TPTP Problem File: SLH0597^1.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_00126_003765__6137918_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1571 ( 552 unt; 293 typ; 0 def)
% Number of atoms : 4130 (1195 equ; 0 cnn)
% Maximal formula atoms : 22 ( 3 avg)
% Number of connectives : 13374 ( 480 ~; 82 |; 303 &;10563 @)
% ( 0 <=>;1946 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 36 ( 35 usr)
% Number of type conns : 936 ( 936 >; 0 *; 0 +; 0 <<)
% Number of symbols : 261 ( 258 usr; 41 con; 0-5 aty)
% Number of variables : 3742 ( 440 ^;3199 !; 103 ?;3742 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:11:40.950
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
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% Explicit typings (258)
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wellor2139422199430923550c_fm_i: set_Pr3972831103112126087c_fm_i > set_Epistemic_fm_i > $o ).
thf(sy_c_Wellorder__Constructions_Owo__rel_Osucc_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
wellor2624885639821348924c_fm_i: set_Pr4658907567593863815c_fm_i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Wellorder__Constructions_Owo__rel_Osucc_001t__Product____Type__Ounit,type,
wellor508239122659082791t_unit: set_Pr5094982260447487303t_unit > product_unit > product_unit ).
thf(sy_c_Wellorder__Constructions_Owo__rel_Osucc_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
wellor6171438539502936604c_fm_i: set_Pr3972831103112126087c_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Wellorder__Constructions_Owo__rel_OworecZSL_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
wellor8883914554786644734c_fm_i: set_Pr4658907567593863815c_fm_i > set_Epistemic_fm_i > ( epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ) > ( ( epistemic_fm_i > set_Epistemic_fm_i ) > epistemic_fm_i > set_Epistemic_fm_i ) > epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Wellorder__Constructions_Owo__rel_Ozero_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
wellor854379368678715862c_fm_i: set_Pr4658907567593863815c_fm_i > epistemic_fm_i ).
thf(sy_c_Wellorder__Constructions_Owo__rel_Ozero_001t__Product____Type__Ounit,type,
wellor829852009790670017t_unit: set_Pr5094982260447487303t_unit > product_unit ).
thf(sy_c_Wellorder__Constructions_Owo__rel_Ozero_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
wellor6421275854403804086c_fm_i: set_Pr3972831103112126087c_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Zorn_OChains_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
chains2242491043985051090c_fm_i: set_Pr4658907567593863815c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Zorn_OChains_001t__Product____Type__Ounit,type,
chains_Product_unit: set_Pr5094982260447487303t_unit > set_set_Product_unit ).
thf(sy_c_Zorn_OChains_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
chains7915783162359121842c_fm_i: set_Pr3972831103112126087c_fm_i > set_se7339729205154126530c_fm_i ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
pred_c1742013745311679565c_fm_i: set_Epistemic_fm_i > ( epistemic_fm_i > epistemic_fm_i > $o ) > set_Epistemic_fm_i > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Product____Type__Ounit,type,
pred_c6915491785556283576t_unit: set_Product_unit > ( product_unit > product_unit > $o ) > set_Product_unit > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
pred_c4586345430894039597c_fm_i: set_se3485332733965609186c_fm_i > ( set_Epistemic_fm_i > set_Epistemic_fm_i > $o ) > set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_member_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member8097212860227194492c_fm_i: ( epistemic_fm_i > epistemic_fm_i ) > set_Ep2190104726579546907c_fm_i > $o ).
thf(sy_c_member_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Product____Type__Ounit_J,type,
member4534395898142662503t_unit: ( epistemic_fm_i > product_unit ) > set_Ep7545435233890007174t_unit > $o ).
thf(sy_c_member_001_062_It__Product____Type__Ounit_Mt__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member3099810079864775633c_fm_i: ( product_unit > epistemic_fm_i ) > set_Pr777577458915813360c_fm_i > $o ).
thf(sy_c_member_001_062_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
member3685380118729357628t_unit: ( product_unit > product_unit ) > set_Pr1882883127215053275t_unit > $o ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
member6642669606046002379c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Epistemic____Logic__Ofm_Itf__i_J_Mt__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member1268636611877661904c_fm_i: produc7666096914482384295c_fm_i > set_Pr4658907567593863815c_fm_i > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
member7821138191923116944t_unit: produc8459935480633519975t_unit > set_Pr5094982260447487303t_unit > $o ).
thf(sy_c_member_001t__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,type,
member7895148549828227280c_fm_i: produc9057620875267856807c_fm_i > set_Pr3972831103112126087c_fm_i > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_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__Product____Type__Ounit_J,type,
member5877623283571906838t_unit: set_Product_unit > set_set_Product_unit > $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_i,type,
i2: i ).
thf(sy_v_p,type,
p: epistemic_fm_i ).
thf(sy_v_q,type,
q: epistemic_fm_i ).
% Relevant facts (1277)
thf(fact_0__092_060open_062consistent_AA_A_123_092_060_094bold_062_092_060not_062_A_IK_Ai_A_Ip_A_092_060_094bold_062_092_060and_062_Aq_J_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_Ap_A_092_060_094bold_062_092_060and_062_AK_Ai_Aq_J_125_092_060close_062,axiom,
episte2285483198712856234tent_i @ a @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ i2 @ ( epistemic_Con_i @ p @ q ) ) @ ( epistemic_Con_i @ ( epistemic_K_i @ i2 @ p ) @ ( epistemic_K_i @ i2 @ q ) ) ) @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) ).
% \<open>consistent A {\<^bold>\<not> (K i (p \<^bold>\<and> q) \<^bold>\<longrightarrow> K i p \<^bold>\<and> K i q)}\<close>
thf(fact_1__092_060open_062Extend_AA_A_123_092_060_094bold_062_092_060not_062_A_IK_Ai_A_Ip_A_092_060_094bold_062_092_060and_062_Aq_J_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_Ap_A_092_060_094bold_062_092_060and_062_AK_Ai_Aq_J_125_A_092_060in_062_A_092_060W_062_A_092_060lparr_062_092_060W_062_A_061_Amcss_AA_M_A_092_060K_062_A_061_A_092_060lambda_062i_AV_O_A_123W_O_Aknown_AV_Ai_A_092_060subseteq_062_AW_125_M_A_092_060pi_062_A_061_Api_092_060rparr_062_A_092_060and_062_A_092_060lparr_062_092_060W_062_A_061_Amcss_AA_M_A_092_060K_062_A_061_A_092_060lambda_062i_AV_O_A_123W_O_Aknown_AV_Ai_A_092_060subseteq_062_AW_125_M_A_092_060pi_062_A_061_Api_092_060rparr_062_M_AExtend_AA_A_123_092_060_094bold_062_092_060not_062_A_IK_Ai_A_Ip_A_092_060_094bold_062_092_060and_062_Aq_J_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_Ap_A_092_060_094bold_062_092_060and_062_AK_Ai_Aq_J_125_A_092_060Turnstile_062_A_092_060_094bold_062_092_060not_062_A_IK_Ai_A_Ip_A_092_060_094bold_062_092_060and_062_Aq_J_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_Ap_A_092_060_094bold_062_092_060and_062_AK_Ai_Aq_J_092_060close_062,axiom,
( ( member1104366573291651755c_fm_i @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ a ) @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ i2 @ ( epistemic_Con_i @ p @ q ) ) @ ( epistemic_Con_i @ ( epistemic_K_i @ i2 @ p ) @ ( epistemic_K_i @ i2 @ q ) ) ) @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) )
@ ( episte2675268102452679275t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ a @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) ) )
& ( episte2498298383968489120c_fm_i
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ a @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) )
@ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ a ) @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ i2 @ ( epistemic_Con_i @ p @ q ) ) @ ( epistemic_Con_i @ ( epistemic_K_i @ i2 @ p ) @ ( epistemic_K_i @ i2 @ q ) ) ) @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) )
@ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ i2 @ ( epistemic_Con_i @ p @ q ) ) @ ( epistemic_Con_i @ ( epistemic_K_i @ i2 @ p ) @ ( epistemic_K_i @ i2 @ q ) ) ) @ epistemic_FF_i ) ) ) ).
% \<open>Extend A {\<^bold>\<not> (K i (p \<^bold>\<and> q) \<^bold>\<longrightarrow> K i p \<^bold>\<and> K i q)} \<in> \<W> \<lparr>\<W> = mcss A, \<K> = \<lambda>i V. {W. known V i \<subseteq> W}, \<pi> = pi\<rparr> \<and> \<lparr>\<W> = mcss A, \<K> = \<lambda>i V. {W. known V i \<subseteq> W}, \<pi> = pi\<rparr>, Extend A {\<^bold>\<not> (K i (p \<^bold>\<and> q) \<^bold>\<longrightarrow> K i p \<^bold>\<and> K i q)} \<Turnstile> \<^bold>\<not> (K i (p \<^bold>\<and> q) \<^bold>\<longrightarrow> K i p \<^bold>\<and> K i q)\<close>
thf(fact_2__092_060open_062_092_060not_062_AA_A_092_060turnstile_062_AK_Ai_A_Ip_A_092_060_094bold_062_092_060and_062_Aq_J_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_Ap_A_092_060_094bold_062_092_060and_062_AK_Ai_Aq_092_060close_062,axiom,
~ ( epistemic_AK_i @ a @ ( epistemic_Imp_i @ ( epistemic_K_i @ i2 @ ( epistemic_Con_i @ p @ q ) ) @ ( epistemic_Con_i @ ( epistemic_K_i @ i2 @ p ) @ ( epistemic_K_i @ i2 @ q ) ) ) ) ).
% \<open>\<not> A \<turnstile> K i (p \<^bold>\<and> q) \<^bold>\<longrightarrow> K i p \<^bold>\<and> K i q\<close>
thf(fact_3_canonical__model_I1_J,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( ( member6642669606046002379c_fm_i @ P2 @ S )
=> ( episte2498298383968489120c_fm_i
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) )
@ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S )
@ P2 ) ) ) ).
% canonical_model(1)
thf(fact_4_truth__lemma,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( member6642669606046002379c_fm_i @ P2 @ V2 )
= ( episte2498298383968489120c_fm_i
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) )
@ V2
@ P2 ) ) ) ) ).
% truth_lemma
thf(fact_5_singleton__insert__inj__eq,axiom,
! [B: product_unit,A2: product_unit,A: set_Product_unit] :
( ( ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit )
= ( insert_Product_unit2 @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_6_singleton__insert__inj__eq,axiom,
! [B: epistemic_fm_i,A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i )
= ( insert7817948997695205106c_fm_i @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_7_singleton__insert__inj__eq_H,axiom,
! [A2: product_unit,A: set_Product_unit,B: product_unit] :
( ( ( insert_Product_unit2 @ A2 @ A )
= ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) )
= ( ( A2 = B )
& ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_8_singleton__insert__inj__eq_H,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) )
= ( ( A2 = B )
& ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_9_canonical__model_I3_J,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( ( member6642669606046002379c_fm_i @ P2 @ S )
=> ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ) ) ).
% canonical_model(3)
thf(fact_10_maximal__Extend,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] : ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ).
% maximal_Extend
thf(fact_11_Extend__subset,axiom,
! [S: set_Epistemic_fm_i,A: epistemic_fm_i > $o] : ( ord_le3843937902494030498c_fm_i @ S @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ).
% Extend_subset
thf(fact_12_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_13_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_14_singleton__conv,axiom,
! [A2: product_unit] :
( ( collect_Product_unit
@ ^ [X: product_unit] : ( X = A2 ) )
= ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ).
% singleton_conv
thf(fact_15_singleton__conv2,axiom,
! [A2: set_Epistemic_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z )
@ A2 ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singleton_conv2
thf(fact_16_singleton__conv2,axiom,
! [A2: epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ( ^ [Y: epistemic_fm_i,Z: epistemic_fm_i] : ( Y = Z )
@ A2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singleton_conv2
thf(fact_17_singleton__conv2,axiom,
! [A2: product_unit] :
( ( collect_Product_unit
@ ( ^ [Y: product_unit,Z: product_unit] : ( Y = Z )
@ A2 ) )
= ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ).
% singleton_conv2
thf(fact_18_Collect__const,axiom,
! [P3: $o] :
( ( P3
=> ( ( collec3087743281813070829c_fm_i
@ ^ [S2: set_Epistemic_fm_i] : P3 )
= top_to3675301018284359218c_fm_i ) )
& ( ~ P3
=> ( ( collec3087743281813070829c_fm_i
@ ^ [S2: set_Epistemic_fm_i] : P3 )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_const
thf(fact_19_Collect__const,axiom,
! [P3: $o] :
( ( P3
=> ( ( collec4904205187116291597c_fm_i
@ ^ [S2: epistemic_fm_i] : P3 )
= top_to8364301187473967186c_fm_i ) )
& ( ~ P3
=> ( ( collec4904205187116291597c_fm_i
@ ^ [S2: epistemic_fm_i] : P3 )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_const
thf(fact_20_Collect__const,axiom,
! [P3: $o] :
( ( P3
=> ( ( collect_Product_unit
@ ^ [S2: product_unit] : P3 )
= top_to1996260823553986621t_unit ) )
& ( ~ P3
=> ( ( collect_Product_unit
@ ^ [S2: product_unit] : P3 )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Collect_const
thf(fact_21_Imp__intro,axiom,
! [M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ Q ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ ( epistemic_Imp_i @ P2 @ Q ) ) ) ).
% Imp_intro
thf(fact_22_insert__subset,axiom,
! [X2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( insert7698009978809854162c_fm_i @ X2 @ A ) @ B2 )
= ( ( member1104366573291651755c_fm_i @ X2 @ B2 )
& ( ord_le5389487502678872194c_fm_i @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_23_insert__subset,axiom,
! [X2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X2 @ A ) @ B2 )
= ( ( member6642669606046002379c_fm_i @ X2 @ B2 )
& ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_24_insert__subset,axiom,
! [X2: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit2 @ X2 @ A ) @ B2 )
= ( ( member_Product_unit @ X2 @ B2 )
& ( ord_le3507040750410214029t_unit @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_25_UNIV__I,axiom,
! [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ top_to3675301018284359218c_fm_i ) ).
% UNIV_I
thf(fact_26_UNIV__I,axiom,
! [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ top_to8364301187473967186c_fm_i ) ).
% UNIV_I
thf(fact_27_UNIV__I,axiom,
! [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_28_empty__Collect__eq,axiom,
! [P3: set_Epistemic_fm_i > $o] :
( ( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ P3 ) )
= ( ! [X: set_Epistemic_fm_i] :
~ ( P3 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_29_empty__Collect__eq,axiom,
! [P3: epistemic_fm_i > $o] :
( ( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ P3 ) )
= ( ! [X: epistemic_fm_i] :
~ ( P3 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_30_empty__Collect__eq,axiom,
! [P3: product_unit > $o] :
( ( bot_bo3957492148770167129t_unit
= ( collect_Product_unit @ P3 ) )
= ( ! [X: product_unit] :
~ ( P3 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_31_Collect__empty__eq,axiom,
! [P3: set_Epistemic_fm_i > $o] :
( ( ( collec3087743281813070829c_fm_i @ P3 )
= bot_bo145720340923748686c_fm_i )
= ( ! [X: set_Epistemic_fm_i] :
~ ( P3 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_32_Collect__empty__eq,axiom,
! [P3: epistemic_fm_i > $o] :
( ( ( collec4904205187116291597c_fm_i @ P3 )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X: epistemic_fm_i] :
~ ( P3 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_33_Collect__empty__eq,axiom,
! [P3: product_unit > $o] :
( ( ( collect_Product_unit @ P3 )
= bot_bo3957492148770167129t_unit )
= ( ! [X: product_unit] :
~ ( P3 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_34_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_35_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_36_all__not__in__conv,axiom,
! [A: set_Product_unit] :
( ( ! [X: product_unit] :
~ ( member_Product_unit @ X @ A ) )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% all_not_in_conv
thf(fact_37_empty__iff,axiom,
! [C: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ C @ bot_bo145720340923748686c_fm_i ) ).
% empty_iff
thf(fact_38_empty__iff,axiom,
! [C: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ C @ bot_bo4194595901900360558c_fm_i ) ).
% empty_iff
thf(fact_39_empty__iff,axiom,
! [C: product_unit] :
~ ( member_Product_unit @ C @ bot_bo3957492148770167129t_unit ) ).
% empty_iff
thf(fact_40_subset__antisym,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_41_subset__antisym,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_42_subsetI,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( member1104366573291651755c_fm_i @ X3 @ B2 ) )
=> ( ord_le5389487502678872194c_fm_i @ A @ B2 ) ) ).
% subsetI
thf(fact_43_subsetI,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( member6642669606046002379c_fm_i @ X3 @ B2 ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ).
% subsetI
thf(fact_44_subsetI,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( member_Product_unit @ X3 @ B2 ) )
=> ( ord_le3507040750410214029t_unit @ A @ B2 ) ) ).
