom,
! [P2: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ( ( P2 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( A2 = X )
& ( P2 @ X ) ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( A2 = X )
& ( P2 @ X ) ) )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_conv_if2
thf(fact_855_Collect__conv__if2,axiom,
! [P2: nat > $o,A2: nat] :
( ( ( P2 @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A2 = X )
& ( P2 @ X ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A2 = X )
& ( P2 @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_856_Collect__conv__if,axiom,
! [P2: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( ( P2 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( X = A2 )
& ( P2 @ X ) ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( X = A2 )
& ( P2 @ X ) ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_conv_if
thf(fact_857_Collect__conv__if,axiom,
! [P2: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ( ( P2 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( X = A2 )
& ( P2 @ X ) ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( X = A2 )
& ( P2 @ X ) ) )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_conv_if
thf(fact_858_Collect__conv__if,axiom,
! [P2: nat > $o,A2: nat] :
( ( ( P2 @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A2 )
& ( P2 @ X ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A2 )
& ( P2 @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_859_singleton__iff,axiom,
! [B4: epistemic_fm_i,A2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B4 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
= ( B4 = A2 ) ) ).
% singleton_iff
thf(fact_860_singleton__iff,axiom,
! [B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B4 @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) )
= ( B4 = A2 ) ) ).
% singleton_iff
thf(fact_861_singleton__iff,axiom,
! [B4: nat,A2: nat] :
( ( member_nat @ B4 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B4 = A2 ) ) ).
% singleton_iff
thf(fact_862_insert__eq__iff,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B4: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ B4 @ B )
=> ( ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= ( insert7817948997695205106c_fm_i @ B4 @ B ) )
= ( ( ( A2 = B4 )
=> ( A = B ) )
& ( ( A2 != B4 )
=> ? [C3: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ B4 @ C3 ) )
& ~ ( member6642669606046002379c_fm_i @ B4 @ C3 )
& ( B
= ( insert7817948997695205106c_fm_i @ A2 @ C3 ) )
& ~ ( member6642669606046002379c_fm_i @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_863_insert__eq__iff,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B4: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ B4 @ B )
=> ( ( ( insert7698009978809854162c_fm_i @ A2 @ A )
= ( insert7698009978809854162c_fm_i @ B4 @ B ) )
= ( ( ( A2 = B4 )
=> ( A = B ) )
& ( ( A2 != B4 )
=> ? [C3: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ B4 @ C3 ) )
& ~ ( member1104366573291651755c_fm_i @ B4 @ C3 )
& ( B
= ( insert7698009978809854162c_fm_i @ A2 @ C3 ) )
& ~ ( member1104366573291651755c_fm_i @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_864_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B4: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B4 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B4 @ B ) )
= ( ( ( A2 = B4 )
=> ( A = B ) )
& ( ( A2 != B4 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B4 @ C3 ) )
& ~ ( member_nat @ B4 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_865_insert__absorb,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_866_insert__absorb,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( insert7698009978809854162c_fm_i @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_867_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_868_insert__ident,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ X3 @ B )
=> ( ( ( insert7817948997695205106c_fm_i @ X3 @ A )
= ( insert7817948997695205106c_fm_i @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_869_insert__ident,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ B )
=> ( ( ( insert7698009978809854162c_fm_i @ X3 @ A )
= ( insert7698009978809854162c_fm_i @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_870_insert__ident,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ~ ( member_nat @ X3 @ B )
=> ( ( ( insert_nat @ X3 @ A )
= ( insert_nat @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_871_singletonD,axiom,
! [B4: epistemic_fm_i,A2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B4 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
=> ( B4 = A2 ) ) ).
% singletonD
thf(fact_872_singletonD,axiom,
! [B4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B4 @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) )
=> ( B4 = A2 ) ) ).
% singletonD
thf(fact_873_singletonD,axiom,
! [B4: nat,A2: nat] :
( ( member_nat @ B4 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B4 = A2 ) ) ).
% singletonD
thf(fact_874_Set_Oset__insert,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ~ ! [B7: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ B7 ) )
=> ( member6642669606046002379c_fm_i @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_875_Set_Oset__insert,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ~ ! [B7: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ X3 @ B7 ) )
=> ( member1104366573291651755c_fm_i @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_876_Set_Oset__insert,axiom,
! [X3: nat,A: set_nat] :
( ( member_nat @ X3 @ A )
=> ~ ! [B7: set_nat] :
( ( A
= ( insert_nat @ X3 @ B7 ) )
=> ( member_nat @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_877_ex__in__conv,axiom,
! [A: set_Epistemic_fm_i] :
( ( ? [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A ) )
= ( A != bot_bo4194595901900360558c_fm_i ) ) ).
% ex_in_conv
thf(fact_878_ex__in__conv,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ? [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A ) )
= ( A != bot_bo145720340923748686c_fm_i ) ) ).
% ex_in_conv
thf(fact_879_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_880_insertI2,axiom,
! [A2: epistemic_fm_i,B: set_Epistemic_fm_i,B4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ B )
=> ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B4 @ B ) ) ) ).
% insertI2
thf(fact_881_insertI2,axiom,
! [A2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,B4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ B )
=> ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B4 @ B ) ) ) ).
% insertI2
thf(fact_882_insertI2,axiom,
! [A2: nat,B: set_nat,B4: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B4 @ B ) ) ) ).
% insertI2
thf(fact_883_insertI1,axiom,
! [A2: epistemic_fm_i,B: set_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A2 @ B ) ) ).
% insertI1
thf(fact_884_insertI1,axiom,
! [A2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] : ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ A2 @ B ) ) ).
% insertI1
thf(fact_885_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_886_equals0I,axiom,
! [A: set_Epistemic_fm_i] :
( ! [Y2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ Y2 @ A )
=> ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% equals0I
thf(fact_887_equals0I,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [Y2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ Y2 @ A )
=> ( A = bot_bo145720340923748686c_fm_i ) ) ).
% equals0I
thf(fact_888_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_889_equals0D,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( A = bot_bo4194595901900360558c_fm_i )
=> ~ ( member6642669606046002379c_fm_i @ A2 @ A ) ) ).
% equals0D
thf(fact_890_equals0D,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( A = bot_bo145720340923748686c_fm_i )
=> ~ ( member1104366573291651755c_fm_i @ A2 @ A ) ) ).
% equals0D
thf(fact_891_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_892_insertE,axiom,
! [A2: epistemic_fm_i,B4: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B4 @ A ) )
=> ( ( A2 != B4 )
=> ( member6642669606046002379c_fm_i @ A2 @ A ) ) ) ).
% insertE
thf(fact_893_insertE,axiom,
! [A2: set_Epistemic_fm_i,B4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B4 @ A ) )
=> ( ( A2 != B4 )
=> ( member1104366573291651755c_fm_i @ A2 @ A ) ) ) ).
% insertE
thf(fact_894_insertE,axiom,
! [A2: nat,B4: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B4 @ A ) )
=> ( ( A2 != B4 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_895_emptyE,axiom,
! [A2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ).
% emptyE
thf(fact_896_emptyE,axiom,
! [A2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ).
% emptyE
thf(fact_897_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_898_empty__def,axiom,
( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : $false ) ) ).
% empty_def
thf(fact_899_empty__def,axiom,
( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : $false ) ) ).
% empty_def
thf(fact_900_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% empty_def
thf(fact_901_insert__Collect,axiom,
! [A2: epistemic_fm_i,P2: epistemic_fm_i > $o] :
( ( insert7817948997695205106c_fm_i @ A2 @ ( collec4904205187116291597c_fm_i @ P2 ) )
= ( collec4904205187116291597c_fm_i
@ ^ [U2: epistemic_fm_i] :
( ( U2 != A2 )
=> ( P2 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_902_insert__Collect,axiom,
! [A2: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( insert7698009978809854162c_fm_i @ A2 @ ( collec3087743281813070829c_fm_i @ P2 ) )
= ( collec3087743281813070829c_fm_i
@ ^ [U2: set_Epistemic_fm_i] :
( ( U2 != A2 )
=> ( P2 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_903_insert__Collect,axiom,
! [A2: nat,P2: nat > $o] :
( ( insert_nat @ A2 @ ( collect_nat @ P2 ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A2 )
=> ( P2 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_904_insert__compr,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A5: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( X = A5 )
| ( member6642669606046002379c_fm_i @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_905_insert__compr,axiom,
( insert7698009978809854162c_fm_i
= ( ^ [A5: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( X = A5 )
| ( member1104366573291651755c_fm_i @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_906_insert__compr,axiom,
( insert_nat
= ( ^ [A5: nat,B2: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A5 )
| ( member_nat @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_907_insert__is__Un,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A5: epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ A5 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% insert_is_Un
thf(fact_908_Un__singleton__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X3: epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
& ( B = bot_bo4194595901900360558c_fm_i ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
& ( B
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_909_singleton__Un__iff,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i )
= ( sup_su1936195050962291414c_fm_i @ A @ B ) )
= ( ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
& ( B = bot_bo4194595901900360558c_fm_i ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
& ( B
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_910_subset__singleton__iff,axiom,
! [X4: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( X4 = bot_bo4194595901900360558c_fm_i )
| ( X4
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% subset_singleton_iff
thf(fact_911_subset__singleton__iff,axiom,
! [X4: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X4 @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) )
= ( ( X4 = bot_bo145720340923748686c_fm_i )
| ( X4
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ) ) ).