% subsetI
thf(fact_45_insert__absorb2,axiom,
! [X2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( insert7817948997695205106c_fm_i @ X2 @ ( insert7817948997695205106c_fm_i @ X2 @ A ) )
= ( insert7817948997695205106c_fm_i @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_46_insert__absorb2,axiom,
! [X2: product_unit,A: set_Product_unit] :
( ( insert_Product_unit2 @ X2 @ ( insert_Product_unit2 @ X2 @ A ) )
= ( insert_Product_unit2 @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_47_insert__iff,axiom,
! [A2: product_unit,B: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ ( insert_Product_unit2 @ B @ A ) )
= ( ( A2 = B )
| ( member_Product_unit @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_48_insert__iff,axiom,
! [A2: epistemic_fm_i,B: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ A ) )
= ( ( A2 = B )
| ( member6642669606046002379c_fm_i @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_49_insert__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B @ A ) )
= ( ( A2 = B )
| ( member1104366573291651755c_fm_i @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_50_insertCI,axiom,
! [A2: product_unit,B2: set_Product_unit,B: product_unit] :
( ( ~ ( member_Product_unit @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_Product_unit @ A2 @ ( insert_Product_unit2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_51_insertCI,axiom,
! [A2: epistemic_fm_i,B2: set_Epistemic_fm_i,B: epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ B2 ) ) ) ).
% insertCI
thf(fact_52_insertCI,axiom,
! [A2: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i] :
( ( ~ ( member1104366573291651755c_fm_i @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B @ B2 ) ) ) ).
% insertCI
thf(fact_53_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_54_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_55_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_56_fm_Oinject_I1_J,axiom,
! [X22: list_char,Y2: list_char] :
( ( ( epistemic_Pro_i @ X22 )
= ( epistemic_Pro_i @ Y2 ) )
= ( X22 = Y2 ) ) ).
% fm.inject(1)
thf(fact_57_frame_Oext__inject,axiom,
! [W3: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More: episte7133455005732253587t_unit,W4: set_se3485332733965609186c_fm_i,K2: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More2: episte7133455005732253587t_unit] :
( ( ( episte8397520288747125024t_unit @ W3 @ K @ More )
= ( episte8397520288747125024t_unit @ W4 @ K2 @ More2 ) )
= ( ( W3 = W4 )
& ( K = K2 )
& ( More = More2 ) ) ) ).
% frame.ext_inject
thf(fact_58_kripke_Oext__inject,axiom,
! [Pi: set_Epistemic_fm_i > list_char > $o,More: product_unit,Pi2: set_Epistemic_fm_i > list_char > $o,More2: product_unit] :
( ( ( episte4955834246032687171t_unit @ Pi @ More )
= ( episte4955834246032687171t_unit @ Pi2 @ More2 ) )
= ( ( Pi = Pi2 )
& ( More = More2 ) ) ) ).
% kripke.ext_inject
thf(fact_59_empty__subsetI,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A ) ).
% empty_subsetI
thf(fact_60_empty__subsetI,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A ) ).
% empty_subsetI
thf(fact_61_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_62_subset__empty,axiom,
! [A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% subset_empty
thf(fact_63_singletonI,axiom,
! [A2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singletonI
thf(fact_64_singletonI,axiom,
! [A2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singletonI
thf(fact_65_singletonI,axiom,
! [A2: product_unit] : ( member_Product_unit @ A2 @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ).
% singletonI
thf(fact_66_Ax,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i] :
( ( A @ P2 )
=> ( epistemic_AK_i @ A @ P2 ) ) ).
% Ax
thf(fact_67_soundness,axiom,
! [A: epistemic_fm_i > $o,P3: episte4846006725945195288t_unit > $o,P2: epistemic_fm_i,M: episte4846006725945195288t_unit,W2: epistemic_fm_i] :
( ! [M2: episte4846006725945195288t_unit,W5: epistemic_fm_i,P4: epistemic_fm_i] :
( ( A @ P4 )
=> ( ( P3 @ M2 )
=> ( ( member6642669606046002379c_fm_i @ W5 @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ W5 @ P4 ) ) ) )
=> ( ( epistemic_AK_i @ A @ P2 )
=> ( ( P3 @ M )
=> ( ( member6642669606046002379c_fm_i @ W2 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W2 @ P2 ) ) ) ) ) ).
% soundness
thf(fact_68_soundness,axiom,
! [A: epistemic_fm_i > $o,P3: episte5489734138418521880t_unit > $o,P2: epistemic_fm_i,M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit,W5: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( A @ P4 )
=> ( ( P3 @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W5 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W5 @ P4 ) ) ) )
=> ( ( epistemic_AK_i @ A @ P2 )
=> ( ( P3 @ M )
=> ( ( member1104366573291651755c_fm_i @ W2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 ) ) ) ) ) ).
% soundness
thf(fact_69_K__trans,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i,R: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P2 @ Q ) @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ Q @ R ) @ ( epistemic_Imp_i @ P2 @ R ) ) ) ) ).
% K_trans
thf(fact_70_R1,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ P2 )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ Q ) )
=> ( epistemic_AK_i @ A @ Q ) ) ) ).
% R1
thf(fact_71_R2,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,I2: i] :
( ( epistemic_AK_i @ A @ P2 )
=> ( epistemic_AK_i @ A @ ( epistemic_K_i @ I2 @ P2 ) ) ) ).
% R2
thf(fact_72_K__imp__trans_H,axiom,
! [A: epistemic_fm_i > $o,Q: epistemic_fm_i,R: epistemic_fm_i,P2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ Q @ R ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P2 @ Q ) @ ( epistemic_Imp_i @ P2 @ R ) ) ) ) ).
% K_imp_trans'
thf(fact_73_K__imp__trans,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i,R: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ Q ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ Q @ R ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ R ) ) ) ) ).
% K_imp_trans
thf(fact_74_completeness_092_060_094sub_062A,axiom,
! [P2: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [M2: episte5489734138418521880t_unit,X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ bot_bo4194595901900360558c_fm_i )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ P2 ) ) )
=> ( epistemic_AK_i @ A @ P2 ) ) ).
% completeness\<^sub>A
thf(fact_75_K__map,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i,I2: i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ Q ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ ( epistemic_K_i @ I2 @ Q ) ) ) ) ).
% K_map
thf(fact_76_K__A2_H,axiom,
! [A: epistemic_fm_i > $o,I2: i,P2: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P2 @ Q ) ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ ( epistemic_K_i @ I2 @ Q ) ) ) ) ).
% K_A2'
thf(fact_77_deriv__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( epistemic_AK_i @ A @ P2 )
=> ( member6642669606046002379c_fm_i @ P2 @ V2 ) ) ) ) ).
% deriv_in_maximal
thf(fact_78_mem__Collect__eq,axiom,
! [A2: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ A2 @ ( collec3087743281813070829c_fm_i @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
! [A2: epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ A2 @ ( collec4904205187116291597c_fm_i @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_80_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_81_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_82_Collect__cong,axiom,
! [P3: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ! [X3: set_Epistemic_fm_i] :
( ( P3 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collec3087743281813070829c_fm_i @ P3 )
= ( collec3087743281813070829c_fm_i @ Q2 ) ) ) ).
% Collect_cong
thf(fact_83_Collect__cong,axiom,
! [P3: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ! [X3: epistemic_fm_i] :
( ( P3 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collec4904205187116291597c_fm_i @ P3 )
= ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ).
% Collect_cong
thf(fact_84_K__multi__imply,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i,B: epistemic_fm_i,C: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ ( epistemic_Imp_i @ B @ C ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ A2 @ B ) @ C ) ) ) ).
% K_multi_imply
thf(fact_85_K__imply__multi,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i,B: epistemic_fm_i,C: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ B ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ C ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ ( epistemic_Con_i @ B @ C ) ) ) ) ) ).
% K_imply_multi
thf(fact_86_K__L__dual,axiom,
! [A: epistemic_fm_i > $o,I2: i,P2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ P2 ) ) ) ).
% K_L_dual
thf(fact_87_K__LK,axiom,
! [A: epistemic_fm_i > $o,I2: i,P2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ epistemic_FF_i ) ) ) ).
% K_LK
thf(fact_88_A2,axiom,
! [A: epistemic_fm_i > $o,I2: i,P2: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I2 @ P2 ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P2 @ Q ) ) ) @ ( epistemic_K_i @ I2 @ Q ) ) ) ).
% A2
thf(fact_89_frame_Oselect__convs_I1_J,axiom,
! [W3: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More: episte7133455005732253587t_unit] :
( ( episte2675268102452679275t_unit @ ( episte8397520288747125024t_unit @ W3 @ K @ More ) )
= W3 ) ).
% frame.select_convs(1)
thf(fact_90_UNIV__witness,axiom,
? [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ top_to3675301018284359218c_fm_i ) ).
% UNIV_witness
thf(fact_91_UNIV__witness,axiom,
? [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ top_to8364301187473967186c_fm_i ) ).
% UNIV_witness
thf(fact_92_UNIV__witness,axiom,
? [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ).
% UNIV_witness
thf(fact_93_UNIV__eq__I,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [X3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( top_to3675301018284359218c_fm_i = A ) ) ).
% UNIV_eq_I
thf(fact_94_UNIV__eq__I,axiom,
! [A: set_Epistemic_fm_i] :
( ! [X3: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( top_to8364301187473967186c_fm_i = A ) ) ).
% UNIV_eq_I
thf(fact_95_UNIV__eq__I,axiom,
! [A: set_Product_unit] :
( ! [X3: product_unit] : ( member_Product_unit @ X3 @ A )
=> ( top_to1996260823553986621t_unit = A ) ) ).
% UNIV_eq_I
thf(fact_96_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_97_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_98_ex__in__conv,axiom,
! [A: set_Product_unit] :
( ( ? [X: product_unit] : ( member_Product_unit @ X @ A ) )
= ( A != bot_bo3957492148770167129t_unit ) ) ).
% ex_in_conv
thf(fact_99_equals0I,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [Y3: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ Y3 @ A )
=> ( A = bot_bo145720340923748686c_fm_i ) ) ).
% equals0I
thf(fact_100_equals0I,axiom,
! [A: set_Epistemic_fm_i] :
( ! [Y3: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ Y3 @ A )
=> ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% equals0I
thf(fact_101_equals0I,axiom,
! [A: set_Product_unit] :
( ! [Y3: product_unit] :
~ ( member_Product_unit @ Y3 @ A )
=> ( A = bot_bo3957492148770167129t_unit ) ) ).
% equals0I
thf(fact_102_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_103_equals0D,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( A = bot_bo4194595901900360558c_fm_i )
=> ~ ( member6642669606046002379c_fm_i @ A2 @ A ) ) ).
% equals0D
thf(fact_104_equals0D,axiom,
! [A: set_Product_unit,A2: product_unit] :
( ( A = bot_bo3957492148770167129t_unit )
=> ~ ( member_Product_unit @ A2 @ A ) ) ).
% equals0D
thf(fact_105_emptyE,axiom,
! [A2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ).
% emptyE
thf(fact_106_emptyE,axiom,
! [A2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ).
% emptyE
thf(fact_107_emptyE,axiom,
! [A2: product_unit] :
~ ( member_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ).
% emptyE
thf(fact_108_Collect__mono__iff,axiom,
! [P3: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) @ ( collec3087743281813070829c_fm_i @ Q2 ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( P3 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_109_Collect__mono__iff,axiom,
! [P3: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) @ ( collec4904205187116291597c_fm_i @ Q2 ) )
= ( ! [X: epistemic_fm_i] :
( ( P3 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_110_Collect__mono__iff,axiom,
! [P3: product_unit > $o,Q2: product_unit > $o] :
( ( ord_le3507040750410214029t_unit @ ( collect_Product_unit @ P3 ) @ ( collect_Product_unit @ Q2 ) )
= ( ! [X: product_unit] :
( ( P3 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_111_set__eq__subset,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z ) )
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B3 )
& ( ord_le3843937902494030498c_fm_i @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_112_set__eq__subset,axiom,
( ( ^ [Y: set_Product_unit,Z: set_Product_unit] : ( Y = Z ) )
= ( ^ [A3: set_Product_unit,B3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A3 @ B3 )
& ( ord_le3507040750410214029t_unit @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_113_subset__trans,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_114_subset__trans,axiom,
! [A: set_Product_unit,B2: set_Product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ C2 )
=> ( ord_le3507040750410214029t_unit @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_115_Collect__mono,axiom,
! [P3: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ! [X3: set_Epistemic_fm_i] :
( ( P3 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) @ ( collec3087743281813070829c_fm_i @ Q2 ) ) ) ).
% Collect_mono
thf(fact_116_Collect__mono,axiom,
! [P3: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ! [X3: epistemic_fm_i] :
( ( P3 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) @ ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ).
% Collect_mono
thf(fact_117_Collect__mono,axiom,
! [P3: product_unit > $o,Q2: product_unit > $o] :
( ! [X3: product_unit] :
( ( P3 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le3507040750410214029t_unit @ ( collect_Product_unit @ P3 ) @ ( collect_Product_unit @ Q2 ) ) ) ).
% Collect_mono
thf(fact_118_subset__refl,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ A ) ).
% subset_refl
thf(fact_119_subset__refl,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ A ) ).
% subset_refl
thf(fact_120_subset__iff,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
! [T: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ T @ A3 )
=> ( member1104366573291651755c_fm_i @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_121_subset__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
! [T: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ T @ A3 )
=> ( member6642669606046002379c_fm_i @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_122_subset__iff,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [A3: set_Product_unit,B3: set_Product_unit] :
! [T: product_unit] :
( ( member_Product_unit @ T @ A3 )
=> ( member_Product_unit @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_123_equalityD2,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A = B2 )
=> ( ord_le3843937902494030498c_fm_i @ B2 @ A ) ) ).
% equalityD2
thf(fact_124_equalityD2,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( A = B2 )
=> ( ord_le3507040750410214029t_unit @ B2 @ A ) ) ).
% equalityD2
thf(fact_125_equalityD1,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A = B2 )
=> ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ).
% equalityD1
thf(fact_126_equalityD1,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( A = B2 )
=> ( ord_le3507040750410214029t_unit @ A @ B2 ) ) ).
% equalityD1
thf(fact_127_subset__eq,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A3 )
=> ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_128_subset__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A3 )
=> ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_129_subset__eq,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [A3: set_Product_unit,B3: set_Product_unit] :
! [X: product_unit] :
( ( member_Product_unit @ X @ A3 )
=> ( member_Product_unit @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_130_equalityE,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A = B2 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ B2 @ A ) ) ) ).
% equalityE
thf(fact_131_equalityE,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( A = B2 )
=> ~ ( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ~ ( ord_le3507040750410214029t_unit @ B2 @ A ) ) ) ).
% equalityE
thf(fact_132_subsetD,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B2 )
=> ( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B2 ) ) ) ).
% subsetD
thf(fact_133_subsetD,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B2 ) ) ) ).
% subsetD
thf(fact_134_subsetD,axiom,
! [A: set_Product_unit,B2: set_Product_unit,C: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ( member_Product_unit @ C @ A )
=> ( member_Product_unit @ C @ B2 ) ) ) ).
% subsetD
thf(fact_135_in__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,X2: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B2 )
=> ( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( member1104366573291651755c_fm_i @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_136_in__mono,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,X2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( member6642669606046002379c_fm_i @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_137_in__mono,axiom,
! [A: set_Product_unit,B2: set_Product_unit,X2: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ( member_Product_unit @ X2 @ A )
=> ( member_Product_unit @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_138_mk__disjoint__insert,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ? [B4: set_Product_unit] :
( ( A
= ( insert_Product_unit2 @ A2 @ B4 ) )
& ~ ( member_Product_unit @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_139_mk__disjoint__insert,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ? [B4: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ A2 @ B4 ) )
& ~ ( member6642669606046002379c_fm_i @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_140_mk__disjoint__insert,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [B4: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ A2 @ B4 ) )
& ~ ( member1104366573291651755c_fm_i @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_141_insert__commute,axiom,
! [X2: epistemic_fm_i,Y4: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( insert7817948997695205106c_fm_i @ X2 @ ( insert7817948997695205106c_fm_i @ Y4 @ A ) )
= ( insert7817948997695205106c_fm_i @ Y4 @ ( insert7817948997695205106c_fm_i @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_142_insert__commute,axiom,
! [X2: product_unit,Y4: product_unit,A: set_Product_unit] :
( ( insert_Product_unit2 @ X2 @ ( insert_Product_unit2 @ Y4 @ A ) )
= ( insert_Product_unit2 @ Y4 @ ( insert_Product_unit2 @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_143_insert__eq__iff,axiom,
! [A2: product_unit,A: set_Product_unit,B: product_unit,B2: set_Product_unit] :
( ~ ( member_Product_unit @ A2 @ A )
=> ( ~ ( member_Product_unit @ B @ B2 )
=> ( ( ( insert_Product_unit2 @ A2 @ A )
= ( insert_Product_unit2 @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_Product_unit] :
( ( A
= ( insert_Product_unit2 @ B @ C3 ) )
& ~ ( member_Product_unit @ B @ C3 )
& ( B2
= ( insert_Product_unit2 @ A2 @ C3 ) )
& ~ ( member_Product_unit @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_144_insert__eq__iff,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ B @ B2 )
=> ( ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= ( insert7817948997695205106c_fm_i @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ B @ C3 ) )
& ~ ( member6642669606046002379c_fm_i @ B @ C3 )
& ( B2
= ( insert7817948997695205106c_fm_i @ A2 @ C3 ) )
& ~ ( member6642669606046002379c_fm_i @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_145_insert__eq__iff,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ B @ B2 )
=> ( ( ( insert7698009978809854162c_fm_i @ A2 @ A )
= ( insert7698009978809854162c_fm_i @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ B @ C3 ) )
& ~ ( member1104366573291651755c_fm_i @ B @ C3 )
& ( B2
= ( insert7698009978809854162c_fm_i @ A2 @ C3 ) )
& ~ ( member1104366573291651755c_fm_i @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_146_insert__absorb,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ( ( insert_Product_unit2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_147_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_148_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_149_insert__ident,axiom,
! [X2: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ~ ( member_Product_unit @ X2 @ A )
=> ( ~ ( member_Product_unit @ X2 @ B2 )
=> ( ( ( insert_Product_unit2 @ X2 @ A )
= ( insert_Product_unit2 @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_150_insert__ident,axiom,
! [X2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ X2 @ B2 )
=> ( ( ( insert7817948997695205106c_fm_i @ X2 @ A )
= ( insert7817948997695205106c_fm_i @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_151_insert__ident,axiom,
! [X2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ X2 @ B2 )
=> ( ( ( insert7698009978809854162c_fm_i @ X2 @ A )
= ( insert7698009978809854162c_fm_i @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_152_Set_Oset__insert,axiom,
! [X2: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ~ ! [B4: set_Product_unit] :
( ( A
= ( insert_Product_unit2 @ X2 @ B4 ) )
=> ( member_Product_unit @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_153_Set_Oset__insert,axiom,
! [X2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ~ ! [B4: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ X2 @ B4 ) )
=> ( member6642669606046002379c_fm_i @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_154_Set_Oset__insert,axiom,
! [X2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ~ ! [B4: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ X2 @ B4 ) )
=> ( member1104366573291651755c_fm_i @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_155_insertI2,axiom,
! [A2: product_unit,B2: set_Product_unit,B: product_unit] :
( ( member_Product_unit @ A2 @ B2 )
=> ( member_Product_unit @ A2 @ ( insert_Product_unit2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_156_insertI2,axiom,
! [A2: epistemic_fm_i,B2: set_Epistemic_fm_i,B: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ B2 )
=> ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ B2 ) ) ) ).
% insertI2
thf(fact_157_insertI2,axiom,
! [A2: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ B2 )
=> ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B @ B2 ) ) ) ).
% insertI2
thf(fact_158_insertI1,axiom,
! [A2: product_unit,B2: set_Product_unit] : ( member_Product_unit @ A2 @ ( insert_Product_unit2 @ A2 @ B2 ) ) ).
% insertI1
thf(fact_159_insertI1,axiom,
! [A2: epistemic_fm_i,B2: set_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) ) ).
% insertI1
thf(fact_160_insertI1,axiom,
! [A2: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i] : ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ A2 @ B2 ) ) ).
% insertI1
thf(fact_161_insertE,axiom,
! [A2: product_unit,B: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ ( insert_Product_unit2 @ B @ A ) )
=> ( ( A2 != B )
=> ( member_Product_unit @ A2 @ A ) ) ) ).
% insertE
thf(fact_162_insertE,axiom,
! [A2: epistemic_fm_i,B: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ A ) )
=> ( ( A2 != B )
=> ( member6642669606046002379c_fm_i @ A2 @ A ) ) ) ).
% insertE
thf(fact_163_insertE,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B @ A ) )
=> ( ( A2 != B )
=> ( member1104366573291651755c_fm_i @ A2 @ A ) ) ) ).
% insertE
thf(fact_164_K__thm,axiom,
! [A: epistemic_fm_i > $o,I2: i,P2: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I2 @ P2 ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ Q @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_Con_i @ P2 @ Q ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ).
% K_thm
thf(fact_165_frame_Ocases__scheme,axiom,
! [R: episte5489734138418521880t_unit] :
~ ! [W6: set_se3485332733965609186c_fm_i,K3: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More3: episte7133455005732253587t_unit] :
( R
!= ( episte8397520288747125024t_unit @ W6 @ K3 @ More3 ) ) ).
% frame.cases_scheme
thf(fact_166_generalization,axiom,
! [P2: epistemic_fm_i,W2: epistemic_fm_i,M: episte4846006725945195288t_unit,I2: i] :
( ! [M2: episte4846006725945195288t_unit,X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ X3 @ P2 ) )
=> ( ( member6642669606046002379c_fm_i @ W2 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W2 @ ( epistemic_K_i @ I2 @ P2 ) ) ) ) ).
% generalization
thf(fact_167_generalization,axiom,
! [P2: epistemic_fm_i,W2: set_Epistemic_fm_i,M: episte5489734138418521880t_unit,I2: i] :
( ! [M2: episte5489734138418521880t_unit,X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ P2 ) )
=> ( ( member1104366573291651755c_fm_i @ W2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ ( epistemic_K_i @ I2 @ P2 ) ) ) ) ).
% generalization
thf(fact_168_kripke_Oext__induct,axiom,
! [P3: episte7133455005732253587t_unit > $o,R: episte7133455005732253587t_unit] :
( ! [Pi3: set_Epistemic_fm_i > list_char > $o,More3: product_unit] : ( P3 @ ( episte4955834246032687171t_unit @ Pi3 @ More3 ) )
=> ( P3 @ R ) ) ).
% kripke.ext_induct
thf(fact_169_UNIV__def,axiom,
( top_to3675301018284359218c_fm_i
= ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : $true ) ) ).
% UNIV_def
thf(fact_170_UNIV__def,axiom,
( top_to8364301187473967186c_fm_i
= ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : $true ) ) ).
% UNIV_def
thf(fact_171_UNIV__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit
@ ^ [X: product_unit] : $true ) ) ).
% UNIV_def
thf(fact_172_empty__def,axiom,
( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : $false ) ) ).
% empty_def
thf(fact_173_empty__def,axiom,
( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : $false ) ) ).
% empty_def
thf(fact_174_empty__def,axiom,
( bot_bo3957492148770167129t_unit
= ( collect_Product_unit
@ ^ [X: product_unit] : $false ) ) ).
% empty_def
thf(fact_175_Collect__subset,axiom,
! [A: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P3 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_176_Collect__subset,axiom,
! [A: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P3 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_177_Collect__subset,axiom,
! [A: set_Product_unit,P3: product_unit > $o] :
( ord_le3507040750410214029t_unit
@ ( collect_Product_unit
@ ^ [X: product_unit] :
( ( member_Product_unit @ X @ A )
& ( P3 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_178_insert__Collect,axiom,
! [A2: product_unit,P3: product_unit > $o] :
( ( insert_Product_unit2 @ A2 @ ( collect_Product_unit @ P3 ) )
= ( collect_Product_unit
@ ^ [U: product_unit] :
( ( U != A2 )
=> ( P3 @ U ) ) ) ) ).
% insert_Collect
thf(fact_179_insert__Collect,axiom,
! [A2: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( insert7698009978809854162c_fm_i @ A2 @ ( collec3087743281813070829c_fm_i @ P3 ) )
= ( collec3087743281813070829c_fm_i
@ ^ [U: set_Epistemic_fm_i] :
( ( U != A2 )
=> ( P3 @ U ) ) ) ) ).
% insert_Collect
thf(fact_180_insert__Collect,axiom,
! [A2: epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( insert7817948997695205106c_fm_i @ A2 @ ( collec4904205187116291597c_fm_i @ P3 ) )
= ( collec4904205187116291597c_fm_i
@ ^ [U: epistemic_fm_i] :
( ( U != A2 )
=> ( P3 @ U ) ) ) ) ).
% insert_Collect
thf(fact_181_insert__compr,axiom,
( insert_Product_unit2
= ( ^ [A4: product_unit,B3: set_Product_unit] :
( collect_Product_unit
@ ^ [X: product_unit] :
( ( X = A4 )
| ( member_Product_unit @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_182_insert__compr,axiom,
( insert7698009978809854162c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( X = A4 )
| ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_183_insert__compr,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A4: epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( X = A4 )
| ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_184_empty__not__UNIV,axiom,
bot_bo4194595901900360558c_fm_i != top_to8364301187473967186c_fm_i ).
% empty_not_UNIV
thf(fact_185_empty__not__UNIV,axiom,
bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).
% empty_not_UNIV
thf(fact_186_subset__UNIV,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ top_to8364301187473967186c_fm_i ) ).
% subset_UNIV
thf(fact_187_subset__UNIV,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% subset_UNIV
thf(fact_188_insert__UNIV,axiom,
! [X2: epistemic_fm_i] :
( ( insert7817948997695205106c_fm_i @ X2 @ top_to8364301187473967186c_fm_i )
= top_to8364301187473967186c_fm_i ) ).
% insert_UNIV
thf(fact_189_insert__UNIV,axiom,
! [X2: product_unit] :
( ( insert_Product_unit2 @ X2 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% insert_UNIV
thf(fact_190_singleton__inject,axiom,
! [A2: epistemic_fm_i,B: epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i )
= ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_191_singleton__inject,axiom,
! [A2: product_unit,B: product_unit] :
( ( ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit )
= ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_192_insert__not__empty,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( insert7817948997695205106c_fm_i @ A2 @ A )
!= bot_bo4194595901900360558c_fm_i ) ).
% insert_not_empty
thf(fact_193_insert__not__empty,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( insert_Product_unit2 @ A2 @ A )
!= bot_bo3957492148770167129t_unit ) ).
% insert_not_empty
thf(fact_194_doubleton__eq__iff,axiom,
! [A2: epistemic_fm_i,B: epistemic_fm_i,C: epistemic_fm_i,D: epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) )
= ( insert7817948997695205106c_fm_i @ C @ ( insert7817948997695205106c_fm_i @ D @ bot_bo4194595901900360558c_fm_i ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_195_doubleton__eq__iff,axiom,
! [A2: product_unit,B: product_unit,C: product_unit,D: product_unit] :
( ( ( insert_Product_unit2 @ A2 @ ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) )
= ( insert_Product_unit2 @ C @ ( insert_Product_unit2 @ D @ bot_bo3957492148770167129t_unit ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_196_singleton__iff,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_197_singleton__iff,axiom,
! [B: epistemic_fm_i,A2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_198_singleton__iff,axiom,
! [B: product_unit,A2: product_unit] :
( ( member_Product_unit @ B @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_199_singletonD,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_200_singletonD,axiom,
! [B: epistemic_fm_i,A2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_201_singletonD,axiom,
! [B: product_unit,A2: product_unit] :
( ( member_Product_unit @ B @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_202_subset__insertI2,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,B: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_203_subset__insertI2,axiom,
! [A: set_Product_unit,B2: set_Product_unit,B: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_204_subset__insertI,axiom,
! [B2: set_Epistemic_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B2 @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_205_subset__insertI,axiom,
! [B2: set_Product_unit,A2: product_unit] : ( ord_le3507040750410214029t_unit @ B2 @ ( insert_Product_unit2 @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_206_subset__insert,axiom,
! [X2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X2 @ B2 ) )
= ( ord_le5389487502678872194c_fm_i @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_207_subset__insert,axiom,
! [X2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X2 @ B2 ) )
= ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_208_subset__insert,axiom,
! [X2: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ~ ( member_Product_unit @ X2 @ A )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ X2 @ B2 ) )
= ( ord_le3507040750410214029t_unit @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_209_insert__mono,axiom,
! [C2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ C2 ) @ ( insert7817948997695205106c_fm_i @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_210_insert__mono,axiom,
! [C2: set_Product_unit,D2: set_Product_unit,A2: product_unit] :
( ( ord_le3507040750410214029t_unit @ C2 @ D2 )
=> ( ord_le3507040750410214029t_unit @ ( insert_Product_unit2 @ A2 @ C2 ) @ ( insert_Product_unit2 @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_211_completeness,axiom,
! [P3: episte5489734138418521880t_unit > $o,P2: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [M2: episte5489734138418521880t_unit] :
( ( P3 @ M2 )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ bot_bo4194595901900360558c_fm_i )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ P2 ) ) ) )
=> ( ( P3
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) )
=> ( epistemic_AK_i @ A @ P2 ) ) ) ).
% completeness
thf(fact_212_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_213_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_214_semantics_Osimps_I5_J,axiom,
! [M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2498298383968489120c_fm_i @ M @ W2 @ ( epistemic_Imp_i @ P2 @ Q ) )
= ( ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ Q ) ) ) ).
% semantics.simps(5)
thf(fact_215_fm_Odistinct_I9_J,axiom,
! [X61: i,X62: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_216_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_217_semantics_Osimps_I1_J,axiom,
! [M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i] :
~ ( episte2498298383968489120c_fm_i @ M @ W2 @ epistemic_FF_i ) ).
% semantics.simps(1)
thf(fact_218_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_219_fm_Odistinct_I15_J,axiom,
! [X22: list_char,X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( epistemic_Pro_i @ X22 )
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_220_fm_Odistinct_I17_J,axiom,
! [X22: list_char,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Pro_i @ X22 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_221_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_222_consistent__hereditary,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( ( ord_le3843937902494030498c_fm_i @ S3 @ S )
=> ( episte2285483198712856234tent_i @ A @ S3 ) ) ) ).
% consistent_hereditary
thf(fact_223_semantics_Osimps_I4_J,axiom,
! [M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2498298383968489120c_fm_i @ M @ W2 @ ( epistemic_Con_i @ P2 @ Q ) )
= ( ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 )
& ( episte2498298383968489120c_fm_i @ M @ W2 @ Q ) ) ) ).
% semantics.simps(4)
thf(fact_224_fm_Odistinct_I1_J,axiom,
! [X22: list_char] :
( epistemic_FF_i
!= ( epistemic_Pro_i @ X22 ) ) ).
% fm.distinct(1)
thf(fact_225_fm_Odistinct_I13_J,axiom,
! [X22: list_char,X41: epistemic_fm_i,X42: epistemic_fm_i] :
( ( epistemic_Pro_i @ X22 )
!= ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_226_ax__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( A @ P2 )
=> ( member6642669606046002379c_fm_i @ P2 @ V2 ) ) ) ) ).
% ax_in_maximal
thf(fact_227_Collect__conv__if2,axiom,
! [P3: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ( ( P3 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( A2 = X )
& ( P3 @ X ) ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
& ( ~ ( P3 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( A2 = X )
& ( P3 @ X ) ) )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_conv_if2
thf(fact_228_Collect__conv__if2,axiom,
! [P3: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( ( P3 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( A2 = X )
& ( P3 @ X ) ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
& ( ~ ( P3 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( A2 = X )
& ( P3 @ X ) ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_conv_if2
thf(fact_229_Collect__conv__if2,axiom,
! [P3: product_unit > $o,A2: product_unit] :
( ( ( P3 @ A2 )
=> ( ( collect_Product_unit
@ ^ [X: product_unit] :
( ( A2 = X )
& ( P3 @ X ) ) )
= ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) )
& ( ~ ( P3 @ A2 )
=> ( ( collect_Product_unit
@ ^ [X: product_unit] :
( ( A2 = X )
& ( P3 @ X ) ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Collect_conv_if2
thf(fact_230_Collect__conv__if,axiom,
! [P3: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ( ( P3 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( X = A2 )
& ( P3 @ X ) ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
& ( ~ ( P3 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( X = A2 )
& ( P3 @ X ) ) )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_conv_if
thf(fact_231_Collect__conv__if,axiom,
! [P3: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( ( P3 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( X = A2 )
& ( P3 @ X ) ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
& ( ~ ( P3 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( X = A2 )
& ( P3 @ X ) ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_conv_if
thf(fact_232_Collect__conv__if,axiom,
! [P3: product_unit > $o,A2: product_unit] :
( ( ( P3 @ A2 )
=> ( ( collect_Product_unit
@ ^ [X: product_unit] :
( ( X = A2 )
& ( P3 @ X ) ) )
= ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) )
& ( ~ ( P3 @ A2 )
=> ( ( collect_Product_unit
@ ^ [X: product_unit] :
( ( X = A2 )
& ( P3 @ X ) ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Collect_conv_if
thf(fact_233_kripke_Ocases__scheme,axiom,
! [R: episte5489734138418521880t_unit] :
~ ! [W6: set_se3485332733965609186c_fm_i,K3: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi3: set_Epistemic_fm_i > list_char > $o,More3: product_unit] :
( R
!= ( episte8397520288747125024t_unit @ W6 @ K3 @ ( episte4955834246032687171t_unit @ Pi3 @ More3 ) ) ) ).