% subset_singleton_iff
thf(fact_912_subset__singletonD,axiom,
! [A: set_Epistemic_fm_i,X3: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
=> ( ( A = bot_bo4194595901900360558c_fm_i )
| ( A
= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% subset_singletonD
thf(fact_913_subset__singletonD,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
=> ( ( A = bot_bo145720340923748686c_fm_i )
| ( A
= ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) ) ) ).
% subset_singletonD
thf(fact_914_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_915_finite_OinsertI,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3304564979551393739c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_916_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_917_insert__subsetI,axiom,
! [X3: nat,A: set_nat,X4: set_nat] :
( ( member_nat @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_918_insert__subsetI,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_919_insert__subsetI,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ X4 @ A )
=> ( ord_le5389487502678872194c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_920_subset__insertI2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,B4: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B4 @ B ) ) ) ).
% subset_insertI2
thf(fact_921_subset__insertI2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,B4: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ B4 @ B ) ) ) ).
% subset_insertI2
thf(fact_922_subset__insertI,axiom,
! [B: set_Epistemic_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A2 @ B ) ) ).
% subset_insertI
thf(fact_923_subset__insertI,axiom,
! [B: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] : ( ord_le5389487502678872194c_fm_i @ B @ ( insert7698009978809854162c_fm_i @ A2 @ B ) ) ).
% subset_insertI
thf(fact_924_subset__insert,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_925_subset__insert,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ B ) )
= ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ) ).
% subset_insert
thf(fact_926_subset__insert,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ B ) )
= ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ) ).
% subset_insert
thf(fact_927_insert__mono,axiom,
! [C2: set_Epistemic_fm_i,D: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ C2 ) @ ( insert7817948997695205106c_fm_i @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_928_insert__mono,axiom,
! [C2: set_se3485332733965609186c_fm_i,D: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C2 @ D )
=> ( ord_le5389487502678872194c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ C2 ) @ ( insert7698009978809854162c_fm_i @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_929_set__ConsD,axiom,
! [Y: epistemic_fm_i,X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member6642669606046002379c_fm_i @ Y @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_930_set__ConsD,axiom,
! [Y: nat,X3: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_931_set__ConsD,axiom,
! [Y: set_Epistemic_fm_i,X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ Y @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member1104366573291651755c_fm_i @ Y @ ( set_se200842218512397079c_fm_i @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_932_list_Oset__cases,axiom,
! [E: epistemic_fm_i,A2: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ E @ ( set_Epistemic_fm_i2 @ A2 ) )
=> ( ! [Z22: list_Epistemic_fm_i] :
( A2
!= ( cons_Epistemic_fm_i @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_i,Z22: list_Epistemic_fm_i] :
( ( A2
= ( cons_Epistemic_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( set_Epistemic_fm_i2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_933_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z22: list_nat] :
( A2
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_934_list_Oset__cases,axiom,
! [E: set_Epistemic_fm_i,A2: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ E @ ( set_se200842218512397079c_fm_i @ A2 ) )
=> ( ! [Z22: list_s8081015415394010888c_fm_i] :
( A2
!= ( cons_s4962720389763977656c_fm_i @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_i,Z22: list_s8081015415394010888c_fm_i] :
( ( A2
= ( cons_s4962720389763977656c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( set_se200842218512397079c_fm_i @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_935_list_Oset__intros_I1_J,axiom,
! [X21: epistemic_fm_i,X22: list_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X21 @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_936_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_937_list_Oset__intros_I1_J,axiom,
! [X21: set_Epistemic_fm_i,X22: list_s8081015415394010888c_fm_i] : ( member1104366573291651755c_fm_i @ X21 @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_938_list_Oset__intros_I2_J,axiom,
! [Y: epistemic_fm_i,X22: list_Epistemic_fm_i,X21: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y @ ( set_Epistemic_fm_i2 @ X22 ) )
=> ( member6642669606046002379c_fm_i @ Y @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_939_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_940_list_Oset__intros_I2_J,axiom,
! [Y: set_Epistemic_fm_i,X22: list_s8081015415394010888c_fm_i,X21: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y @ ( set_se200842218512397079c_fm_i @ X22 ) )
=> ( member1104366573291651755c_fm_i @ Y @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_941_Int__insert__right,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_942_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_943_Int__insert__right,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_944_Int__insert__left,axiom,
! [A2: epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_945_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_946_Int__insert__left,axiom,
! [A2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B ) @ C2 )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_947_finite_OemptyI,axiom,
finite3304564979551393739c_fm_i @ bot_bo4194595901900360558c_fm_i ).
% finite.emptyI
thf(fact_948_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_949_infinite__imp__nonempty,axiom,
! [S: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ( S != bot_bo4194595901900360558c_fm_i ) ) ).
% infinite_imp_nonempty
thf(fact_950_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_951_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_952_subset__emptyI,axiom,
! [A: set_Epistemic_fm_i] :
( ! [X2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) ) ).
% subset_emptyI
thf(fact_953_subset__emptyI,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [X2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A @ bot_bo145720340923748686c_fm_i ) ) ).
% subset_emptyI
thf(fact_954_filter_Osimps_I2_J,axiom,
! [P2: epistemic_fm_i > $o,X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( P2 @ X3 )
=> ( ( filter7636273843821131039c_fm_i @ P2 @ ( cons_Epistemic_fm_i @ X3 @ Xs ) )
= ( cons_Epistemic_fm_i @ X3 @ ( filter7636273843821131039c_fm_i @ P2 @ Xs ) ) ) )
& ( ~ ( P2 @ X3 )
=> ( ( filter7636273843821131039c_fm_i @ P2 @ ( cons_Epistemic_fm_i @ X3 @ Xs ) )
= ( filter7636273843821131039c_fm_i @ P2 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_955_filter_Osimps_I2_J,axiom,
! [P2: nat > $o,X3: nat,Xs: list_nat] :
( ( ( P2 @ X3 )
=> ( ( filter_nat @ P2 @ ( cons_nat @ X3 @ Xs ) )
= ( cons_nat @ X3 @ ( filter_nat @ P2 @ Xs ) ) ) )
& ( ~ ( P2 @ X3 )
=> ( ( filter_nat @ P2 @ ( cons_nat @ X3 @ Xs ) )
= ( filter_nat @ P2 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_956_filter_Osimps_I2_J,axiom,
! [P2: set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( ( P2 @ X3 )
=> ( ( filter3188398074982218495c_fm_i @ P2 @ ( cons_s4962720389763977656c_fm_i @ X3 @ Xs ) )
= ( cons_s4962720389763977656c_fm_i @ X3 @ ( filter3188398074982218495c_fm_i @ P2 @ Xs ) ) ) )
& ( ~ ( P2 @ X3 )
=> ( ( filter3188398074982218495c_fm_i @ P2 @ ( cons_s4962720389763977656c_fm_i @ X3 @ Xs ) )
= ( filter3188398074982218495c_fm_i @ P2 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_957_disjoint__iff__not__equal,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ! [Y4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y4 @ B )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_958_Int__empty__right,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ bot_bo145720340923748686c_fm_i )
= bot_bo145720340923748686c_fm_i ) ).
% Int_empty_right
thf(fact_959_Int__empty__left,axiom,
! [B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ bot_bo145720340923748686c_fm_i @ B )
= bot_bo145720340923748686c_fm_i ) ).
% Int_empty_left
thf(fact_960_disjoint__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
=> ~ ( member6642669606046002379c_fm_i @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_961_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_962_disjoint__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ~ ( member1104366573291651755c_fm_i @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_963_Int__emptyI,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ~ ( member6642669606046002379c_fm_i @ X2 @ B ) )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i ) ) ).
% Int_emptyI
thf(fact_964_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_965_Int__emptyI,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ~ ( member1104366573291651755c_fm_i @ X2 @ B ) )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i ) ) ).
% Int_emptyI
thf(fact_966_Un__empty__left,axiom,
! [B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ B )
= B ) ).