% kripke.cases_scheme
thf(fact_234_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_235_subset__singleton__iff,axiom,
! [X4: set_Product_unit,A2: product_unit] :
( ( ord_le3507040750410214029t_unit @ X4 @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) )
= ( ( X4 = bot_bo3957492148770167129t_unit )
| ( X4
= ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% subset_singleton_iff
thf(fact_236_subset__singletonD,axiom,
! [A: set_Epistemic_fm_i,X2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) )
=> ( ( A = bot_bo4194595901900360558c_fm_i )
| ( A
= ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% subset_singletonD
thf(fact_237_subset__singletonD,axiom,
! [A: set_Product_unit,X2: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
=> ( ( A = bot_bo3957492148770167129t_unit )
| ( A
= ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% subset_singletonD
thf(fact_238_consequent__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( member6642669606046002379c_fm_i @ P2 @ V2 )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P2 @ Q ) @ V2 )
=> ( member6642669606046002379c_fm_i @ Q @ V2 ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_239_maximal__extension,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ~ ! [W7: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ V2 @ W7 )
=> ( ( episte2285483198712856234tent_i @ A @ W7 )
=> ~ ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W7 ) ) ) ) ).
% maximal_extension
thf(fact_240_kripke_Ocases,axiom,
! [R: episte5489734138418521880t_unit] :
~ ! [W6: set_se3485332733965609186c_fm_i,K3: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi3: set_Epistemic_fm_i > list_char > $o] :
( R
!= ( episte8397520288747125024t_unit @ W6 @ K3 @ ( episte4955834246032687171t_unit @ Pi3 @ product_Unity ) ) ) ).
% kripke.cases
thf(fact_241_distribution,axiom,
! [M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i,I2: i,P2: epistemic_fm_i,Q: epistemic_fm_i] : ( episte2498298383968489120c_fm_i @ M @ W2 @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I2 @ P2 ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P2 @ Q ) ) ) @ ( epistemic_K_i @ I2 @ Q ) ) ) ).
% distribution
thf(fact_242_exactly__one__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( member6642669606046002379c_fm_i @ P2 @ V2 )
= ( ~ ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ V2 ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_243_consistent__Extend,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( episte2285483198712856234tent_i @ A @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ) ).
% consistent_Extend
thf(fact_244_canonical__model_I2_J,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( ( member6642669606046002379c_fm_i @ P2 @ S )
=> ( episte2285483198712856234tent_i @ A @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ) ) ).
% canonical_model(2)
thf(fact_245_maximal_H__Extend,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] : ( maxima2621630817641868953c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ).
% maximal'_Extend
thf(fact_246_maximal__maximal_H,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S )
= ( maxima2621630817641868953c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ).
% maximal_maximal'
thf(fact_247_the__elem__eq,axiom,
! [X2: epistemic_fm_i] :
( ( the_el2173195912186371479c_fm_i @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) )
= X2 ) ).
% the_elem_eq
thf(fact_248_the__elem__eq,axiom,
! [X2: product_unit] :
( ( the_el608902216710161154t_unit @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
= X2 ) ).
% the_elem_eq
thf(fact_249_iso__tuple__UNIV__I,axiom,
! [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ top_to3675301018284359218c_fm_i ) ).
% iso_tuple_UNIV_I
thf(fact_250_iso__tuple__UNIV__I,axiom,
! [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ top_to8364301187473967186c_fm_i ) ).
% iso_tuple_UNIV_I
thf(fact_251_iso__tuple__UNIV__I,axiom,
! [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ).
% iso_tuple_UNIV_I
thf(fact_252_dual__order_Orefl,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_253_dual__order_Orefl,axiom,
! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_254_order__refl,axiom,
! [X2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X2 @ X2 ) ).
% order_refl
thf(fact_255_order__refl,axiom,
! [X2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ X2 @ X2 ) ).
% order_refl
thf(fact_256_is__singletonI,axiom,
! [X2: epistemic_fm_i] : ( is_sin4923117429174228502c_fm_i @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% is_singletonI
thf(fact_257_is__singletonI,axiom,
! [X2: product_unit] : ( is_sin2160648248035936513t_unit @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) ).
% is_singletonI
thf(fact_258_truth,axiom,
! [M: episte4846006725945195288t_unit,W2: epistemic_fm_i,I2: i,P2: epistemic_fm_i] :
( ( episte540185425077706131t_unit @ M )
=> ( ( member6642669606046002379c_fm_i @ W2 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ P2 ) ) ) ) ).
% truth
thf(fact_259_truth,axiom,
! [M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i,I2: i,P2: epistemic_fm_i] :
( ( episte251305198166243731t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ P2 ) ) ) ) ).
% truth
thf(fact_260_Extend__subset_H,axiom,
! [S: set_Epistemic_fm_i,A: epistemic_fm_i > $o] :
( ( ( field_Epistemic_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) )
!= bot_bo4194595901900360558c_fm_i )
=> ( ord_le3843937902494030498c_fm_i @ S @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ) ).
% Extend_subset'
thf(fact_261_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
thf(fact_262_top__unit__def,axiom,
top_top_Product_unit = product_Unity ).
% top_unit_def
thf(fact_263_top__set__def,axiom,
( top_to3675301018284359218c_fm_i
= ( collec3087743281813070829c_fm_i @ top_to393988326734912811fm_i_o ) ) ).
% top_set_def
thf(fact_264_top__set__def,axiom,
( top_to8364301187473967186c_fm_i
= ( collec4904205187116291597c_fm_i @ top_to2858054242046946059fm_i_o ) ) ).
% top_set_def
thf(fact_265_top__set__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% top_set_def
thf(fact_266_bot__set__def,axiom,
( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ bot_bo6089404950617257231fm_i_o ) ) ).
% bot_set_def
thf(fact_267_bot__set__def,axiom,
( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ bot_bo2580527446789968623fm_i_o ) ) ).
% bot_set_def
thf(fact_268_bot__set__def,axiom,
( bot_bo3957492148770167129t_unit
= ( collect_Product_unit @ bot_bo4642748612307482820unit_o ) ) ).
% bot_set_def
thf(fact_269_less__eq__set__def,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: 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 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_270_less__eq__set__def,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: 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 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_271_less__eq__set__def,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [A3: set_Product_unit,B3: set_Product_unit] :
( ord_le2788506356509622416unit_o
@ ^ [X: product_unit] : ( member_Product_unit @ X @ A3 )
@ ^ [X: product_unit] : ( member_Product_unit @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_272_is__singleton__the__elem,axiom,
( is_sin4923117429174228502c_fm_i
= ( ^ [A3: set_Epistemic_fm_i] :
( A3
= ( insert7817948997695205106c_fm_i @ ( the_el2173195912186371479c_fm_i @ A3 ) @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% is_singleton_the_elem
thf(fact_273_is__singleton__the__elem,axiom,
( is_sin2160648248035936513t_unit
= ( ^ [A3: set_Product_unit] :
( A3
= ( insert_Product_unit2 @ ( the_el608902216710161154t_unit @ A3 ) @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% is_singleton_the_elem
thf(fact_274_is__singletonI_H,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( member1104366573291651755c_fm_i @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_sin2358732406619996150c_fm_i @ A ) ) ) ).
% is_singletonI'
thf(fact_275_is__singletonI_H,axiom,
! [A: set_Epistemic_fm_i] :
( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i,Y3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( member6642669606046002379c_fm_i @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_sin4923117429174228502c_fm_i @ A ) ) ) ).
% is_singletonI'
thf(fact_276_is__singletonI_H,axiom,
! [A: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,Y3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ( member_Product_unit @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_sin2160648248035936513t_unit @ A ) ) ) ).
% is_singletonI'
thf(fact_277_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z ) )
= ( ^ [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X @ Y5 )
& ( ord_le3843937902494030498c_fm_i @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_278_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Product_unit,Z: set_Product_unit] : ( Y = Z ) )
= ( ^ [X: set_Product_unit,Y5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X @ Y5 )
& ( ord_le3507040750410214029t_unit @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_279_ord__eq__le__trans,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2 = B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_280_ord__eq__le__trans,axiom,
! [A2: set_Product_unit,B: set_Product_unit,C: set_Product_unit] :
( ( A2 = B )
=> ( ( ord_le3507040750410214029t_unit @ B @ C )
=> ( ord_le3507040750410214029t_unit @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_281_ord__le__eq__trans,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( B = C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_282_ord__le__eq__trans,axiom,
! [A2: set_Product_unit,B: set_Product_unit,C: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ B )
=> ( ( B = C )
=> ( ord_le3507040750410214029t_unit @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_283_order__antisym,axiom,
! [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y4 )
=> ( ( ord_le3843937902494030498c_fm_i @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_284_order__antisym,axiom,
! [X2: set_Product_unit,Y4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X2 @ Y4 )
=> ( ( ord_le3507040750410214029t_unit @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_285_order_Otrans,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% order.trans
thf(fact_286_order_Otrans,axiom,
! [A2: set_Product_unit,B: set_Product_unit,C: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ B )
=> ( ( ord_le3507040750410214029t_unit @ B @ C )
=> ( ord_le3507040750410214029t_unit @ A2 @ C ) ) ) ).
% order.trans
thf(fact_287_order__trans,axiom,
! [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y4 )
=> ( ( ord_le3843937902494030498c_fm_i @ Y4 @ Z2 )
=> ( ord_le3843937902494030498c_fm_i @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_288_order__trans,axiom,
! [X2: set_Product_unit,Y4: set_Product_unit,Z2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X2 @ Y4 )
=> ( ( ord_le3507040750410214029t_unit @ Y4 @ Z2 )
=> ( ord_le3507040750410214029t_unit @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_289_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z ) )
= ( ^ [A4: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B5 @ A4 )
& ( ord_le3843937902494030498c_fm_i @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_290_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Product_unit,Z: set_Product_unit] : ( Y = Z ) )
= ( ^ [A4: set_Product_unit,B5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ B5 @ A4 )
& ( ord_le3507040750410214029t_unit @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_291_dual__order_Oantisym,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_292_dual__order_Oantisym,axiom,
! [B: set_Product_unit,A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ B @ A2 )
=> ( ( ord_le3507040750410214029t_unit @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_293_dual__order_Otrans,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ B )
=> ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_294_dual__order_Otrans,axiom,
! [B: set_Product_unit,A2: set_Product_unit,C: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ B @ A2 )
=> ( ( ord_le3507040750410214029t_unit @ C @ B )
=> ( ord_le3507040750410214029t_unit @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_295_antisym,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_296_antisym,axiom,
! [A2: set_Product_unit,B: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ B )
=> ( ( ord_le3507040750410214029t_unit @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_297_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z ) )
= ( ^ [A4: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A4 @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_298_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Product_unit,Z: set_Product_unit] : ( Y = Z ) )
= ( ^ [A4: set_Product_unit,B5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A4 @ B5 )
& ( ord_le3507040750410214029t_unit @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_299_order__subst1,axiom,
! [A2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ B ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_300_order__subst1,axiom,
! [A2: set_Epistemic_fm_i,F: set_Product_unit > set_Epistemic_fm_i,B: set_Product_unit,C: set_Product_unit] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ B ) )
=> ( ( ord_le3507040750410214029t_unit @ B @ C )
=> ( ! [X3: set_Product_unit,Y3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X3 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_301_order__subst1,axiom,
! [A2: set_Product_unit,F: set_Epistemic_fm_i > set_Product_unit,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3507040750410214029t_unit @ A2 @ ( F @ B ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y3 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3507040750410214029t_unit @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_302_order__subst1,axiom,
! [A2: set_Product_unit,F: set_Product_unit > set_Product_unit,B: set_Product_unit,C: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ ( F @ B ) )
=> ( ( ord_le3507040750410214029t_unit @ B @ C )
=> ( ! [X3: set_Product_unit,Y3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X3 @ Y3 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3507040750410214029t_unit @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_303_order__subst2,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F @ B ) @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_304_order__subst2,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Product_unit,C: set_Product_unit] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ord_le3507040750410214029t_unit @ ( F @ B ) @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y3 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3507040750410214029t_unit @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_305_order__subst2,axiom,
! [A2: set_Product_unit,B: set_Product_unit,F: set_Product_unit > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3507040750410214029t_unit @ A2 @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F @ B ) @ C )
=> ( ! [X3: set_Product_unit,Y3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X3 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_306_order__subst2,axiom,
! [A2: set_Product_unit,B: set_Product_unit,F: set_Product_unit > set_Product_unit,C: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ B )
=> ( ( ord_le3507040750410214029t_unit @ ( F @ B ) @ C )
=> ( ! [X3: set_Product_unit,Y3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X3 @ Y3 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3507040750410214029t_unit @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_307_order__eq__refl,axiom,
! [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( X2 = Y4 )
=> ( ord_le3843937902494030498c_fm_i @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_308_order__eq__refl,axiom,
! [X2: set_Product_unit,Y4: set_Product_unit] :
( ( X2 = Y4 )
=> ( ord_le3507040750410214029t_unit @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_309_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_310_ord__eq__le__subst,axiom,
! [A2: set_Product_unit,F: set_Epistemic_fm_i > set_Product_unit,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y3 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3507040750410214029t_unit @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_311_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_i,F: set_Product_unit > set_Epistemic_fm_i,B: set_Product_unit,C: set_Product_unit] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3507040750410214029t_unit @ B @ C )
=> ( ! [X3: set_Product_unit,Y3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X3 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_312_ord__eq__le__subst,axiom,
! [A2: set_Product_unit,F: set_Product_unit > set_Product_unit,B: set_Product_unit,C: set_Product_unit] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3507040750410214029t_unit @ B @ C )
=> ( ! [X3: set_Product_unit,Y3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X3 @ Y3 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3507040750410214029t_unit @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_313_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_314_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Product_unit,C: set_Product_unit] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y3 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3507040750410214029t_unit @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_315_ord__le__eq__subst,axiom,
! [A2: set_Product_unit,B: set_Product_unit,F: set_Product_unit > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3507040750410214029t_unit @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_Product_unit,Y3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X3 @ Y3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_316_ord__le__eq__subst,axiom,
! [A2: set_Product_unit,B: set_Product_unit,F: set_Product_unit > set_Product_unit,C: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_Product_unit,Y3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X3 @ Y3 )
=> ( ord_le3507040750410214029t_unit @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3507040750410214029t_unit @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_317_order__antisym__conv,axiom,
! [Y4: set_Epistemic_fm_i,X2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y4 @ X2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_318_order__antisym__conv,axiom,
! [Y4: set_Product_unit,X2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ Y4 @ X2 )
=> ( ( ord_le3507040750410214029t_unit @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_319_old_Ounit_Oexhaust,axiom,
! [Y4: product_unit] : ( Y4 = product_Unity ) ).
% old.unit.exhaust
thf(fact_320_UNIV__unit,axiom,
( top_to1996260823553986621t_unit
= ( insert_Product_unit2 @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ).
% UNIV_unit
thf(fact_321_is__singletonE,axiom,
! [A: set_Epistemic_fm_i] :
( ( is_sin4923117429174228502c_fm_i @ A )
=> ~ ! [X3: epistemic_fm_i] :
( A
!= ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% is_singletonE
thf(fact_322_is__singletonE,axiom,
! [A: set_Product_unit] :
( ( is_sin2160648248035936513t_unit @ A )
=> ~ ! [X3: product_unit] :
( A
!= ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ).
% is_singletonE
thf(fact_323_is__singleton__def,axiom,
( is_sin4923117429174228502c_fm_i
= ( ^ [A3: set_Epistemic_fm_i] :
? [X: epistemic_fm_i] :
( A3
= ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% is_singleton_def
thf(fact_324_is__singleton__def,axiom,
( is_sin2160648248035936513t_unit
= ( ^ [A3: set_Product_unit] :
? [X: product_unit] :
( A3
= ( insert_Product_unit2 @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% is_singleton_def
thf(fact_325_bot_Oextremum,axiom,
! [A2: product_unit] : ( ord_le3221252021190050221t_unit @ bot_bot_Product_unit @ A2 ) ).
% bot.extremum
thf(fact_326_bot_Oextremum,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A2 ) ).
% bot.extremum
thf(fact_327_bot_Oextremum,axiom,
! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A2 ) ).
% bot.extremum
thf(fact_328_bot_Oextremum__unique,axiom,
! [A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ bot_bot_Product_unit )
= ( A2 = bot_bot_Product_unit ) ) ).
% bot.extremum_unique
thf(fact_329_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_330_bot_Oextremum__unique,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ bot_bo3957492148770167129t_unit )
= ( A2 = bot_bo3957492148770167129t_unit ) ) ).
% bot.extremum_unique
thf(fact_331_bot_Oextremum__uniqueI,axiom,
! [A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ bot_bot_Product_unit )
=> ( A2 = bot_bot_Product_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_332_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_333_bot_Oextremum__uniqueI,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ bot_bo3957492148770167129t_unit )
=> ( A2 = bot_bo3957492148770167129t_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_334_top__greatest,axiom,
! [A2: product_unit] : ( ord_le3221252021190050221t_unit @ A2 @ top_top_Product_unit ) ).
% top_greatest
thf(fact_335_top__greatest,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ top_to8364301187473967186c_fm_i ) ).
% top_greatest
thf(fact_336_top__greatest,axiom,
! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ top_to1996260823553986621t_unit ) ).
% top_greatest
thf(fact_337_top_Oextremum__unique,axiom,
! [A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ top_top_Product_unit @ A2 )
= ( A2 = top_top_Product_unit ) ) ).
% top.extremum_unique
thf(fact_338_top_Oextremum__unique,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ top_to8364301187473967186c_fm_i @ A2 )
= ( A2 = top_to8364301187473967186c_fm_i ) ) ).
% top.extremum_unique
thf(fact_339_top_Oextremum__unique,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
= ( A2 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_unique
thf(fact_340_top_Oextremum__uniqueI,axiom,
! [A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ top_top_Product_unit @ A2 )
=> ( A2 = top_top_Product_unit ) ) ).
% top.extremum_uniqueI
thf(fact_341_top_Oextremum__uniqueI,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ top_to8364301187473967186c_fm_i @ A2 )
=> ( A2 = top_to8364301187473967186c_fm_i ) ) ).
% top.extremum_uniqueI
thf(fact_342_top_Oextremum__uniqueI,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
=> ( A2 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_343_Field__card__of,axiom,
! [A: set_Epistemic_fm_i] :
( ( field_Epistemic_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ A ) )
= A ) ).
% Field_card_of
thf(fact_344_Field__card__of,axiom,
! [A: set_Product_unit] :
( ( field_Product_unit @ ( bNF_Ca7083678892733797993t_unit @ A ) )
= A ) ).
% Field_card_of
thf(fact_345_Field__empty,axiom,
( ( field_Epistemic_fm_i @ bot_bo3668964140382854387c_fm_i )
= bot_bo4194595901900360558c_fm_i ) ).
% Field_empty
thf(fact_346_Field__empty,axiom,
( ( field_Product_unit @ bot_bo1654022762305692595t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Field_empty
thf(fact_347_reflexive_092_060_094sub_062T,axiom,
! [A: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( episte251305198166243731t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) ) ) ).
% reflexive\<^sub>T
thf(fact_348_Extend__bound,axiom,
! [N: epistemic_fm_i,A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ N @ ( field_Epistemic_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ N ) @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S ) ) ) ).
% Extend_bound
thf(fact_349_maximal_H__def,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima2621630817641868953c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( field_Epistemic_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) ) )
=> ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ X @ S ) ) ) ) ) ).
% maximal'_def
thf(fact_350_default__unit__def,axiom,
defaul566961228789861419t_unit = product_Unity ).
% default_unit_def
thf(fact_351_extend__subset,axiom,
! [N: epistemic_fm_i,S: set_Epistemic_fm_i,A: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ N @ ( field_Epistemic_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) ) )
=> ( ord_le3843937902494030498c_fm_i @ S @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ N ) ) ) ).
% extend_subset
thf(fact_352_Euclidean_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [A: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( episte6275404236303465413t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) ) ) ).
% Euclidean\<^sub>K\<^sub>5
thf(fact_353_Un__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
| ( member6642669606046002379c_fm_i @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_354_Un__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B2 ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
| ( member1104366573291651755c_fm_i @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_355_UnCI,axiom,
! [C: epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ C @ B2 )
=> ( member6642669606046002379c_fm_i @ C @ A ) )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% UnCI
thf(fact_356_UnCI,axiom,
! [C: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ~ ( member1104366573291651755c_fm_i @ C @ B2 )
=> ( member1104366573291651755c_fm_i @ C @ A ) )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B2 ) ) ) ).
% UnCI
thf(fact_357_Un__empty,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A @ B2 )
= bot_bo4194595901900360558c_fm_i )
= ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B2 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% Un_empty
thf(fact_358_Un__empty,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ( sup_su793286257634532545t_unit @ A @ B2 )
= bot_bo3957492148770167129t_unit )
= ( ( A = bot_bo3957492148770167129t_unit )
& ( B2 = bot_bo3957492148770167129t_unit ) ) ) ).
% Un_empty
thf(fact_359_Un__subset__iff,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ C2 )
= ( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
& ( ord_le3843937902494030498c_fm_i @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_360_Un__subset__iff,axiom,
! [A: set_Product_unit,B2: set_Product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( sup_su793286257634532545t_unit @ A @ B2 ) @ C2 )
= ( ( ord_le3507040750410214029t_unit @ A @ C2 )
& ( ord_le3507040750410214029t_unit @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_361_Un__insert__right,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% Un_insert_right
thf(fact_362_Un__insert__right,axiom,
! [A: set_Product_unit,A2: product_unit,B2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A @ ( insert_Product_unit2 @ A2 @ B2 ) )
= ( insert_Product_unit2 @ A2 @ ( sup_su793286257634532545t_unit @ A @ B2 ) ) ) ).
% Un_insert_right
thf(fact_363_Un__insert__left,axiom,
! [A2: epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_364_Un__insert__left,axiom,
! [A2: product_unit,B2: set_Product_unit,C2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ A2 @ B2 ) @ C2 )
= ( insert_Product_unit2 @ A2 @ ( sup_su793286257634532545t_unit @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_365_Collect__disj__eq,axiom,
! [P3: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P3 @ X )
| ( Q2 @ X ) ) )
= ( sup_su2582925890723967158c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) @ ( collec3087743281813070829c_fm_i @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_366_Collect__disj__eq,axiom,
! [P3: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P3 @ X )
| ( Q2 @ X ) ) )
= ( sup_su1936195050962291414c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) @ ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_367_Un__def,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A3 )
| ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_368_Un__def,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A3 )
| ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_369_UnI2,axiom,
! [C: epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ B2 )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% UnI2
thf(fact_370_UnI2,axiom,
! [C: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ B2 )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B2 ) ) ) ).
% UnI2
thf(fact_371_UnI1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ) ).
% UnI1
thf(fact_372_UnI1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B2 ) ) ) ).
% UnI1
thf(fact_373_UnE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) )
=> ( ~ ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B2 ) ) ) ).
% UnE
thf(fact_374_UnE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B2 ) )
=> ( ~ ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B2 ) ) ) ).
% UnE
thf(fact_375_Un__UNIV__right,axiom,
! [A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ top_to8364301187473967186c_fm_i )
= top_to8364301187473967186c_fm_i ) ).
% Un_UNIV_right
thf(fact_376_Un__UNIV__right,axiom,
! [A: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% Un_UNIV_right
thf(fact_377_Un__UNIV__left,axiom,
! [B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ top_to8364301187473967186c_fm_i @ B2 )
= top_to8364301187473967186c_fm_i ) ).
% Un_UNIV_left
thf(fact_378_Un__UNIV__left,axiom,
! [B2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ B2 )
= top_to1996260823553986621t_unit ) ).
% Un_UNIV_left
thf(fact_379_Un__empty__right,axiom,
! [A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= A ) ).
% Un_empty_right
thf(fact_380_Un__empty__right,axiom,
! [A: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A @ bot_bo3957492148770167129t_unit )
= A ) ).
% Un_empty_right
thf(fact_381_Un__empty__left,axiom,
! [B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_382_Un__empty__left,axiom,
! [B2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_383_subset__Un__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_384_subset__Un__eq,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [A3: set_Product_unit,B3: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_385_subset__UnE,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) )
=> ~ ! [A5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A5 @ A )
=> ! [B6: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B6 @ B2 )
=> ( C2
!= ( sup_su1936195050962291414c_fm_i @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_386_subset__UnE,axiom,
! [C2: set_Product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ C2 @ ( sup_su793286257634532545t_unit @ A @ B2 ) )
=> ~ ! [A5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A5 @ A )
=> ! [B6: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ B6 @ B2 )
=> ( C2
!= ( sup_su793286257634532545t_unit @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_387_Un__absorb2,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_388_Un__absorb2,axiom,
! [B2: set_Product_unit,A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ B2 @ A )
=> ( ( sup_su793286257634532545t_unit @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_389_Un__absorb1,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_390_Un__absorb1,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ( sup_su793286257634532545t_unit @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_391_Un__upper2,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B2 @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ).
% Un_upper2
thf(fact_392_Un__upper2,axiom,
! [B2: set_Product_unit,A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ B2 @ ( sup_su793286257634532545t_unit @ A @ B2 ) ) ).
% Un_upper2
thf(fact_393_Un__upper1,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) ) ).
% Un_upper1
thf(fact_394_Un__upper1,axiom,
! [A: set_Product_unit,B2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ ( sup_su793286257634532545t_unit @ A @ B2 ) ) ).
% Un_upper1
thf(fact_395_Un__least,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_396_Un__least,axiom,
! [A: set_Product_unit,C2: set_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ C2 )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ C2 )
=> ( ord_le3507040750410214029t_unit @ ( sup_su793286257634532545t_unit @ A @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_397_Un__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B2 ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_398_Un__mono,axiom,
! [A: set_Product_unit,C2: set_Product_unit,B2: set_Product_unit,D2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ C2 )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ D2 )
=> ( ord_le3507040750410214029t_unit @ ( sup_su793286257634532545t_unit @ A @ B2 ) @ ( sup_su793286257634532545t_unit @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_399_insert__def,axiom,
( insert_Product_unit2
= ( ^ [A4: product_unit] :
( sup_su793286257634532545t_unit
@ ( collect_Product_unit
@ ^ [X: product_unit] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_400_insert__def,axiom,
( insert7698009978809854162c_fm_i
= ( ^ [A4: set_Epistemic_fm_i] :
( sup_su2582925890723967158c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_401_insert__def,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A4: epistemic_fm_i] :
( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_402_soundness__Ax5,axiom,
! [P2: epistemic_fm_i,M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i] :
( ( epistemic_Ax5_i @ P2 )
=> ( ( episte6275404236303465413t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 ) ) ) ) ).
% soundness_Ax5
thf(fact_403_S5_H__4,axiom,
! [A: epistemic_fm_i > $o,I2: i,P2: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P2 ) ) ) ) ) ) ).
% S5'_4
thf(fact_404_singleton__Un__iff,axiom,
! [X2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i )
= ( sup_su1936195050962291414c_fm_i @ A @ B2 ) )
= ( ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B2
= ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) )
& ( B2 = bot_bo4194595901900360558c_fm_i ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) )
& ( B2
= ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_405_singleton__Un__iff,axiom,
! [X2: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ( ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit )
= ( sup_su793286257634532545t_unit @ A @ B2 ) )
= ( ( ( A = bot_bo3957492148770167129t_unit )
& ( B2
= ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) )
| ( ( A
= ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
& ( B2 = bot_bo3957492148770167129t_unit ) )
| ( ( A
= ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
& ( B2
= ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_406_Un__singleton__iff,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,X2: epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A @ B2 )
= ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B2
= ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) )
& ( B2 = bot_bo4194595901900360558c_fm_i ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) )
& ( B2
= ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_407_Un__singleton__iff,axiom,
! [A: set_Product_unit,B2: set_Product_unit,X2: product_unit] :
( ( ( sup_su793286257634532545t_unit @ A @ B2 )
= ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
= ( ( ( A = bot_bo3957492148770167129t_unit )
& ( B2
= ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) )
| ( ( A
= ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
& ( B2 = bot_bo3957492148770167129t_unit ) )
| ( ( A
= ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
& ( B2
= ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_408_insert__is__Un,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A4: epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ A4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% insert_is_Un
thf(fact_409_insert__is__Un,axiom,
( insert_Product_unit2
= ( ^ [A4: product_unit] : ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ A4 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% insert_is_Un
thf(fact_410_S5_H__B,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,I2: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% S5'_B
thf(fact_411_AxT_Ointros,axiom,
! [I2: i,P2: epistemic_fm_i] : ( epistemic_AxT_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ P2 ) ) ).
% AxT.intros
thf(fact_412_AxT_Osimps,axiom,
( epistemic_AxT_i
= ( ^ [A4: epistemic_fm_i] :
? [I: i,P: epistemic_fm_i] :
( A4
= ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P ) @ P ) ) ) ) ).
% AxT.simps
thf(fact_413_AxT_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_AxT_i @ A2 )
=> ~ ! [I3: i,P4: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P4 ) @ P4 ) ) ) ).
% AxT.cases
thf(fact_414_top__empty__eq,axiom,
( top_to393988326734912811fm_i_o
= ( ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ top_to3675301018284359218c_fm_i ) ) ) ).
% top_empty_eq
thf(fact_415_top__empty__eq,axiom,
( top_to2858054242046946059fm_i_o
= ( ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ top_to8364301187473967186c_fm_i ) ) ) ).
% top_empty_eq
thf(fact_416_top__empty__eq,axiom,
( top_to2465898995584390880unit_o
= ( ^ [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ) ) ).
% top_empty_eq
thf(fact_417_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_418_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_419_bot__empty__eq,axiom,
( bot_bo4642748612307482820unit_o
= ( ^ [X: product_unit] : ( member_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ).
% bot_empty_eq
thf(fact_420_soundness__AxT,axiom,
! [P2: epistemic_fm_i,M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i] :
( ( epistemic_AxT_i @ P2 )
=> ( ( episte251305198166243731t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 ) ) ) ) ).
% soundness_AxT
thf(fact_421_Ax5_Ointros,axiom,
! [I2: i,P2: epistemic_fm_i] : ( epistemic_Ax5_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% Ax5.intros
thf(fact_422_Ax5_Osimps,axiom,
( epistemic_Ax5_i
= ( ^ [A4: epistemic_fm_i] :
? [I: i,P: epistemic_fm_i] :
( A4
= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_423_Ax5_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_Ax5_i @ A2 )
=> ~ ! [I3: i,P4: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% Ax5.cases
thf(fact_424_pred__subset__eq,axiom,
! [R2: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( ord_le5706303257367236315fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ R2 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ S ) )
= ( ord_le5389487502678872194c_fm_i @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_425_pred__subset__eq,axiom,
! [R2: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ R2 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ S ) )
= ( ord_le3843937902494030498c_fm_i @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_426_pred__subset__eq,axiom,
! [R2: set_Product_unit,S: set_Product_unit] :
( ( ord_le2788506356509622416unit_o
@ ^ [X: product_unit] : ( member_Product_unit @ X @ R2 )
@ ^ [X: product_unit] : ( member_Product_unit @ X @ S ) )
= ( ord_le3507040750410214029t_unit @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_427_consistent__extend,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,N: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( episte2285483198712856234tent_i @ A @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ N ) ) ) ).
% consistent_extend
thf(fact_428_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_429_T__L,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,I2: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% T_L
thf(fact_430_consistent__consequent,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( member6642669606046002379c_fm_i @ P2 @ V2 )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ Q ) )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q @ bot_bo4194595901900360558c_fm_i ) @ V2 ) ) ) ) ) ).
% consistent_consequent
thf(fact_431_AxT__reflexive,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,I2: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( member1104366573291651755c_fm_i @ V2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P ) @ V2 ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_432_K5__L,axiom,
! [A: epistemic_fm_i > $o,I2: i,P2: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ P2 ) ) ) ) ).
% K5_L
thf(fact_433_mono__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i,S4: set_Pr4658907567593863815c_fm_i] :
( ( ord_le3932863852503339559c_fm_i @ R @ S4 )
=> ( ord_le3843937902494030498c_fm_i @ ( field_Epistemic_fm_i @ R ) @ ( field_Epistemic_fm_i @ S4 ) ) ) ).
% mono_Field
thf(fact_434_mono__Field,axiom,
! [R: set_Pr5094982260447487303t_unit,S4: set_Pr5094982260447487303t_unit] :
( ( ord_le930853871177014503t_unit @ R @ S4 )
=> ( ord_le3507040750410214029t_unit @ ( field_Product_unit @ R ) @ ( field_Product_unit @ S4 ) ) ) ).
% mono_Field
thf(fact_435_Ax5__Euclidean,axiom,
! [A: epistemic_fm_i > $o,U2: set_Epistemic_fm_i,V2: set_Epistemic_fm_i,W8: set_Epistemic_fm_i,I2: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ U2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ U2 )
=> ( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( episte2285483198712856234tent_i @ A @ W8 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W8 )
=> ( ( member1104366573291651755c_fm_i @ V2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P ) @ U2 ) ) ) ) )
=> ( ( member1104366573291651755c_fm_i @ W8
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P ) @ U2 ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ W8
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P ) @ V2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_436_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X2 @ top_to8364301187473967186c_fm_i )
= top_to8364301187473967186c_fm_i ) ).