% Un_empty_left
thf(fact_967_Un__empty__right,axiom,
! [A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= A ) ).
% Un_empty_right
thf(fact_968_insert__def,axiom,
( insert7698009978809854162c_fm_i
= ( ^ [A5: set_Epistemic_fm_i] :
( sup_su2582925890723967158c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( X = A5 ) ) ) ) ) ).
% insert_def
thf(fact_969_insert__def,axiom,
( insert_nat
= ( ^ [A5: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X: nat] : ( X = A5 ) ) ) ) ) ).
% insert_def
thf(fact_970_insert__def,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A5: epistemic_fm_i] :
( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : ( X = A5 ) ) ) ) ) ).
% insert_def
thf(fact_971_maximal__def,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S )
= ( ! [P: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ P @ S ) ) ) ) ).
% maximal_def
thf(fact_972_set__subset__Cons,axiom,
! [Xs: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_973_set__subset__Cons,axiom,
! [Xs: list_Epistemic_fm_i,X3: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_974_set__subset__Cons,axiom,
! [Xs: list_s8081015415394010888c_fm_i,X3: set_Epistemic_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_975_finite__has__maximal,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ? [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_976_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_977_finite__has__maximal,axiom,
! [A: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ? [X2: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X2 @ A )
& ! [Xa: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ Xa @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_978_finite__has__minimal,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ? [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_979_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_980_finite__has__minimal,axiom,
! [A: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ? [X2: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X2 @ A )
& ! [Xa: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ Xa @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_981_consistent__consequent,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( member6642669606046002379c_fm_i @ P4 @ V )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ Q3 ) )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q3 @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_consequent
thf(fact_982_imply_Osimps_I2_J,axiom,
! [P4: epistemic_fm_i,Ps: list_Epistemic_fm_i,Q3: epistemic_fm_i] :
( ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ Ps ) @ Q3 )
= ( epistemic_Imp_i @ P4 @ ( epistemic_imply_i @ Ps @ Q3 ) ) ) ).
% imply.simps(2)
thf(fact_983_K__imply__head,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Ps: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ Ps ) @ P4 ) ) ).
% K_imply_head
thf(fact_984_K__imply__Cons,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,Q3: epistemic_fm_i,P4: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q3 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ Ps ) @ Q3 ) ) ) ).
% K_imply_Cons
thf(fact_985_K__swap,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Q3: epistemic_fm_i,G: list_Epistemic_fm_i,R2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ ( cons_Epistemic_fm_i @ Q3 @ G ) ) @ R2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ Q3 @ ( cons_Epistemic_fm_i @ P4 @ G ) ) @ R2 ) ) ) ).
% K_swap
thf(fact_986_consistent__consequent_H,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( member6642669606046002379c_fm_i @ P4 @ V )
=> ( ! [G4: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G4 @ H3 @ ( epistemic_Imp_i @ P4 @ Q3 ) )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q3 @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_consequent'
thf(fact_987_conjunct_Osimps_I2_J,axiom,
! [P4: epistemic_fm_i,Ps: list_Epistemic_fm_i] :
( ( stalnaker_conjunct_i @ ( cons_Epistemic_fm_i @ P4 @ Ps ) )
= ( epistemic_Con_i @ P4 @ ( stalnaker_conjunct_i @ Ps ) ) ) ).
% conjunct.simps(2)
thf(fact_988_inconsistent__subset,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P4 @ bot_bo4194595901900360558c_fm_i ) @ V ) )
=> ~ ! [V2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ V2 ) @ V )
=> ~ ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ V2 ) @ epistemic_FF_i ) ) ) ) ) ).
% inconsistent_subset
thf(fact_989_K__distrib__K__imp,axiom,
! [A: epistemic_fm_i > $o,I: i,G: list_Epistemic_fm_i,Q3: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_K_i @ I @ ( epistemic_imply_i @ G @ Q3 ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ I ) @ G ) @ ( epistemic_K_i @ I @ Q3 ) ) ) ) ).
% K_distrib_K_imp
thf(fact_990_K__ImpI,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,G: list_Epistemic_fm_i,Q3: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ G ) @ Q3 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ ( epistemic_Imp_i @ P4 @ Q3 ) ) ) ) ).
% K_ImpI
thf(fact_991_K__mp,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Q3: epistemic_fm_i,G: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ ( cons_Epistemic_fm_i @ ( epistemic_Imp_i @ P4 @ Q3 ) @ G ) ) @ Q3 ) ) ).
% K_mp
thf(fact_992_exists__finite__inconsistent,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,V: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ V ) )
=> ~ ! [W: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ W ) @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ V ) )
=> ( ~ ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ W )
=> ( ( finite3304564979551393739c_fm_i @ W )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ W ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_993_finite__ranking__induct,axiom,
! [S: set_se3485332733965609186c_fm_i,P2: set_se3485332733965609186c_fm_i > $o,F: set_Epistemic_fm_i > nat] :
( ( finite7933139204641697195c_fm_i @ S )
=> ( ( P2 @ bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i,S5: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ S5 )
=> ( ! [Y5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y5 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P2 @ S5 )
=> ( P2 @ ( insert7698009978809854162c_fm_i @ X2 @ S5 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_994_finite__ranking__induct,axiom,
! [S: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o,F: epistemic_fm_i > nat] :
( ( finite3304564979551393739c_fm_i @ S )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X2: epistemic_fm_i,S5: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ S5 )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P2 @ S5 )
=> ( P2 @ ( insert7817948997695205106c_fm_i @ X2 @ S5 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_995_finite__ranking__induct,axiom,
! [S: set_nat,P2: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X2: nat,S5: set_nat] :
( ( finite_finite_nat @ S5 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P2 @ S5 )
=> ( P2 @ ( insert_nat @ X2 @ S5 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_996_extendS__def,axiom,
! [A: epistemic_fm_i > $o,N: epistemic_fm_i,Prev: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S @ N @ Prev )
= ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) ) )
& ( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S @ N @ Prev )
= Prev ) ) ) ).
% extendS_def
thf(fact_997_MCS_Omaximal__def,axiom,
! [Consistent: set_se3485332733965609186c_fm_i > $o,S: set_se3485332733965609186c_fm_i] :
( ( maxima5540402109999912942c_fm_i @ Consistent )
=> ( ( maxima7741914846562630577c_fm_i @ Consistent @ S )
= ( ! [P: set_Epistemic_fm_i] :
( ( Consistent @ ( sup_su2582925890723967158c_fm_i @ ( insert7698009978809854162c_fm_i @ P @ bot_bo145720340923748686c_fm_i ) @ S ) )
=> ( member1104366573291651755c_fm_i @ P @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_998_MCS_Omaximal__def,axiom,
! [Consistent: set_nat > $o,S: set_nat] :
( ( maxima5350474959086842340CS_nat @ Consistent )
=> ( ( maxima3892657229671227617al_nat @ Consistent @ S )
= ( ! [P: nat] :
( ( Consistent @ ( sup_sup_set_nat @ ( insert_nat @ P @ bot_bot_set_nat ) @ S ) )
=> ( member_nat @ P @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_999_MCS_Omaximal__def,axiom,
! [Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ( ( maxima3264069618988350929c_fm_i @ Consistent @ S )
= ( ! [P: epistemic_fm_i] :
( ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ P @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_1000_bot__set__def,axiom,
( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ bot_bo2580527446789968623fm_i_o ) ) ).
% bot_set_def
thf(fact_1001_bot__set__def,axiom,
( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ bot_bo6089404950617257231fm_i_o ) ) ).
% bot_set_def
thf(fact_1002_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1003_bot__empty__eq,axiom,
( bot_bo2580527446789968623fm_i_o
= ( ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% bot_empty_eq
thf(fact_1004_bot__empty__eq,axiom,
( bot_bo6089404950617257231fm_i_o
= ( ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ bot_bo145720340923748686c_fm_i ) ) ) ).
% bot_empty_eq
thf(fact_1005_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1006_MCS__axioms,axiom,
! [A: epistemic_fm_i > $o] : ( maxima1924290493700099598c_fm_i @ ( episte2285483198712856234tent_i @ A ) ) ).
% MCS_axioms
thf(fact_1007_arg__min__least,axiom,
! [S: set_se3485332733965609186c_fm_i,Y: set_Epistemic_fm_i,F: set_Epistemic_fm_i > nat] :
( ( finite7933139204641697195c_fm_i @ S )
=> ( ( S != bot_bo145720340923748686c_fm_i )
=> ( ( member1104366573291651755c_fm_i @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic8104409813523524473_i_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1008_arg__min__least,axiom,
! [S: set_Epistemic_fm_i,Y: epistemic_fm_i,F: epistemic_fm_i > nat] :
( ( finite3304564979551393739c_fm_i @ S )
=> ( ( S != bot_bo4194595901900360558c_fm_i )
=> ( ( member6642669606046002379c_fm_i @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7487310264250698137_i_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1009_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1010_MCS__Lim__Ord_OextendS__def,axiom,
! [R2: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,N: epistemic_fm_i,Prev: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R2 @ Consistent )
=> ( ( ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ Consistent @ S @ N @ Prev )
= ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) ) )
& ( ~ ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ Consistent @ S @ N @ Prev )
= Prev ) ) ) ) ).