% boolean_algebra.disj_one_right
thf(fact_437_boolean__algebra_Odisj__one__right,axiom,
! [X2: product_unit] :
( ( sup_sup_Product_unit @ X2 @ top_top_Product_unit )
= top_top_Product_unit ) ).
% boolean_algebra.disj_one_right
thf(fact_438_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X2 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% boolean_algebra.disj_one_right
thf(fact_439_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ top_to8364301187473967186c_fm_i @ X2 )
= top_to8364301187473967186c_fm_i ) ).
% boolean_algebra.disj_one_left
thf(fact_440_boolean__algebra_Odisj__one__left,axiom,
! [X2: product_unit] :
( ( sup_sup_Product_unit @ top_top_Product_unit @ X2 )
= top_top_Product_unit ) ).
% boolean_algebra.disj_one_left
thf(fact_441_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X2 )
= top_to1996260823553986621t_unit ) ).
% boolean_algebra.disj_one_left
thf(fact_442_sup__top__right,axiom,
! [X2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X2 @ top_to8364301187473967186c_fm_i )
= top_to8364301187473967186c_fm_i ) ).
% sup_top_right
thf(fact_443_sup__top__right,axiom,
! [X2: product_unit] :
( ( sup_sup_Product_unit @ X2 @ top_top_Product_unit )
= top_top_Product_unit ) ).
% sup_top_right
thf(fact_444_sup__top__right,axiom,
! [X2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X2 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% sup_top_right
thf(fact_445_sup__top__left,axiom,
! [X2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ top_to8364301187473967186c_fm_i @ X2 )
= top_to8364301187473967186c_fm_i ) ).
% sup_top_left
thf(fact_446_sup__top__left,axiom,
! [X2: product_unit] :
( ( sup_sup_Product_unit @ top_top_Product_unit @ X2 )
= top_top_Product_unit ) ).
% sup_top_left
thf(fact_447_sup__top__left,axiom,
! [X2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X2 )
= top_to1996260823553986621t_unit ) ).
% sup_top_left
thf(fact_448_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_449_sup__bot_Oright__neutral,axiom,
! [A2: product_unit] :
( ( sup_sup_Product_unit @ A2 @ bot_bot_Product_unit )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_450_sup__bot_Oright__neutral,axiom,
! [A2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A2 @ bot_bo3957492148770167129t_unit )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_451_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( sup_su1936195050962291414c_fm_i @ A2 @ B ) )
= ( ( A2 = bot_bo4194595901900360558c_fm_i )
& ( B = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_452_sup__bot_Oneutr__eq__iff,axiom,
! [A2: product_unit,B: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ A2 @ B ) )
= ( ( A2 = bot_bot_Product_unit )
& ( B = bot_bot_Product_unit ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_453_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Product_unit,B: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( sup_su793286257634532545t_unit @ A2 @ B ) )
= ( ( A2 = bot_bo3957492148770167129t_unit )
& ( B = bot_bo3957492148770167129t_unit ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_454_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_455_sup__bot_Oleft__neutral,axiom,
! [A2: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_456_sup__bot_Oleft__neutral,axiom,
! [A2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_457_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A2 @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ( A2 = bot_bo4194595901900360558c_fm_i )
& ( B = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_458_sup__bot_Oeq__neutr__iff,axiom,
! [A2: product_unit,B: product_unit] :
( ( ( sup_sup_Product_unit @ A2 @ B )
= bot_bot_Product_unit )
= ( ( A2 = bot_bot_Product_unit )
& ( B = bot_bot_Product_unit ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_459_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Product_unit,B: set_Product_unit] :
( ( ( sup_su793286257634532545t_unit @ A2 @ B )
= bot_bo3957492148770167129t_unit )
= ( ( A2 = bot_bo3957492148770167129t_unit )
& ( B = bot_bo3957492148770167129t_unit ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_460_sup_Oright__idem,axiom,
! [A2: product_unit,B: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ A2 @ B ) @ B )
= ( sup_sup_Product_unit @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_461_sup__left__idem,axiom,
! [X2: product_unit,Y4: product_unit] :
( ( sup_sup_Product_unit @ X2 @ ( sup_sup_Product_unit @ X2 @ Y4 ) )
= ( sup_sup_Product_unit @ X2 @ Y4 ) ) ).
% sup_left_idem
thf(fact_462_sup_Oleft__idem,axiom,
! [A2: product_unit,B: product_unit] :
( ( sup_sup_Product_unit @ A2 @ ( sup_sup_Product_unit @ A2 @ B ) )
= ( sup_sup_Product_unit @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_463_sup__idem,axiom,
! [X2: product_unit] :
( ( sup_sup_Product_unit @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_464_sup_Oidem,axiom,
! [A2: product_unit] :
( ( sup_sup_Product_unit @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_465_le__sup__iff,axiom,
! [X2: product_unit,Y4: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X2 @ Y4 ) @ Z2 )
= ( ( ord_le3221252021190050221t_unit @ X2 @ Z2 )
& ( ord_le3221252021190050221t_unit @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_466_le__sup__iff,axiom,
! [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X2 @ Y4 ) @ Z2 )
= ( ( ord_le3843937902494030498c_fm_i @ X2 @ Z2 )
& ( ord_le3843937902494030498c_fm_i @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_467_le__sup__iff,axiom,
! [X2: set_Product_unit,Y4: set_Product_unit,Z2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( sup_su793286257634532545t_unit @ X2 @ Y4 ) @ Z2 )
= ( ( ord_le3507040750410214029t_unit @ X2 @ Z2 )
& ( ord_le3507040750410214029t_unit @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_468_sup_Obounded__iff,axiom,
! [B: product_unit,C: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B @ C ) @ A2 )
= ( ( ord_le3221252021190050221t_unit @ B @ A2 )
& ( ord_le3221252021190050221t_unit @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_469_sup_Obounded__iff,axiom,
! [B: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B @ C ) @ A2 )
= ( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
& ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_470_sup_Obounded__iff,axiom,
! [B: set_Product_unit,C: set_Product_unit,A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( sup_su793286257634532545t_unit @ B @ C ) @ A2 )
= ( ( ord_le3507040750410214029t_unit @ B @ A2 )
& ( ord_le3507040750410214029t_unit @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_471_sup__bot__left,axiom,
! [X2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_472_sup__bot__left,axiom,
! [X2: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_473_sup__bot__left,axiom,
! [X2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_474_sup__bot__right,axiom,
! [X2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i )
= X2 ) ).
% sup_bot_right
thf(fact_475_sup__bot__right,axiom,
! [X2: product_unit] :
( ( sup_sup_Product_unit @ X2 @ bot_bot_Product_unit )
= X2 ) ).
% sup_bot_right
thf(fact_476_sup__bot__right,axiom,
! [X2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X2 @ bot_bo3957492148770167129t_unit )
= X2 ) ).
% sup_bot_right
thf(fact_477_bot__eq__sup__iff,axiom,
! [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( sup_su1936195050962291414c_fm_i @ X2 @ Y4 ) )
= ( ( X2 = bot_bo4194595901900360558c_fm_i )
& ( Y4 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% bot_eq_sup_iff
thf(fact_478_bot__eq__sup__iff,axiom,
! [X2: product_unit,Y4: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ X2 @ Y4 ) )
= ( ( X2 = bot_bot_Product_unit )
& ( Y4 = bot_bot_Product_unit ) ) ) ).
% bot_eq_sup_iff
thf(fact_479_bot__eq__sup__iff,axiom,
! [X2: set_Product_unit,Y4: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( sup_su793286257634532545t_unit @ X2 @ Y4 ) )
= ( ( X2 = bot_bo3957492148770167129t_unit )
& ( Y4 = bot_bo3957492148770167129t_unit ) ) ) ).
% bot_eq_sup_iff
thf(fact_480_sup__eq__bot__iff,axiom,
! [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ X2 @ Y4 )
= bot_bo4194595901900360558c_fm_i )
= ( ( X2 = bot_bo4194595901900360558c_fm_i )
& ( Y4 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_eq_bot_iff
thf(fact_481_sup__eq__bot__iff,axiom,
! [X2: product_unit,Y4: product_unit] :
( ( ( sup_sup_Product_unit @ X2 @ Y4 )
= bot_bot_Product_unit )
= ( ( X2 = bot_bot_Product_unit )
& ( Y4 = bot_bot_Product_unit ) ) ) ).
% sup_eq_bot_iff
thf(fact_482_sup__eq__bot__iff,axiom,
! [X2: set_Product_unit,Y4: set_Product_unit] :
( ( ( sup_su793286257634532545t_unit @ X2 @ Y4 )
= bot_bo3957492148770167129t_unit )
= ( ( X2 = bot_bo3957492148770167129t_unit )
& ( Y4 = bot_bo3957492148770167129t_unit ) ) ) ).
% sup_eq_bot_iff
thf(fact_483_sup__unit__def,axiom,
( sup_sup_Product_unit
= ( ^ [Uu: product_unit,Uv: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_484_sup__set__def,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: 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 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_485_sup__set__def,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: 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 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_486_sup__Un__eq,axiom,
! [R2: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( sup_su2844414564337698183fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ R2 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ S ) )
= ( ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ ( sup_su1936195050962291414c_fm_i @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_487_sup__Un__eq,axiom,
! [R2: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( sup_su6723579903606884263fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ R2 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ S ) )
= ( ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ ( sup_su2582925890723967158c_fm_i @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_488_sup__left__commute,axiom,
! [X2: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ X2 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) )
= ( sup_sup_Product_unit @ Y4 @ ( sup_sup_Product_unit @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_489_sup_Oleft__commute,axiom,
! [B: product_unit,A2: product_unit,C: product_unit] :
( ( sup_sup_Product_unit @ B @ ( sup_sup_Product_unit @ A2 @ C ) )
= ( sup_sup_Product_unit @ A2 @ ( sup_sup_Product_unit @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_490_boolean__algebra__cancel_Osup2,axiom,
! [B2: product_unit,K4: product_unit,B: product_unit,A2: product_unit] :
( ( B2
= ( sup_sup_Product_unit @ K4 @ B ) )
=> ( ( sup_sup_Product_unit @ A2 @ B2 )
= ( sup_sup_Product_unit @ K4 @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_491_boolean__algebra__cancel_Osup1,axiom,
! [A: product_unit,K4: product_unit,A2: product_unit,B: product_unit] :
( ( A
= ( sup_sup_Product_unit @ K4 @ A2 ) )
=> ( ( sup_sup_Product_unit @ A @ B )
= ( sup_sup_Product_unit @ K4 @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_492_sup__commute,axiom,
( sup_sup_Product_unit
= ( ^ [X: product_unit,Y5: product_unit] : ( sup_sup_Product_unit @ Y5 @ X ) ) ) ).
% sup_commute
thf(fact_493_sup_Ocommute,axiom,
( sup_sup_Product_unit
= ( ^ [A4: product_unit,B5: product_unit] : ( sup_sup_Product_unit @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_494_sup__assoc,axiom,
! [X2: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ X2 @ Y4 ) @ Z2 )
= ( sup_sup_Product_unit @ X2 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_495_sup_Oassoc,axiom,
! [A2: product_unit,B: product_unit,C: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ A2 @ B ) @ C )
= ( sup_sup_Product_unit @ A2 @ ( sup_sup_Product_unit @ B @ C ) ) ) ).
% sup.assoc
thf(fact_496_inf__sup__aci_I5_J,axiom,
( sup_sup_Product_unit
= ( ^ [X: product_unit,Y5: product_unit] : ( sup_sup_Product_unit @ Y5 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_497_inf__sup__aci_I6_J,axiom,
! [X2: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ ( sup_sup_Product_unit @ X2 @ Y4 ) @ Z2 )
= ( sup_sup_Product_unit @ X2 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_498_inf__sup__aci_I7_J,axiom,
! [X2: product_unit,Y4: product_unit,Z2: product_unit] :
( ( sup_sup_Product_unit @ X2 @ ( sup_sup_Product_unit @ Y4 @ Z2 ) )
= ( sup_sup_Product_unit @ Y4 @ ( sup_sup_Product_unit @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_499_inf__sup__aci_I8_J,axiom,
! [X2: product_unit,Y4: product_unit] :
( ( sup_sup_Product_unit @ X2 @ ( sup_sup_Product_unit @ X2 @ Y4 ) )
= ( sup_sup_Product_unit @ X2 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_500_sup_OcoboundedI2,axiom,
! [C: product_unit,B: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ B )
=> ( ord_le3221252021190050221t_unit @ C @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_501_sup_OcoboundedI2,axiom,
! [C: set_Epistemic_fm_i,B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ B )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_502_sup_OcoboundedI2,axiom,
! [C: set_Product_unit,B: set_Product_unit,A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ C @ B )
=> ( ord_le3507040750410214029t_unit @ C @ ( sup_su793286257634532545t_unit @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_503_sup_OcoboundedI1,axiom,
! [C: product_unit,A2: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ A2 )
=> ( ord_le3221252021190050221t_unit @ C @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_504_sup_OcoboundedI1,axiom,
! [C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_505_sup_OcoboundedI1,axiom,
! [C: set_Product_unit,A2: set_Product_unit,B: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ C @ A2 )
=> ( ord_le3507040750410214029t_unit @ C @ ( sup_su793286257634532545t_unit @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_506_sup_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A4: product_unit,B5: product_unit] :
( ( sup_sup_Product_unit @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_507_sup_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_508_sup_Oabsorb__iff2,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [A4: set_Product_unit,B5: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_509_sup_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B5: product_unit,A4: product_unit] :
( ( sup_sup_Product_unit @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_510_sup_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B5: set_Epistemic_fm_i,A4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_511_sup_Oabsorb__iff1,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [B5: set_Product_unit,A4: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_512_sup_Ocobounded2,axiom,
! [B: product_unit,A2: product_unit] : ( ord_le3221252021190050221t_unit @ B @ ( sup_sup_Product_unit @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_513_sup_Ocobounded2,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_514_sup_Ocobounded2,axiom,
! [B: set_Product_unit,A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ B @ ( sup_su793286257634532545t_unit @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_515_sup_Ocobounded1,axiom,
! [A2: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ A2 @ ( sup_sup_Product_unit @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_516_sup_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_517_sup_Ocobounded1,axiom,
! [A2: set_Product_unit,B: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ ( sup_su793286257634532545t_unit @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_518_sup_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B5: product_unit,A4: product_unit] :
( A4
= ( sup_sup_Product_unit @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_519_sup_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B5: set_Epistemic_fm_i,A4: set_Epistemic_fm_i] :
( A4
= ( sup_su1936195050962291414c_fm_i @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_520_sup_Oorder__iff,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [B5: set_Product_unit,A4: set_Product_unit] :
( A4
= ( sup_su793286257634532545t_unit @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_521_sup_OboundedI,axiom,
! [B: product_unit,A2: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A2 )
=> ( ( ord_le3221252021190050221t_unit @ C @ A2 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_522_sup_OboundedI,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_523_sup_OboundedI,axiom,
! [B: set_Product_unit,A2: set_Product_unit,C: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ B @ A2 )
=> ( ( ord_le3507040750410214029t_unit @ C @ A2 )
=> ( ord_le3507040750410214029t_unit @ ( sup_su793286257634532545t_unit @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_524_sup_OboundedE,axiom,
! [B: product_unit,C: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B @ C ) @ A2 )
=> ~ ( ( ord_le3221252021190050221t_unit @ B @ A2 )
=> ~ ( ord_le3221252021190050221t_unit @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_525_sup_OboundedE,axiom,
! [B: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B @ C ) @ A2 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_526_sup_OboundedE,axiom,
! [B: set_Product_unit,C: set_Product_unit,A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( sup_su793286257634532545t_unit @ B @ C ) @ A2 )
=> ~ ( ( ord_le3507040750410214029t_unit @ B @ A2 )
=> ~ ( ord_le3507040750410214029t_unit @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_527_sup__absorb2,axiom,
! [X2: product_unit,Y4: product_unit] :
( ( ord_le3221252021190050221t_unit @ X2 @ Y4 )
=> ( ( sup_sup_Product_unit @ X2 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_528_sup__absorb2,axiom,
! [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y4 )
=> ( ( sup_su1936195050962291414c_fm_i @ X2 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_529_sup__absorb2,axiom,
! [X2: set_Product_unit,Y4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X2 @ Y4 )
=> ( ( sup_su793286257634532545t_unit @ X2 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_530_sup__absorb1,axiom,
! [Y4: product_unit,X2: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y4 @ X2 )
=> ( ( sup_sup_Product_unit @ X2 @ Y4 )
= X2 ) ) ).
% sup_absorb1
thf(fact_531_sup__absorb1,axiom,
! [Y4: set_Epistemic_fm_i,X2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y4 @ X2 )
=> ( ( sup_su1936195050962291414c_fm_i @ X2 @ Y4 )
= X2 ) ) ).
% sup_absorb1
thf(fact_532_sup__absorb1,axiom,
! [Y4: set_Product_unit,X2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ Y4 @ X2 )
=> ( ( sup_su793286257634532545t_unit @ X2 @ Y4 )
= X2 ) ) ).
% sup_absorb1
thf(fact_533_sup_Oabsorb2,axiom,
! [A2: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ B )
=> ( ( sup_sup_Product_unit @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_534_sup_Oabsorb2,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_535_sup_Oabsorb2,axiom,
! [A2: set_Product_unit,B: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ B )
=> ( ( sup_su793286257634532545t_unit @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_536_sup_Oabsorb1,axiom,
! [B: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A2 )
=> ( ( sup_sup_Product_unit @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_537_sup_Oabsorb1,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_538_sup_Oabsorb1,axiom,
! [B: set_Product_unit,A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ B @ A2 )
=> ( ( sup_su793286257634532545t_unit @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_539_sup__unique,axiom,
! [F: product_unit > product_unit > product_unit,X2: product_unit,Y4: product_unit] :
( ! [X3: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: product_unit,Y3: product_unit] : ( ord_le3221252021190050221t_unit @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: product_unit,Y3: product_unit,Z3: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y3 @ X3 )
=> ( ( ord_le3221252021190050221t_unit @ Z3 @ X3 )
=> ( ord_le3221252021190050221t_unit @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_Product_unit @ X2 @ Y4 )
= ( F @ X2 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_540_sup__unique,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y3 @ X3 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z3 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su1936195050962291414c_fm_i @ X2 @ Y4 )
= ( F @ X2 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_541_sup__unique,axiom,
! [F: set_Product_unit > set_Product_unit > set_Product_unit,X2: set_Product_unit,Y4: set_Product_unit] :
( ! [X3: set_Product_unit,Y3: set_Product_unit] : ( ord_le3507040750410214029t_unit @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_Product_unit,Y3: set_Product_unit] : ( ord_le3507040750410214029t_unit @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_Product_unit,Y3: set_Product_unit,Z3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ Y3 @ X3 )
=> ( ( ord_le3507040750410214029t_unit @ Z3 @ X3 )
=> ( ord_le3507040750410214029t_unit @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su793286257634532545t_unit @ X2 @ Y4 )
= ( F @ X2 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_542_sup_OorderI,axiom,
! [A2: product_unit,B: product_unit] :
( ( A2
= ( sup_sup_Product_unit @ A2 @ B ) )
=> ( ord_le3221252021190050221t_unit @ B @ A2 ) ) ).
% sup.orderI
thf(fact_543_sup_OorderI,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A2
= ( sup_su1936195050962291414c_fm_i @ A2 @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ B @ A2 ) ) ).
% sup.orderI
thf(fact_544_sup_OorderI,axiom,
! [A2: set_Product_unit,B: set_Product_unit] :
( ( A2
= ( sup_su793286257634532545t_unit @ A2 @ B ) )
=> ( ord_le3507040750410214029t_unit @ B @ A2 ) ) ).
% sup.orderI
thf(fact_545_sup_OorderE,axiom,
! [B: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A2 )
=> ( A2
= ( sup_sup_Product_unit @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_546_sup_OorderE,axiom,
! [B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A2 )
=> ( A2
= ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_547_sup_OorderE,axiom,
! [B: set_Product_unit,A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ B @ A2 )
=> ( A2
= ( sup_su793286257634532545t_unit @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_548_le__iff__sup,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X: product_unit,Y5: product_unit] :
( ( sup_sup_Product_unit @ X @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_549_le__iff__sup,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_550_le__iff__sup,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [X: set_Product_unit,Y5: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_551_sup__least,axiom,
! [Y4: product_unit,X2: product_unit,Z2: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y4 @ X2 )
=> ( ( ord_le3221252021190050221t_unit @ Z2 @ X2 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ Y4 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_552_sup__least,axiom,
! [Y4: set_Epistemic_fm_i,X2: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y4 @ X2 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z2 @ X2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y4 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_553_sup__least,axiom,
! [Y4: set_Product_unit,X2: set_Product_unit,Z2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ Y4 @ X2 )
=> ( ( ord_le3507040750410214029t_unit @ Z2 @ X2 )
=> ( ord_le3507040750410214029t_unit @ ( sup_su793286257634532545t_unit @ Y4 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_554_sup__mono,axiom,
! [A2: product_unit,C: product_unit,B: product_unit,D: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ C )
=> ( ( ord_le3221252021190050221t_unit @ B @ D )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A2 @ B ) @ ( sup_sup_Product_unit @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_555_sup__mono,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) @ ( sup_su1936195050962291414c_fm_i @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_556_sup__mono,axiom,
! [A2: set_Product_unit,C: set_Product_unit,B: set_Product_unit,D: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ C )
=> ( ( ord_le3507040750410214029t_unit @ B @ D )
=> ( ord_le3507040750410214029t_unit @ ( sup_su793286257634532545t_unit @ A2 @ B ) @ ( sup_su793286257634532545t_unit @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_557_sup_Omono,axiom,
! [C: product_unit,A2: product_unit,D: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ A2 )
=> ( ( ord_le3221252021190050221t_unit @ D @ B )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ C @ D ) @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_558_sup_Omono,axiom,
! [C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,D: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ D @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ C @ D ) @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_559_sup_Omono,axiom,
! [C: set_Product_unit,A2: set_Product_unit,D: set_Product_unit,B: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ C @ A2 )
=> ( ( ord_le3507040750410214029t_unit @ D @ B )
=> ( ord_le3507040750410214029t_unit @ ( sup_su793286257634532545t_unit @ C @ D ) @ ( sup_su793286257634532545t_unit @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_560_le__supI2,axiom,
! [X2: product_unit,B: product_unit,A2: product_unit] :
( ( ord_le3221252021190050221t_unit @ X2 @ B )
=> ( ord_le3221252021190050221t_unit @ X2 @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_561_le__supI2,axiom,
! [X2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ B )
=> ( ord_le3843937902494030498c_fm_i @ X2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_562_le__supI2,axiom,
! [X2: set_Product_unit,B: set_Product_unit,A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X2 @ B )
=> ( ord_le3507040750410214029t_unit @ X2 @ ( sup_su793286257634532545t_unit @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_563_le__supI1,axiom,
! [X2: product_unit,A2: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ X2 @ A2 )
=> ( ord_le3221252021190050221t_unit @ X2 @ ( sup_sup_Product_unit @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_564_le__supI1,axiom,
! [X2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ X2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_565_le__supI1,axiom,
! [X2: set_Product_unit,A2: set_Product_unit,B: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ X2 @ A2 )
=> ( ord_le3507040750410214029t_unit @ X2 @ ( sup_su793286257634532545t_unit @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_566_sup__ge2,axiom,
! [Y4: product_unit,X2: product_unit] : ( ord_le3221252021190050221t_unit @ Y4 @ ( sup_sup_Product_unit @ X2 @ Y4 ) ) ).
% sup_ge2
thf(fact_567_sup__ge2,axiom,
! [Y4: set_Epistemic_fm_i,X2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y4 @ ( sup_su1936195050962291414c_fm_i @ X2 @ Y4 ) ) ).
% sup_ge2
thf(fact_568_sup__ge2,axiom,
! [Y4: set_Product_unit,X2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ Y4 @ ( sup_su793286257634532545t_unit @ X2 @ Y4 ) ) ).
% sup_ge2
thf(fact_569_sup__ge1,axiom,
! [X2: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ X2 @ ( sup_sup_Product_unit @ X2 @ Y4 ) ) ).
% sup_ge1
thf(fact_570_sup__ge1,axiom,
! [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X2 @ ( sup_su1936195050962291414c_fm_i @ X2 @ Y4 ) ) ).
% sup_ge1
thf(fact_571_sup__ge1,axiom,
! [X2: set_Product_unit,Y4: set_Product_unit] : ( ord_le3507040750410214029t_unit @ X2 @ ( sup_su793286257634532545t_unit @ X2 @ Y4 ) ) ).
% sup_ge1
thf(fact_572_le__supI,axiom,
! [A2: product_unit,X2: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A2 @ X2 )
=> ( ( ord_le3221252021190050221t_unit @ B @ X2 )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A2 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_573_le__supI,axiom,
! [A2: set_Epistemic_fm_i,X2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ X2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ X2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_574_le__supI,axiom,
! [A2: set_Product_unit,X2: set_Product_unit,B: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ X2 )
=> ( ( ord_le3507040750410214029t_unit @ B @ X2 )
=> ( ord_le3507040750410214029t_unit @ ( sup_su793286257634532545t_unit @ A2 @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_575_le__supE,axiom,
! [A2: product_unit,B: product_unit,X2: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A2 @ B ) @ X2 )
=> ~ ( ( ord_le3221252021190050221t_unit @ A2 @ X2 )
=> ~ ( ord_le3221252021190050221t_unit @ B @ X2 ) ) ) ).
% le_supE
thf(fact_576_le__supE,axiom,
! [A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B ) @ X2 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A2 @ X2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ B @ X2 ) ) ) ).
% le_supE
thf(fact_577_le__supE,axiom,
! [A2: set_Product_unit,B: set_Product_unit,X2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( sup_su793286257634532545t_unit @ A2 @ B ) @ X2 )
=> ~ ( ( ord_le3507040750410214029t_unit @ A2 @ X2 )
=> ~ ( ord_le3507040750410214029t_unit @ B @ X2 ) ) ) ).
% le_supE
thf(fact_578_inf__sup__ord_I3_J,axiom,
! [X2: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ X2 @ ( sup_sup_Product_unit @ X2 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_579_inf__sup__ord_I3_J,axiom,
! [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X2 @ ( sup_su1936195050962291414c_fm_i @ X2 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_580_inf__sup__ord_I3_J,axiom,
! [X2: set_Product_unit,Y4: set_Product_unit] : ( ord_le3507040750410214029t_unit @ X2 @ ( sup_su793286257634532545t_unit @ X2 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_581_inf__sup__ord_I4_J,axiom,
! [Y4: product_unit,X2: product_unit] : ( ord_le3221252021190050221t_unit @ Y4 @ ( sup_sup_Product_unit @ X2 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_582_inf__sup__ord_I4_J,axiom,
! [Y4: set_Epistemic_fm_i,X2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y4 @ ( sup_su1936195050962291414c_fm_i @ X2 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_583_inf__sup__ord_I4_J,axiom,
! [Y4: set_Product_unit,X2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ Y4 @ ( sup_su793286257634532545t_unit @ X2 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_584_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_585_boolean__algebra_Odisj__zero__right,axiom,
! [X2: product_unit] :
( ( sup_sup_Product_unit @ X2 @ bot_bot_Product_unit )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_586_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X2 @ bot_bo3957492148770167129t_unit )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_587_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_588_AxB__symmetric_H,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,W8: set_Epistemic_fm_i,I2: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( episte2285483198712856234tent_i @ A @ W8 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W8 )
=> ( ( member1104366573291651755c_fm_i @ W8
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P ) @ V2 ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ V2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P ) @ W8 ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_589_exists__finite__inconsistent,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,V2: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ V2 ) )
=> ~ ! [W7: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ W7 ) @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ V2 ) )
=> ( ~ ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ W7 )
=> ( ( finite3304564979551393739c_fm_i @ W7 )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ W7 ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_590_Ax4__transitive,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,W8: set_Epistemic_fm_i,I2: i,U2: set_Epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V2 )
=> ( ( member1104366573291651755c_fm_i @ W8
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P ) @ V2 ) ) ) ) )
=> ( ( member1104366573291651755c_fm_i @ U2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P ) @ W8 ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ U2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I2 @ P ) @ V2 ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_591_Collect__empty__eq__bot,axiom,
! [P3: set_Epistemic_fm_i > $o] :
( ( ( collec3087743281813070829c_fm_i @ P3 )
= bot_bo145720340923748686c_fm_i )
= ( P3 = bot_bo6089404950617257231fm_i_o ) ) ).
% Collect_empty_eq_bot
thf(fact_592_Collect__empty__eq__bot,axiom,
! [P3: epistemic_fm_i > $o] :
( ( ( collec4904205187116291597c_fm_i @ P3 )
= bot_bo4194595901900360558c_fm_i )
= ( P3 = bot_bo2580527446789968623fm_i_o ) ) ).
% Collect_empty_eq_bot
thf(fact_593_Collect__empty__eq__bot,axiom,
! [P3: product_unit > $o] :
( ( ( collect_Product_unit @ P3 )
= bot_bo3957492148770167129t_unit )
= ( P3 = bot_bo4642748612307482820unit_o ) ) ).
% Collect_empty_eq_bot
thf(fact_594_symmetric_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [A: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( episte80897895827131294t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) ) ) ).
% symmetric\<^sub>K\<^sub>B
thf(fact_595_consistent__consequent_H,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( member6642669606046002379c_fm_i @ P2 @ V2 )
=> ( ! [G: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G @ H @ ( epistemic_Imp_i @ P2 @ Q ) )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q @ bot_bo4194595901900360558c_fm_i ) @ V2 ) ) ) ) ) ).
% consistent_consequent'
thf(fact_596_infinite__UNIV__fm,axiom,
~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ).
% infinite_UNIV_fm
thf(fact_597_eval_Osimps_I5_J,axiom,
! [G2: list_char > $o,H2: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_eval_i @ G2 @ H2 @ ( epistemic_Imp_i @ P2 @ Q ) )
= ( ( epistemic_eval_i @ G2 @ H2 @ P2 )
=> ( epistemic_eval_i @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_598_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H2: epistemic_fm_i > $o,I2: i,P2: epistemic_fm_i] :
( ( epistemic_eval_i @ Ux @ H2 @ ( epistemic_K_i @ I2 @ P2 ) )
= ( H2 @ ( epistemic_K_i @ I2 @ P2 ) ) ) ).
% eval.simps(6)
thf(fact_599_A1,axiom,
! [P2: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [G: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G @ H @ P2 )
=> ( epistemic_AK_i @ A @ P2 ) ) ).
% A1
thf(fact_600_eval_Osimps_I1_J,axiom,
! [Uu2: list_char > $o,Uv2: epistemic_fm_i > $o] :
~ ( epistemic_eval_i @ Uu2 @ Uv2 @ epistemic_FF_i ) ).
% eval.simps(1)
thf(fact_601_tautology,axiom,
! [P2: epistemic_fm_i,M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i] :
( ! [G: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G @ H @ P2 )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 ) ) ).
% tautology
thf(fact_602_eval_Osimps_I4_J,axiom,
! [G2: list_char > $o,H2: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_eval_i @ G2 @ H2 @ ( epistemic_Con_i @ P2 @ Q ) )
= ( ( epistemic_eval_i @ G2 @ H2 @ P2 )
& ( epistemic_eval_i @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(4)
thf(fact_603_eval_Osimps_I2_J,axiom,
! [G2: list_char > $o,Uw: epistemic_fm_i > $o,X2: list_char] :
( ( epistemic_eval_i @ G2 @ Uw @ ( epistemic_Pro_i @ X2 ) )
= ( G2 @ X2 ) ) ).
% eval.simps(2)
thf(fact_604_soundness__AxB,axiom,
! [P2: epistemic_fm_i,M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i] :
( ( epistemic_AxB_i @ P2 )
=> ( ( episte80897895827131294t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 ) ) ) ) ).
% soundness_AxB
thf(fact_605_inconsistent__finite,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ S )
=> ? [S5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S5 @ S )
& ( finite3304564979551393739c_fm_i @ S5 )
& ~ ( episte2285483198712856234tent_i @ A @ S5 ) ) ) ).
% inconsistent_finite
thf(fact_606_S5__S5_H,axiom,
! [P2: epistemic_fm_i] :
( ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_AxB_i @ P )
| ( epistemic_Ax4_i @ P ) )
@ P2 )
=> ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_Ax5_i @ P ) )
@ P2 ) ) ).
% S5_S5'
thf(fact_607_S5_H__S5,axiom,
! [P2: epistemic_fm_i] :
( ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_Ax5_i @ P ) )
@ P2 )
=> ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_AxB_i @ P )
| ( epistemic_Ax4_i @ P ) )
@ P2 ) ) ).
% S5'_S5
thf(fact_608_Ax4_Ointros,axiom,
! [I2: i,P2: epistemic_fm_i] : ( epistemic_Ax4_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P2 ) ) ) ) ).