% MCS_Lim_Ord.extendS_def
thf(fact_1011_MCS__Lim__Ord_Oconsistent__hereditary,axiom,
! [R2: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i,S2: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R2 @ Consistent )
=> ( ( Consistent @ S )
=> ( ( ord_le3843937902494030498c_fm_i @ S2 @ S )
=> ( Consistent @ S2 ) ) ) ) ).
% MCS_Lim_Ord.consistent_hereditary
thf(fact_1012_MCS__Lim__Ord_Oconsistent__hereditary,axiom,
! [R2: set_Pr3972831103112126087c_fm_i,Consistent: set_se3485332733965609186c_fm_i > $o,S: set_se3485332733965609186c_fm_i,S2: set_se3485332733965609186c_fm_i] :
( ( maxima9073963094320882813c_fm_i @ R2 @ Consistent )
=> ( ( Consistent @ S )
=> ( ( ord_le5389487502678872194c_fm_i @ S2 @ S )
=> ( Consistent @ S2 ) ) ) ) ).
% MCS_Lim_Ord.consistent_hereditary
thf(fact_1013_is__singletonI_H,axiom,
! [A: set_Epistemic_fm_i] :
( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ! [X2: epistemic_fm_i,Y2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ( member6642669606046002379c_fm_i @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_sin4923117429174228502c_fm_i @ A ) ) ) ).
% is_singletonI'
thf(fact_1014_is__singletonI_H,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ( member1104366573291651755c_fm_i @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_sin2358732406619996150c_fm_i @ A ) ) ) ).
% is_singletonI'
thf(fact_1015_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_1016_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R2: set_Pr1261947904930325089at_nat,Consistent: set_nat > $o,S: set_nat] :
( ( maxima4501515429357695765rd_nat @ R2 @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S3: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ S )
& ( finite_finite_nat @ S3 )
& ~ ( Consistent @ S3 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_1017_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R2: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R2 @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S3 @ S )
& ( finite3304564979551393739c_fm_i @ S3 )
& ~ ( Consistent @ S3 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_1018_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R2: set_Pr3972831103112126087c_fm_i,Consistent: set_se3485332733965609186c_fm_i > $o,S: set_se3485332733965609186c_fm_i] :
( ( maxima9073963094320882813c_fm_i @ R2 @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ S3 @ S )
& ( finite7933139204641697195c_fm_i @ S3 )
& ~ ( Consistent @ S3 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_1019_set__rec,axiom,
( set_Epistemic_fm_i2
= ( rec_li5834409330748533462c_fm_i @ bot_bo4194595901900360558c_fm_i
@ ^ [X: epistemic_fm_i,Uu2: list_Epistemic_fm_i] : ( insert7817948997695205106c_fm_i @ X ) ) ) ).
% set_rec
thf(fact_1020_set__rec,axiom,
( set_nat2
= ( rec_list_set_nat_nat @ bot_bot_set_nat
@ ^ [X: nat,Uu2: list_nat] : ( insert_nat @ X ) ) ) ).
% set_rec
thf(fact_1021_set__rec,axiom,
( set_se200842218512397079c_fm_i
= ( rec_li386023649751468758c_fm_i @ bot_bo145720340923748686c_fm_i
@ ^ [X: set_Epistemic_fm_i,Uu2: list_s8081015415394010888c_fm_i] : ( insert7698009978809854162c_fm_i @ X ) ) ) ).
% set_rec
thf(fact_1022_Sup__fin_Oinsert,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A ) )
= ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1023_Sup__fin_Oinsert,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1024_Inf__fin_Oinsert,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A ) )
= ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1025_Inf__fin_Oinsert,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X3 @ A ) )
= ( inf_in161960956874937808c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1026_consistent__disjuncts,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P4: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Dis_i @ P4 @ Q3 ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P4 @ bot_bo4194595901900360558c_fm_i ) @ V ) )
| ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q3 @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_disjuncts
thf(fact_1027_inf__Sup__absorb,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ ( lattic3744900211830597177c_fm_i @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_1028_inf__Sup__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_1029_inf__Sup__absorb,axiom,
! [A: set_se7339729205154126530c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A2 @ ( lattic5920028816461561881c_fm_i @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_1030_sup__Inf__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_1031_sup__Inf__absorb,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_1032_eval_Osimps_I3_J,axiom,
! [G3: list_char > $o,H2: epistemic_fm_i > $o,P4: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( epistemic_eval_i @ G3 @ H2 @ ( epistemic_Dis_i @ P4 @ Q3 ) )
= ( ( epistemic_eval_i @ G3 @ H2 @ P4 )
| ( epistemic_eval_i @ G3 @ H2 @ Q3 ) ) ) ).
% eval.simps(3)
thf(fact_1033_fm_Odistinct_I19_J,axiom,
! [X31: epistemic_fm_i,X32: epistemic_fm_i,X41: epistemic_fm_i,X42: epistemic_fm_i] :
( ( epistemic_Dis_i @ X31 @ X32 )
!= ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.distinct(19)
thf(fact_1034_fm_Odistinct_I3_J,axiom,
! [X31: epistemic_fm_i,X32: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_Dis_i @ X31 @ X32 ) ) ).
% fm.distinct(3)
thf(fact_1035_fm_Odistinct_I23_J,axiom,
! [X31: epistemic_fm_i,X32: epistemic_fm_i,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Dis_i @ X31 @ X32 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(23)
thf(fact_1036_fm_Odistinct_I21_J,axiom,
! [X31: epistemic_fm_i,X32: epistemic_fm_i,X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( epistemic_Dis_i @ X31 @ X32 )
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(21)
thf(fact_1037_Inf__fin__le__Sup__fin,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1038_Inf__fin__le__Sup__fin,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1039_Inf__fin__le__Sup__fin,axiom,
! [A: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ord_le5389487502678872194c_fm_i @ ( lattic7860428167660714879c_fm_i @ A ) @ ( lattic5920028816461561881c_fm_i @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1040_Sup__fin_OcoboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1041_Sup__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1042_Sup__fin_OcoboundedI,axiom,
! [A: set_se7339729205154126530c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ A2 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ ( lattic5920028816461561881c_fm_i @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1043_Inf__fin_OcoboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1044_Inf__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1045_Inf__fin_OcoboundedI,axiom,
! [A: set_se7339729205154126530c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ A2 @ A )
=> ( ord_le5389487502678872194c_fm_i @ ( lattic7860428167660714879c_fm_i @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1046_Inf__fin_Oin__idem,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ X3 @ ( lattic8495942116038605215c_fm_i @ A ) )
= ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1047_Inf__fin_Oin__idem,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X3 @ A )
=> ( ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1048_Inf__fin_Oin__idem,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ X3 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ A ) )
= ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1049_Sup__fin_Oin__idem,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X3 @ A )
=> ( ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1050_Sup__fin_Oin__idem,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ X3 @ ( lattic3744900211830597177c_fm_i @ A ) )
= ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1051_Inf__fin_OboundedE,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ ( lattic8495942116038605215c_fm_i @ A ) )
=> ! [A8: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A8 @ A )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1052_Inf__fin_OboundedE,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A )
=> ( ord_less_eq_nat @ X3 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1053_Inf__fin_OboundedE,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ A ) )
=> ! [A8: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ A8 @ A )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1054_Inf__fin_OboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [A6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ A6 ) )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1055_Inf__fin_OboundedI,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( ord_less_eq_nat @ X3 @ A6 ) )
=> ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1056_Inf__fin_OboundedI,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ! [A6: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ A6 @ A )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ A6 ) )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1057_Sup__fin_OboundedE,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X3 )
=> ! [A8: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A8 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A8 @ X3 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1058_Sup__fin_OboundedE,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X3 )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A )
=> ( ord_less_eq_nat @ A8 @ X3 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1059_Sup__fin_OboundedE,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ ( lattic5920028816461561881c_fm_i @ A ) @ X3 )
=> ! [A8: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ A8 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A8 @ X3 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1060_Sup__fin_OboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [A6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A6 @ X3 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X3 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1061_Sup__fin_OboundedI,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( ord_less_eq_nat @ A6 @ X3 ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X3 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1062_Sup__fin_OboundedI,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ! [A6: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ A6 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A6 @ X3 ) )
=> ( ord_le5389487502678872194c_fm_i @ ( lattic5920028816461561881c_fm_i @ A ) @ X3 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1063_Inf__fin_Obounded__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ ( lattic8495942116038605215c_fm_i @ A ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ X ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1064_Inf__fin_Obounded__iff,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1065_Inf__fin_Obounded__iff,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ A ) )
= ( ! [X: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ A )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ X ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1066_Sup__fin_Obounded__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X3 )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ( ord_le3843937902494030498c_fm_i @ X @ X3 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1067_Sup__fin_Obounded__iff,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X3 )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1068_Sup__fin_Obounded__iff,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ ( lattic5920028816461561881c_fm_i @ A ) @ X3 )
= ( ! [X: set_se3485332733965609186c_fm_i] :
( ( member1461078328125707403c_fm_i @ X @ A )
=> ( ord_le5389487502678872194c_fm_i @ X @ X3 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1069_K__DisE,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,G: list_Epistemic_fm_i,R2: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ G ) @ R2 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ Q3 @ G ) @ R2 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ ( epistemic_Dis_i @ P4 @ Q3 ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ R2 ) ) ) ) ) ).
% K_DisE
thf(fact_1070_K__DisL,axiom,
! [A: epistemic_fm_i > $o,P4: epistemic_fm_i,Ps: list_Epistemic_fm_i,Q3: epistemic_fm_i,P7: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P4 @ Ps ) @ Q3 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P7 @ Ps ) @ Q3 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ ( epistemic_Dis_i @ P4 @ P7 ) @ Ps ) @ Q3 ) ) ) ) ).
% K_DisL
thf(fact_1071_Inf__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1072_Inf__fin_Osubset__imp,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A @ B )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( finite9087983574947184523c_fm_i @ B )
=> ( ord_le5389487502678872194c_fm_i @ ( lattic7860428167660714879c_fm_i @ B ) @ ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1073_Inf__fin_Osubset__imp,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ B ) @ ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1074_Sup__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1075_Sup__fin_Osubset__imp,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( ord_le2772500130360476258c_fm_i @ A @ B )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( finite9087983574947184523c_fm_i @ B )
=> ( ord_le5389487502678872194c_fm_i @ ( lattic5920028816461561881c_fm_i @ A ) @ ( lattic5920028816461561881c_fm_i @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1076_Sup__fin_Osubset__imp,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1077_Inf__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1078_Inf__fin_Osubset,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( B != bot_bo4781621276559889198c_fm_i )
=> ( ( ord_le2772500130360476258c_fm_i @ B @ A )
=> ( ( inf_in161960956874937808c_fm_i @ ( lattic7860428167660714879c_fm_i @ B ) @ ( lattic7860428167660714879c_fm_i @ A ) )
= ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1079_Inf__fin_Osubset,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( B != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ ( lattic8495942116038605215c_fm_i @ B ) @ ( lattic8495942116038605215c_fm_i @ A ) )
= ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1080_Sup__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1081_Sup__fin_Osubset,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( B != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ ( lattic3744900211830597177c_fm_i @ B ) @ ( lattic3744900211830597177c_fm_i @ A ) )
= ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1082_Inf__fin_Oclosed,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X2 @ Y2 ) @ ( insert7698009978809854162c_fm_i @ X2 @ ( insert7698009978809854162c_fm_i @ Y2 @ bot_bo145720340923748686c_fm_i ) ) )
=> ( member1104366573291651755c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_1083_Inf__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : ( member_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( insert_nat @ X2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_1084_Inf__fin_Oclosed,axiom,
! [A: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ! [X2: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] : ( member1461078328125707403c_fm_i @ ( inf_in161960956874937808c_fm_i @ X2 @ Y2 ) @ ( insert7111703059596746418c_fm_i @ X2 @ ( insert7111703059596746418c_fm_i @ Y2 @ bot_bo4781621276559889198c_fm_i ) ) )
=> ( member1461078328125707403c_fm_i @ ( lattic7860428167660714879c_fm_i @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_1085_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( lattic8495942116038605215c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) )
= ( inf_in3450601097109690352c_fm_i @ X3 @ ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1086_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X3 @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A ) )
= ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1087_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ~ ( member1461078328125707403c_fm_i @ X3 @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X3 @ A ) )
= ( inf_in161960956874937808c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1088_Sup__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : ( member_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( insert_nat @ X2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1089_Sup__fin_Oclosed,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X2: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X2 @ Y2 ) @ ( insert7698009978809854162c_fm_i @ X2 @ ( insert7698009978809854162c_fm_i @ Y2 @ bot_bo145720340923748686c_fm_i ) ) )
=> ( member1104366573291651755c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1090_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X3 @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A ) )
= ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1091_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1092_Inf__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1093_Inf__fin_Ounion,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( finite9087983574947184523c_fm_i @ B )
=> ( ( B != bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( sup_su2317974602288960150c_fm_i @ A @ B ) )
= ( inf_in161960956874937808c_fm_i @ ( lattic7860428167660714879c_fm_i @ A ) @ ( lattic7860428167660714879c_fm_i @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1094_Sup__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1095_Sup__fin_Ounion,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ( B != bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) )
= ( sup_su1936195050962291414c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1096_fm_Oexhaust,axiom,
! [Y: epistemic_fm_i] :
( ( Y != epistemic_FF_i )
=> ( ! [X23: list_char] :
( Y
!= ( epistemic_Pro_i @ X23 ) )
=> ( ! [X312: epistemic_fm_i,X322: epistemic_fm_i] :
( Y
!= ( epistemic_Dis_i @ X312 @ X322 ) )
=> ( ! [X412: epistemic_fm_i,X422: epistemic_fm_i] :
( Y
!= ( epistemic_Con_i @ X412 @ X422 ) )
=> ( ! [X512: epistemic_fm_i,X522: epistemic_fm_i] :
( Y
!= ( epistemic_Imp_i @ X512 @ X522 ) )
=> ~ ! [X612: i,X622: epistemic_fm_i] :
( Y
!= ( epistemic_K_i @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_1097_subset__singleton__iff__Uniq,axiom,
! [A: set_nat] :
( ( ? [A5: nat] : ( ord_less_eq_set_nat @ A @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) )
= ( uniq_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1098_subset__singleton__iff__Uniq,axiom,
! [A: set_Epistemic_fm_i] :
( ( ? [A5: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A5 @ bot_bo4194595901900360558c_fm_i ) ) )
= ( uniq_Epistemic_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1099_subset__singleton__iff__Uniq,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ? [A5: set_Epistemic_fm_i] : ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A5 @ bot_bo145720340923748686c_fm_i ) ) )
= ( uniq_s5891897140743797767c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1100_fm_Odistinct_I15_J,axiom,
! [X24: list_char,X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( epistemic_Pro_i @ X24 )
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_1101_fm_Odistinct_I17_J,axiom,
! [X24: list_char,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Pro_i @ X24 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_1102_fm_Odistinct_I1_J,axiom,
! [X24: list_char] :
( epistemic_FF_i
!= ( epistemic_Pro_i @ X24 ) ) ).
% fm.distinct(1)
thf(fact_1103_fm_Odistinct_I13_J,axiom,
! [X24: list_char,X41: epistemic_fm_i,X42: epistemic_fm_i] :
( ( epistemic_Pro_i @ X24 )
!= ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_1104_eval_Osimps_I2_J,axiom,
! [G3: list_char > $o,Uw: epistemic_fm_i > $o,X3: list_char] :
( ( epistemic_eval_i @ G3 @ Uw @ ( epistemic_Pro_i @ X3 ) )
= ( G3 @ X3 ) ) ).
% eval.simps(2)
thf(fact_1105_fm_Orel__induct,axiom,
! [R: i > i > $o,X3: epistemic_fm_i,Y: epistemic_fm_i,Q: epistemic_fm_i > epistemic_fm_i > $o] :
( ( epistemic_rel_fm_i_i @ R @ X3 @ Y )
=> ( ( Q @ epistemic_FF_i @ epistemic_FF_i )
=> ( ! [A22: list_char,B22: list_char] :
( ( A22 = B22 )
=> ( Q @ ( epistemic_Pro_i @ A22 ) @ ( epistemic_Pro_i @ B22 ) ) )
=> ( ! [A31: epistemic_fm_i,A32: epistemic_fm_i,B31: epistemic_fm_i,B32: epistemic_fm_i] :
( ( Q @ A31 @ B31 )
=> ( ( Q @ A32 @ B32 )
=> ( Q @ ( epistemic_Dis_i @ A31 @ A32 ) @ ( epistemic_Dis_i @ B31 @ B32 ) ) ) )
=> ( ! [A41: epistemic_fm_i,A42: epistemic_fm_i,B41: epistemic_fm_i,B42: epistemic_fm_i] :
( ( Q @ A41 @ B41 )
=> ( ( Q @ A42 @ B42 )
=> ( Q @ ( epistemic_Con_i @ A41 @ A42 ) @ ( epistemic_Con_i @ B41 @ B42 ) ) ) )
=> ( ! [A51: epistemic_fm_i,A52: epistemic_fm_i,B51: epistemic_fm_i,B52: epistemic_fm_i] :
( ( Q @ A51 @ B51 )
=> ( ( Q @ A52 @ B52 )
=> ( Q @ ( epistemic_Imp_i @ A51 @ A52 ) @ ( epistemic_Imp_i @ B51 @ B52 ) ) ) )
=> ( ! [A61: i,A62: epistemic_fm_i,B61: i,B62: epistemic_fm_i] :
( ( R @ A61 @ B61 )
=> ( ( Q @ A62 @ B62 )
=> ( Q @ ( epistemic_K_i @ A61 @ A62 ) @ ( epistemic_K_i @ B61 @ B62 ) ) ) )
=> ( Q @ X3 @ Y ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_1106_fm_Orel__cases,axiom,
! [R: i > i > $o,A2: epistemic_fm_i,B4: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R @ A2 @ B4 )
=> ( ( ( A2 = epistemic_FF_i )
=> ( B4 != epistemic_FF_i ) )
=> ( ! [X2: list_char] :
( ( A2
= ( epistemic_Pro_i @ X2 ) )
=> ! [Y2: list_char] :
( ( B4
= ( epistemic_Pro_i @ Y2 ) )
=> ( X2 != Y2 ) ) )
=> ( ! [X1: epistemic_fm_i,X2a: epistemic_fm_i] :
( ( A2
= ( epistemic_Dis_i @ X1 @ X2a ) )
=> ! [Y1: epistemic_fm_i,Y2a: epistemic_fm_i] :
( ( B4
= ( epistemic_Dis_i @ Y1 @ Y2a ) )
=> ( ( epistemic_rel_fm_i_i @ R @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_i_i @ R @ X2a @ Y2a ) ) ) )
=> ( ! [X1a: epistemic_fm_i,X2b: epistemic_fm_i] :
( ( A2
= ( epistemic_Con_i @ X1a @ X2b ) )
=> ! [Y1a: epistemic_fm_i,Y2b: epistemic_fm_i] :
( ( B4
= ( epistemic_Con_i @ Y1a @ Y2b ) )
=> ( ( epistemic_rel_fm_i_i @ R @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_i_i @ R @ X2b @ Y2b ) ) ) )
=> ( ! [X1b: epistemic_fm_i,X2c: epistemic_fm_i] :
( ( A2
= ( epistemic_Imp_i @ X1b @ X2c ) )
=> ! [Y1b: epistemic_fm_i,Y2c: epistemic_fm_i] :
( ( B4
= ( epistemic_Imp_i @ Y1b @ Y2c ) )
=> ( ( epistemic_rel_fm_i_i @ R @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_i_i @ R @ X2c @ Y2c ) ) ) )
=> ~ ! [X1c: i,X2d: epistemic_fm_i] :
( ( A2
= ( epistemic_K_i @ X1c @ X2d ) )
=> ! [Y1c: i,Y2d: epistemic_fm_i] :
( ( B4
= ( epistemic_K_i @ Y1c @ Y2d ) )
=> ( ( R @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_i_i @ R @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_1107_fm_Octr__transfer_I1_J,axiom,
! [R: i > i > $o] : ( epistemic_rel_fm_i_i @ R @ epistemic_FF_i @ epistemic_FF_i ) ).
% fm.ctr_transfer(1)
thf(fact_1108_fm_Orel__intros_I4_J,axiom,
! [R: i > i > $o,X41: epistemic_fm_i,Y41: epistemic_fm_i,X42: epistemic_fm_i,Y42: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R @ X41 @ Y41 )
=> ( ( epistemic_rel_fm_i_i @ R @ X42 @ Y42 )
=> ( epistemic_rel_fm_i_i @ R @ ( epistemic_Con_i @ X41 @ X42 ) @ ( epistemic_Con_i @ Y41 @ Y42 ) ) ) ) ).
% fm.rel_intros(4)
thf(fact_1109_fm_Orel__inject_I4_J,axiom,
! [R: i > i > $o,X41: epistemic_fm_i,X42: epistemic_fm_i,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R @ ( epistemic_Con_i @ X41 @ X42 ) @ ( epistemic_Con_i @ Y41 @ Y42 ) )
= ( ( epistemic_rel_fm_i_i @ R @ X41 @ Y41 )
& ( epistemic_rel_fm_i_i @ R @ X42 @ Y42 ) ) ) ).
% fm.rel_inject(4)
thf(fact_1110_fm_Orel__intros_I6_J,axiom,
! [R: i > i > $o,X61: i,Y61: i,X62: epistemic_fm_i,Y62: epistemic_fm_i] :
( ( R @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_i_i @ R @ X62 @ Y62 )
=> ( epistemic_rel_fm_i_i @ R @ ( epistemic_K_i @ X61 @ X62 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_1111_fm_Orel__inject_I6_J,axiom,
! [R: i > i > $o,X61: i,X62: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R @ ( epistemic_K_i @ X61 @ X62 ) @ ( epistemic_K_i @ Y61 @ Y62 ) )
= ( ( R @ X61 @ Y61 )
& ( epistemic_rel_fm_i_i @ R @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_1112_fm_Orel__intros_I5_J,axiom,
! [R: i > i > $o,X51: epistemic_fm_i,Y51: epistemic_fm_i,X52: epistemic_fm_i,Y52: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R @ X51 @ Y51 )
=> ( ( epistemic_rel_fm_i_i @ R @ X52 @ Y52 )
=> ( epistemic_rel_fm_i_i @ R @ ( epistemic_Imp_i @ X51 @ X52 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ) ) ).
% fm.rel_intros(5)
thf(fact_1113_fm_Orel__inject_I5_J,axiom,
! [R: i > i > $o,X51: epistemic_fm_i,X52: epistemic_fm_i,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R @ ( epistemic_Imp_i @ X51 @ X52 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) )
= ( ( epistemic_rel_fm_i_i @ R @ X51 @ Y51 )
& ( epistemic_rel_fm_i_i @ R @ X52 @ Y52 ) ) ) ).
% fm.rel_inject(5)
thf(fact_1114_fm_Orel__distinct_I30_J,axiom,
! [R: i > i > $o,Y61: i,Y62: epistemic_fm_i,X51: epistemic_fm_i,X52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ ( epistemic_K_i @ Y61 @ Y62 ) @ ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.rel_distinct(30)
thf(fact_1115_fm_Orel__distinct_I29_J,axiom,
! [R: i > i > $o,X51: epistemic_fm_i,X52: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ ( epistemic_Imp_i @ X51 @ X52 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(29)
thf(fact_1116_fm_Orel__distinct_I7_J,axiom,
! [R: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ epistemic_FF_i @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(7)
thf(fact_1117_fm_Orel__distinct_I8_J,axiom,
! [R: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ ( epistemic_Imp_i @ Y51 @ Y52 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(8)
thf(fact_1118_fm_Orel__distinct_I10_J,axiom,
! [R: i > i > $o,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ ( epistemic_K_i @ Y61 @ Y62 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(10)
thf(fact_1119_fm_Orel__distinct_I9_J,axiom,
! [R: i > i > $o,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ epistemic_FF_i @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(9)
thf(fact_1120_fm_Orel__distinct_I26_J,axiom,
! [R: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i,X41: epistemic_fm_i,X42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ ( epistemic_Imp_i @ Y51 @ Y52 ) @ ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.rel_distinct(26)
thf(fact_1121_fm_Orel__distinct_I25_J,axiom,
! [R: i > i > $o,X41: epistemic_fm_i,X42: epistemic_fm_i,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ ( epistemic_Con_i @ X41 @ X42 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(25)
thf(fact_1122_fm_Orel__distinct_I28_J,axiom,
! [R: i > i > $o,Y61: i,Y62: epistemic_fm_i,X41: epistemic_fm_i,X42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ ( epistemic_K_i @ Y61 @ Y62 ) @ ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.rel_distinct(28)
thf(fact_1123_fm_Orel__distinct_I27_J,axiom,
! [R: i > i > $o,X41: epistemic_fm_i,X42: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ ( epistemic_Con_i @ X41 @ X42 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(27)
thf(fact_1124_fm_Orel__distinct_I5_J,axiom,
! [R: i > i > $o,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ epistemic_FF_i @ ( epistemic_Con_i @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(5)
thf(fact_1125_fm_Orel__distinct_I6_J,axiom,
! [R: i > i > $o,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R @ ( epistemic_Con_i @ Y41 @ Y42 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(6)
thf(fact_1126_Sup__fin_Oinsert__remove,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A ) )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X3 @ A ) )
= ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1127_Sup__fin_Oinsert__remove,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) )
= X3 ) )
& ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
!= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( lattic3744900211830597177c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1128_Sup__fin_Oremove,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X3 @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1129_Sup__fin_Oremove,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ A )
= X3 ) )
& ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
!= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ A )
= ( sup_su1936195050962291414c_fm_i @ X3 @ ( lattic3744900211830597177c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1130_Inf__fin_Oremove,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( lattic8495942116038605215c_fm_i @ A )
= X3 ) )
& ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
!= bot_bo145720340923748686c_fm_i )
=> ( ( lattic8495942116038605215c_fm_i @ A )
= ( inf_in3450601097109690352c_fm_i @ X3 @ ( lattic8495942116038605215c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1131_Inf__fin_Oremove,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X3 @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1132_Inf__fin_Oremove,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ X3 @ A )
=> ( ( ( ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X3 @ bot_bo4781621276559889198c_fm_i ) )
= bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ A )
= X3 ) )
& ( ( ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X3 @ bot_bo4781621276559889198c_fm_i ) )
!= bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ A )
= ( inf_in161960956874937808c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X3 @ bot_bo4781621276559889198c_fm_i ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1133_Diff__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
& ~ ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1134_Diff__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
& ~ ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1135_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1136_DiffI,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1137_DiffI,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ C @ B )
=> ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1138_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1139_finite__Diff2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
= ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% finite_Diff2
thf(fact_1140_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_1141_finite__Diff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1142_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1143_Diff__insert0,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ B ) )
= ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1144_Diff__insert0,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ B ) )
= ( minus_1073301138667672009c_fm_i @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1145_Diff__insert0,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1146_insert__Diff1,axiom,
! [X3: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ B )
= ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1147_insert__Diff1,axiom,
! [X3: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ B )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ B )
= ( minus_1073301138667672009c_fm_i @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1148_insert__Diff1,axiom,
! [X3: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X3 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1149_Un__Diff__cancel2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ B @ A ) @ A )
= ( sup_su1936195050962291414c_fm_i @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_1150_Un__Diff__cancel,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ A ) )
= ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_1151_Diff__eq__empty__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1152_Diff__eq__empty__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( minus_1073301138667672009c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i )
= ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1153_finite__Diff__insert,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) ) )
= ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1154_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1155_Diff__disjoint,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ A ) )
= bot_bo145720340923748686c_fm_i ) ).
% Diff_disjoint
thf(fact_1156_Diff__infinite__finite,axiom,
! [T2: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ T2 )
=> ( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1157_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1158_Diff__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,D: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ D @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1159_Diff__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,D: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ C2 )
=> ( ( ord_le5389487502678872194c_fm_i @ D @ B )
=> ( ord_le5389487502678872194c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1160_Diff__subset,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1161_Diff__subset,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1162_double__diff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C2 )
=> ( ( minus_1543973939109951465c_fm_i @ B @ ( minus_1543973939109951465c_fm_i @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1163_double__diff,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ C2 )
=> ( ( minus_1073301138667672009c_fm_i @ B @ ( minus_1073301138667672009c_fm_i @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1164_insert__Diff__if,axiom,
! [X3: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ X3 @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ B )
= ( minus_1543973939109951465c_fm_i @ A @ B ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ X3 @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ B )
= ( insert7817948997695205106c_fm_i @ X3 @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1165_insert__Diff__if,axiom,
! [X3: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ X3 @ B )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ B )
= ( minus_1073301138667672009c_fm_i @ A @ B ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ X3 @ B )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ B )
= ( insert7698009978809854162c_fm_i @ X3 @ ( minus_1073301138667672009c_fm_i @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1166_insert__Diff__if,axiom,
! [X3: nat,B: set_nat,A: set_nat] :
( ( ( member_nat @ X3 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat @ X3 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( insert_nat @ X3 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1167_Int__Diff,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_1168_Diff__Int2,axiom,
! [A: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) )
= ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_1169_Diff__Diff__Int,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_1170_Diff__Int__distrib,axiom,
! [C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ C2 @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ C2 @ A ) @ ( inf_in161960956874937808c_fm_i @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_1171_Diff__Int__distrib2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ C2 )
= ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1172_Un__Diff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C2 )
= ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ C2 ) @ ( minus_1543973939109951465c_fm_i @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_1173_DiffD2,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
=> ~ ( member6642669606046002379c_fm_i @ C @ B ) ) ).
% DiffD2
thf(fact_1174_DiffD2,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
=> ~ ( member1104366573291651755c_fm_i @ C @ B ) ) ).
% DiffD2
thf(fact_1175_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_1176_DiffD1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
=> ( member6642669606046002379c_fm_i @ C @ A ) ) ).
% DiffD1
thf(fact_1177_DiffD1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
=> ( member1104366573291651755c_fm_i @ C @ A ) ) ).
% DiffD1
thf(fact_1178_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_1179_DiffE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% DiffE
thf(fact_1180_DiffE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
=> ~ ( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% DiffE
thf(fact_1181_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_1182_set__diff__eq,axiom,
( minus_1543973939109951465c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A3 )
& ~ ( member6642669606046002379c_fm_i @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1183_set__diff__eq,axiom,
( minus_1073301138667672009c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A3 )
& ~ ( member1104366573291651755c_fm_i @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1184_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ~ ( member_nat @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1185_diff__shunt__var,axiom,
! [X3: set_Epistemic_fm_i,Y: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ X3 @ Y )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ X3 @ Y ) ) ).
% diff_shunt_var
thf(fact_1186_diff__shunt__var,axiom,
! [X3: set_se3485332733965609186c_fm_i,Y: set_se3485332733965609186c_fm_i] :
( ( ( minus_1073301138667672009c_fm_i @ X3 @ Y )
= bot_bo145720340923748686c_fm_i )
= ( ord_le5389487502678872194c_fm_i @ X3 @ Y ) ) ).
% diff_shunt_var
thf(fact_1187_Diff__insert__absorb,axiom,
! [X3: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X3 @ A ) @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1188_Diff__insert__absorb,axiom,
! [X3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X3 @ A ) @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1189_Diff__insert__absorb,axiom,
! [X3: nat,A: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1190_insert__Diff,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( insert7817948997695205106c_fm_i @ A2 @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1191_insert__Diff,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( insert7698009978809854162c_fm_i @ A2 @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1192_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1193_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X3: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
& ~ ( member_nat @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1194_subset__Diff__insert,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X3: epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ X3 @ C2 ) ) )
= ( ( ord_le3843937902494030498c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ C2 ) )
& ~ ( member6642669606046002379c_fm_i @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1195_subset__Diff__insert,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ ( insert7698009978809854162c_fm_i @ X3 @ C2 ) ) )
= ( ( ord_le5389487502678872194c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ C2 ) )
& ~ ( member1104366573291651755c_fm_i @ X3 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1196_Diff__triv,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i )
=> ( ( minus_1073301138667672009c_fm_i @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1197_Int__Diff__disjoint,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= bot_bo145720340923748686c_fm_i ) ).
% Int_Diff_disjoint
thf(fact_1198_Diff__partition,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1199_Diff__partition,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( sup_su2582925890723967158c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1200_Diff__subset__conv,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ C2 )
= ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1201_Diff__subset__conv,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ C2 )
= ( ord_le5389487502678872194c_fm_i @ A @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1202_Diff__Un,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ A @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) )
= ( inf_in161960956874937808c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1203_Diff__Un,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1204_Diff__Int,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) )
= ( sup_su2582925890723967158c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1205_Diff__Int,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1206_Int__Diff__Un,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1207_Int__Diff__Un,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1208_Un__Diff__Int,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1209_Un__Diff__Int,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1210_finite__empty__induct,axiom,
! [A: set_se3485332733965609186c_fm_i,P2: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( P2 @ A )
=> ( ! [A6: set_Epistemic_fm_i,A7: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A7 )
=> ( ( member1104366573291651755c_fm_i @ A6 @ A7 )
=> ( ( P2 @ A7 )
=> ( P2 @ ( minus_1073301138667672009c_fm_i @ A7 @ ( insert7698009978809854162c_fm_i @ A6 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) )
=> ( P2 @ bot_bo145720340923748686c_fm_i ) ) ) ) ).
% finite_empty_induct
thf(fact_1211_finite__empty__induct,axiom,
! [A: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( P2 @ A )
=> ( ! [A6: epistemic_fm_i,A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( member6642669606046002379c_fm_i @ A6 @ A7 )
=> ( ( P2 @ A7 )
=> ( P2 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ A6 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ( P2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% finite_empty_induct
thf(fact_1212_finite__empty__induct,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P2 @ A )
=> ( ! [A6: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( member_nat @ A6 @ A7 )
=> ( ( P2 @ A7 )
=> ( P2 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1213_infinite__coinduct,axiom,
! [X4: set_Epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ( X4 @ A )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( X4 @ A7 )
=> ? [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A7 )
& ( ( X4 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) )
| ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ~ ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% infinite_coinduct
thf(fact_1214_infinite__coinduct,axiom,
! [X4: set_nat > $o,A: set_nat] :
( ( X4 @ A )
=> ( ! [A7: set_nat] :
( ( X4 @ A7 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A7 )
& ( ( X4 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_1215_infinite__remove,axiom,
! [S: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ S @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% infinite_remove
thf(fact_1216_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1217_subset__insert__iff,axiom,
! [A: set_nat,X3: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B ) )
= ( ( ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1218_subset__insert__iff,axiom,
! [A: set_Epistemic_fm_i,X3: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ B ) )
= ( ( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) @ B ) )
& ( ~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1219_subset__insert__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ B ) )
= ( ( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ord_le5389487502678872194c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) @ B ) )
& ( ~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1220_Diff__single__insert,axiom,
! [A: set_Epistemic_fm_i,X3: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) @ B )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1221_Diff__single__insert,axiom,
! [A: set_se3485332733965609186c_fm_i,X3: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) @ B )
=> ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X3 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1222_minus__coset__filter,axiom,
! [A: set_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( coset_Epistemic_fm_i @ Xs ) )
= ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_1223_minus__coset__filter,axiom,
! [A: set_nat,Xs: list_nat] :
( ( minus_minus_set_nat @ A @ ( coset_nat @ Xs ) )
= ( set_nat2
@ ( filter_nat
@ ^ [X: nat] : ( member_nat @ X @ A )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_1224_minus__coset__filter,axiom,
! [A: set_se3485332733965609186c_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ A @ ( coset_1342939534705115381c_fm_i @ Xs ) )
= ( set_se200842218512397079c_fm_i
@ ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_1225_finite__remove__induct,axiom,
! [B: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( P2 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1226_finite__remove__induct,axiom,
! [B: set_Epistemic_fm_i,P2: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( A7 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A7 @ B )
=> ( ! [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A7 )
=> ( P2 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1227_finite__remove__induct,axiom,
! [B: set_se3485332733965609186c_fm_i,P2: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ B )
=> ( ( P2 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A7: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A7 )
=> ( ( A7 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ A7 @ B )
=> ( ! [X6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X6 @ A7 )
=> ( P2 @ ( minus_1073301138667672009c_fm_i @ A7 @ ( insert7698009978809854162c_fm_i @ X6 @ bot_bo145720340923748686c_fm_i ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1228_remove__induct,axiom,
! [P2: set_nat > $o,B: set_nat] :
( ( P2 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P2 @ B ) )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( P2 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_1229_remove__induct,axiom,
! [P2: set_Epistemic_fm_i > $o,B: set_Epistemic_fm_i] :
( ( P2 @ bot_bo4194595901900360558c_fm_i )
=> ( ( ~ ( finite3304564979551393739c_fm_i @ B )
=> ( P2 @ B ) )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( A7 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A7 @ B )
=> ( ! [X6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X6 @ A7 )
=> ( P2 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X6 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_1230_remove__induct,axiom,
! [P2: set_se3485332733965609186c_fm_i > $o,B: set_se3485332733965609186c_fm_i] :
( ( P2 @ bot_bo145720340923748686c_fm_i )
=> ( ( ~ ( finite7933139204641697195c_fm_i @ B )
=> ( P2 @ B ) )
=> ( ! [A7: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A7 )
=> ( ( A7 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ A7 @ B )
=> ( ! [X6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X6 @ A7 )
=> ( P2 @ ( minus_1073301138667672009c_fm_i @ A7 @ ( insert7698009978809854162c_fm_i @ X6 @ bot_bo145720340923748686c_fm_i ) ) ) )
=> ( P2 @ A7 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_1231_set__minus__filter__out,axiom,
! [Xs: list_Epistemic_fm_i,Y: epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( insert7817948997695205106c_fm_i @ Y @ bot_bo4194595901900360558c_fm_i ) )
= ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [X: epistemic_fm_i] : ( X != Y )
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_1232_set__minus__filter__out,axiom,
! [Xs: list_nat,Y: nat] :
( ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
= ( set_nat2
@ ( filter_nat
@ ^ [X: nat] : ( X != Y )
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_1233_set__minus__filter__out,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Y: set_Epistemic_fm_i] :
( ( minus_1073301138667672009c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( insert7698009978809854162c_fm_i @ Y @ bot_bo145720340923748686c_fm_i ) )
= ( set_se200842218512397079c_fm_i
@ ( filter3188398074982218495c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( X != Y )
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_1234_Inf__fin_Oinsert__remove,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A ) )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X3 @ A ) )
= ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1235_Inf__fin_Oinsert__remove,axiom,
! [A: set_se7339729205154126530c_fm_i,X3: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( ( ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X3 @ bot_bo4781621276559889198c_fm_i ) )
= bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X3 @ A ) )
= X3 ) )
& ( ( ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X3 @ bot_bo4781621276559889198c_fm_i ) )
!= bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X3 @ A ) )
= ( inf_in161960956874937808c_fm_i @ X3 @ ( lattic7860428167660714879c_fm_i @ ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X3 @ bot_bo4781621276559889198c_fm_i ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1236_set__removeAll,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( remove817376155714978094c_fm_i @ X3 @ Xs ) )
= ( minus_1543973939109951465c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% set_removeAll
thf(fact_1237_set__removeAll,axiom,
! [X3: nat,Xs: list_nat] :
( ( set_nat2 @ ( removeAll_nat @ X3 @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% set_removeAll
thf(fact_1238_set__removeAll,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ ( remove6226062834164189966c_fm_i @ X3 @ Xs ) )
= ( minus_1073301138667672009c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) ) ) ).
% set_removeAll
thf(fact_1239_removeAll__id,axiom,
! [X3: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remove817376155714978094c_fm_i @ X3 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1240_removeAll__id,axiom,
! [X3: nat,Xs: list_nat] :
( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X3 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1241_removeAll__id,axiom,
! [X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X3 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( remove6226062834164189966c_fm_i @ X3 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1242_removeAll__filter__not,axiom,
! [P2: nat > $o,X3: nat,Xs: list_nat] :
( ~ ( P2 @ X3 )
=> ( ( removeAll_nat @ X3 @ ( filter_nat @ P2 @ Xs ) )
= ( filter_nat @ P2 @ Xs ) ) ) ).
% removeAll_filter_not
thf(fact_1243_removeAll__filter__not,axiom,
! [P2: set_Epistemic_fm_i > $o,X3: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ~ ( P2 @ X3 )
=> ( ( remove6226062834164189966c_fm_i @ X3 @ ( filter3188398074982218495c_fm_i @ P2 @ Xs ) )
= ( filter3188398074982218495c_fm_i @ P2 @ Xs ) ) ) ).
% removeAll_filter_not
thf(fact_1244_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1245_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1246_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B4: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B4 ) )
=> ? [X2: nat] :
( ( P2 @ X2 )
& ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1247_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1248_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1249_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1250_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1251_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1252_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1253_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B4: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B4 ) )
= ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1254_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1255_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1256_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1257_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1258_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1259_finite__less__ub,axiom,
! [F: nat > nat,U3: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U3 ) ) ) ) ).
% finite_less_ub
thf(fact_1260_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_eq_nat @ X @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1261_bounded__Max__nat,axiom,
! [P2: nat > $o,X3: nat,M3: nat] :
( ( P2 @ X3 )
=> ( ! [X2: nat] :
( ( P2 @ X2 )
=> ( ord_less_eq_nat @ X2 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P2 @ M4 )
=> ~ ! [X6: nat] :
( ( P2 @ X6 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1262_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1263_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X2 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1264_nat__induct__at__least,axiom,
! [M: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P2 @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1265_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P2 @ M5 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_1266_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1267_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1268_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1269_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1270_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1271_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1272_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1273_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1274_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1275_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1276_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
% Conjectures (1)
thf(conj_0,conjecture,
( ord_le3843937902494030498c_fm_i
@ ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [P: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ u ) ) ) ).
%------------------------------------------------------------------------------