% Ax4.intros
thf(fact_609_Ax4_Osimps,axiom,
( epistemic_Ax4_i
= ( ^ [A4: epistemic_fm_i] :
? [I: i,P: epistemic_fm_i] :
( A4
= ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P ) @ ( epistemic_K_i @ I @ ( epistemic_K_i @ I @ P ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_610_Ax4_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_Ax4_i @ A2 )
=> ~ ! [I3: i,P4: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P4 ) @ ( epistemic_K_i @ I3 @ ( epistemic_K_i @ I3 @ P4 ) ) ) ) ) ).
% Ax4.cases
thf(fact_611_KB4__5,axiom,
! [A: epistemic_fm_i > $o,I2: i,P2: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% KB4_5
thf(fact_612_duality__taut,axiom,
! [I2: i,P2: epistemic_fm_i,Q: epistemic_fm_i,G3: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H3 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ Q @ epistemic_FF_i ) ) ) @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ Q @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ epistemic_FF_i ) ) ) ) ).
% duality_taut
thf(fact_613_AxB_Ointros,axiom,
! [P2: epistemic_fm_i,I2: i] : ( epistemic_AxB_i @ ( epistemic_Imp_i @ P2 @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% AxB.intros
thf(fact_614_AxB_Osimps,axiom,
( epistemic_AxB_i
= ( ^ [A4: epistemic_fm_i] :
? [P: epistemic_fm_i,I: i] :
( A4
= ( epistemic_Imp_i @ P @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% AxB.simps
thf(fact_615_AxB_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_AxB_i @ A2 )
=> ~ ! [P4: epistemic_fm_i,I3: i] :
( A2
!= ( epistemic_Imp_i @ P4 @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% AxB.cases
thf(fact_616_AK_Osimps,axiom,
( epistemic_AK_i
= ( ^ [A3: epistemic_fm_i > $o,A4: epistemic_fm_i] :
( ? [P: epistemic_fm_i] :
( ( A4 = P )
& ! [G4: list_char > $o,H4: epistemic_fm_i > $o] : ( epistemic_eval_i @ G4 @ H4 @ P ) )
| ? [I: i,P: epistemic_fm_i,Q3: epistemic_fm_i] :
( A4
= ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P @ Q3 ) ) ) @ ( epistemic_K_i @ I @ Q3 ) ) )
| ? [P: epistemic_fm_i] :
( ( A4 = P )
& ( A3 @ P ) )
| ? [P: epistemic_fm_i,Q3: epistemic_fm_i] :
( ( A4 = Q3 )
& ( epistemic_AK_i @ A3 @ P )
& ( epistemic_AK_i @ A3 @ ( epistemic_Imp_i @ P @ Q3 ) ) )
| ? [P: epistemic_fm_i,I: i] :
( ( A4
= ( epistemic_K_i @ I @ P ) )
& ( epistemic_AK_i @ A3 @ P ) ) ) ) ) ).
% AK.simps
thf(fact_617_AK_Ocases,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ A2 )
=> ( ~ ! [G3: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H3 @ A2 )
=> ( ! [I3: i,P4: epistemic_fm_i,Q4: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P4 ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P4 @ Q4 ) ) ) @ ( epistemic_K_i @ I3 @ Q4 ) ) )
=> ( ~ ( A @ A2 )
=> ( ! [P4: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ P4 )
=> ~ ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P4 @ A2 ) ) )
=> ~ ! [P4: epistemic_fm_i] :
( ? [I3: i] :
( A2
= ( epistemic_K_i @ I3 @ P4 ) )
=> ~ ( epistemic_AK_i @ A @ P4 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_618_eval__semantics,axiom,
! [Pi4: set_Epistemic_fm_i > list_char > $o,W2: set_Epistemic_fm_i,W8: set_se3485332733965609186c_fm_i,R: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,P2: epistemic_fm_i] :
( ( epistemic_eval_i @ ( Pi4 @ W2 ) @ ( episte2498298383968489120c_fm_i @ ( episte8397520288747125024t_unit @ W8 @ R @ ( episte4955834246032687171t_unit @ Pi4 @ product_Unity ) ) @ W2 ) @ P2 )
= ( episte2498298383968489120c_fm_i @ ( episte8397520288747125024t_unit @ W8 @ R @ ( episte4955834246032687171t_unit @ Pi4 @ product_Unity ) ) @ W2 @ P2 ) ) ).
% eval_semantics
thf(fact_619_K4__L,axiom,
! [A: epistemic_fm_i > $o,I2: i,P2: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% K4_L
thf(fact_620_finite__Collect__subsets,axiom,
! [A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [B3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_621_finite__Collect__subsets,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( finite1772178364199683094t_unit
@ ( collec7787489603430924120t_unit
@ ^ [B3: set_Product_unit] : ( ord_le3507040750410214029t_unit @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_622_finite__Collect__not,axiom,
! [P3: set_Epistemic_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) )
=> ( ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
~ ( P3 @ X ) ) )
= ( finite7933139204641697195c_fm_i @ top_to3675301018284359218c_fm_i ) ) ) ).
% finite_Collect_not
thf(fact_623_finite__Collect__not,axiom,
! [P3: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) )
=> ( ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
~ ( P3 @ X ) ) )
= ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_Collect_not
thf(fact_624_finite__Collect__not,axiom,
! [P3: product_unit > $o] :
( ( finite4290736615968046902t_unit @ ( collect_Product_unit @ P3 ) )
=> ( ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [X: product_unit] :
~ ( P3 @ X ) ) )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Collect_not
thf(fact_625_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_626_finite__insert,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ ( insert_Product_unit2 @ A2 @ A ) )
= ( finite4290736615968046902t_unit @ A ) ) ).
% finite_insert
thf(fact_627_finite__Plus__UNIV__iff,axiom,
( ( finite7028649318107984188c_fm_i @ top_to795506832365754251c_fm_i )
= ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_628_finite__Plus__UNIV__iff,axiom,
( ( finite6693373646013997607t_unit @ top_to3751750602860535030t_unit )
= ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_629_finite__Plus__UNIV__iff,axiom,
( ( finite5258787827736110737c_fm_i @ top_to6207264864741117024c_fm_i )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_630_finite__Plus__UNIV__iff,axiom,
( ( finite3146551501593861116t_unit @ top_to2771918933716375115t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_631_finite__option__UNIV,axiom,
( ( finite4430841290055254033c_fm_i @ top_to7034302889105699736c_fm_i )
= ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% finite_option_UNIV
thf(fact_632_finite__option__UNIV,axiom,
( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% finite_option_UNIV
thf(fact_633_finite__Collect__disjI,axiom,
! [P3: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P3 @ X )
| ( Q2 @ X ) ) ) )
= ( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) )
& ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_634_finite__Collect__disjI,axiom,
! [P3: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P3 @ X )
| ( Q2 @ X ) ) ) )
= ( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) )
& ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_635_finite__Collect__conjI,axiom,
! [P3: set_Epistemic_fm_i > $o,Q2: set_Epistemic_fm_i > $o] :
( ( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) )
| ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ Q2 ) ) )
=> ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( P3 @ X )
& ( Q2 @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_636_finite__Collect__conjI,axiom,
! [P3: epistemic_fm_i > $o,Q2: epistemic_fm_i > $o] :
( ( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) )
| ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ Q2 ) ) )
=> ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( P3 @ X )
& ( Q2 @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_637_not__finite__existsD,axiom,
! [P3: set_Epistemic_fm_i > $o] :
( ~ ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P3 ) )
=> ? [X_1: set_Epistemic_fm_i] : ( P3 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_638_not__finite__existsD,axiom,
! [P3: epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P3 ) )
=> ? [X_1: epistemic_fm_i] : ( P3 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_639_finite__has__minimal2,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( ord_le3843937902494030498c_fm_i @ X3 @ A2 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_640_finite__has__minimal2,axiom,
! [A: set_set_Product_unit,A2: set_Product_unit] :
( ( finite1772178364199683094t_unit @ A )
=> ( ( member5877623283571906838t_unit @ A2 @ A )
=> ? [X3: set_Product_unit] :
( ( member5877623283571906838t_unit @ X3 @ A )
& ( ord_le3507040750410214029t_unit @ X3 @ A2 )
& ! [Xa: set_Product_unit] :
( ( member5877623283571906838t_unit @ Xa @ A )
=> ( ( ord_le3507040750410214029t_unit @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_641_finite__has__maximal2,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( ord_le3843937902494030498c_fm_i @ A2 @ X3 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_642_finite__has__maximal2,axiom,
! [A: set_set_Product_unit,A2: set_Product_unit] :
( ( finite1772178364199683094t_unit @ A )
=> ( ( member5877623283571906838t_unit @ A2 @ A )
=> ? [X3: set_Product_unit] :
( ( member5877623283571906838t_unit @ X3 @ A )
& ( ord_le3507040750410214029t_unit @ A2 @ X3 )
& ! [Xa: set_Product_unit] :
( ( member5877623283571906838t_unit @ Xa @ A )
=> ( ( ord_le3507040750410214029t_unit @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_643_Finite__Set_Ofinite__set,axiom,
( ( finite7933139204641697195c_fm_i @ top_to3675301018284359218c_fm_i )
= ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% Finite_Set.finite_set
thf(fact_644_Finite__Set_Ofinite__set,axiom,
( ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% Finite_Set.finite_set
thf(fact_645_finite__prod,axiom,
( ( finite4491501931600271312c_fm_i @ top_to9141131093895489495c_fm_i )
= ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_prod
thf(fact_646_finite__prod,axiom,
( ( finite5137985727275662011t_unit @ top_to7296822901894720322t_unit )
= ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_647_finite__prod,axiom,
( ( finite3703399908997775141c_fm_i @ top_to528965126920526508c_fm_i )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ) ).
% finite_prod
thf(fact_648_finite__prod,axiom,
( ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_649_finite__Prod__UNIV,axiom,
( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( finite4491501931600271312c_fm_i @ top_to9141131093895489495c_fm_i ) ) ) ).
% finite_Prod_UNIV
thf(fact_650_finite__Prod__UNIV,axiom,
( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite5137985727275662011t_unit @ top_to7296822901894720322t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_651_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( finite3703399908997775141c_fm_i @ top_to528965126920526508c_fm_i ) ) ) ).
% finite_Prod_UNIV
thf(fact_652_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_653_finite__UNIV,axiom,
finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ).
% finite_UNIV
thf(fact_654_ex__new__if__finite,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ~ ( finite7933139204641697195c_fm_i @ top_to3675301018284359218c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ A )
=> ? [A6: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ A6 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_655_ex__new__if__finite,axiom,
! [A: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ( finite3304564979551393739c_fm_i @ A )
=> ? [A6: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ A6 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_656_ex__new__if__finite,axiom,
! [A: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ A )
=> ? [A6: product_unit] :
~ ( member_Product_unit @ A6 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_657_finite_OemptyI,axiom,
finite3304564979551393739c_fm_i @ bot_bo4194595901900360558c_fm_i ).
% finite.emptyI
thf(fact_658_finite_OemptyI,axiom,
finite4290736615968046902t_unit @ bot_bo3957492148770167129t_unit ).
% finite.emptyI
thf(fact_659_infinite__imp__nonempty,axiom,
! [S: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ( S != bot_bo4194595901900360558c_fm_i ) ) ).
% infinite_imp_nonempty
thf(fact_660_infinite__imp__nonempty,axiom,
! [S: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ S )
=> ( S != bot_bo3957492148770167129t_unit ) ) ).
% infinite_imp_nonempty
thf(fact_661_rev__finite__subset,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% rev_finite_subset
thf(fact_662_rev__finite__subset,axiom,
! [B2: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( finite4290736615968046902t_unit @ A ) ) ) ).
% rev_finite_subset
thf(fact_663_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_664_infinite__super,axiom,
! [S: set_Product_unit,T2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ S @ T2 )
=> ( ~ ( finite4290736615968046902t_unit @ S )
=> ~ ( finite4290736615968046902t_unit @ T2 ) ) ) ).
% infinite_super
thf(fact_665_finite__subset,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( finite3304564979551393739c_fm_i @ B2 )
=> ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% finite_subset
thf(fact_666_finite__subset,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( finite4290736615968046902t_unit @ A ) ) ) ).
% finite_subset
thf(fact_667_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_668_finite_OinsertI,axiom,
! [A: set_Product_unit,A2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( finite4290736615968046902t_unit @ ( insert_Product_unit2 @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_669_finite__has__maximal,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
& ! [Xa: product_unit] :
( ( member_Product_unit @ Xa @ A )
=> ( ( ord_le3221252021190050221t_unit @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_670_finite__has__maximal,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_671_finite__has__maximal,axiom,
! [A: set_set_Product_unit] :
( ( finite1772178364199683094t_unit @ A )
=> ( ( A != bot_bo6284674557190688313t_unit )
=> ? [X3: set_Product_unit] :
( ( member5877623283571906838t_unit @ X3 @ A )
& ! [Xa: set_Product_unit] :
( ( member5877623283571906838t_unit @ Xa @ A )
=> ( ( ord_le3507040750410214029t_unit @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_672_finite__has__minimal,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
& ! [Xa: product_unit] :
( ( member_Product_unit @ Xa @ A )
=> ( ( ord_le3221252021190050221t_unit @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_673_finite__has__minimal,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_674_finite__has__minimal,axiom,
! [A: set_set_Product_unit] :
( ( finite1772178364199683094t_unit @ A )
=> ( ( A != bot_bo6284674557190688313t_unit )
=> ? [X3: set_Product_unit] :
( ( member5877623283571906838t_unit @ X3 @ A )
& ! [Xa: set_Product_unit] :
( ( member5877623283571906838t_unit @ Xa @ A )
=> ( ( ord_le3507040750410214029t_unit @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_675_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_676_finite_Ocases,axiom,
! [A2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ( A2 != bot_bo3957492148770167129t_unit )
=> ~ ! [A7: set_Product_unit] :
( ? [A6: product_unit] :
( A2
= ( insert_Product_unit2 @ A6 @ A7 ) )
=> ~ ( finite4290736615968046902t_unit @ A7 ) ) ) ) ).
% finite.cases
thf(fact_677_finite_Osimps,axiom,
( finite3304564979551393739c_fm_i
= ( ^ [A4: set_Epistemic_fm_i] :
( ( A4 = bot_bo4194595901900360558c_fm_i )
| ? [A3: set_Epistemic_fm_i,B5: epistemic_fm_i] :
( ( A4
= ( insert7817948997695205106c_fm_i @ B5 @ A3 ) )
& ( finite3304564979551393739c_fm_i @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_678_finite_Osimps,axiom,
( finite4290736615968046902t_unit
= ( ^ [A4: set_Product_unit] :
( ( A4 = bot_bo3957492148770167129t_unit )
| ? [A3: set_Product_unit,B5: product_unit] :
( ( A4
= ( insert_Product_unit2 @ B5 @ A3 ) )
& ( finite4290736615968046902t_unit @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_679_finite__induct,axiom,
! [F2: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( P3 @ bot_bo145720340923748686c_fm_i )
=> ( ! [X3: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ F3 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert7698009978809854162c_fm_i @ X3 @ F3 ) ) ) ) )
=> ( P3 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_680_finite__induct,axiom,
! [F2: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( P3 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ~ ( member6642669606046002379c_fm_i @ X3 @ F3 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert7817948997695205106c_fm_i @ X3 @ F3 ) ) ) ) )
=> ( P3 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_681_finite__induct,axiom,
! [F2: set_Product_unit,P3: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ( P3 @ bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ~ ( member_Product_unit @ X3 @ F3 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert_Product_unit2 @ X3 @ F3 ) ) ) ) )
=> ( P3 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_682_finite__ne__induct,axiom,
! [F2: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( F2 != bot_bo145720340923748686c_fm_i )
=> ( ! [X3: set_Epistemic_fm_i] : ( P3 @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
=> ( ! [X3: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( F3 != bot_bo145720340923748686c_fm_i )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ F3 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert7698009978809854162c_fm_i @ X3 @ F3 ) ) ) ) ) )
=> ( P3 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_683_finite__ne__induct,axiom,
! [F2: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( F2 != bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i] : ( P3 @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
=> ( ! [X3: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( F3 != bot_bo4194595901900360558c_fm_i )
=> ( ~ ( member6642669606046002379c_fm_i @ X3 @ F3 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert7817948997695205106c_fm_i @ X3 @ F3 ) ) ) ) ) )
=> ( P3 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_684_finite__ne__induct,axiom,
! [F2: set_Product_unit,P3: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ( F2 != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] : ( P3 @ ( insert_Product_unit2 @ X3 @ bot_bo3957492148770167129t_unit ) )
=> ( ! [X3: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ( F3 != bot_bo3957492148770167129t_unit )
=> ( ~ ( member_Product_unit @ X3 @ F3 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert_Product_unit2 @ X3 @ F3 ) ) ) ) ) )
=> ( P3 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_685_infinite__finite__induct,axiom,
! [P3: set_se3485332733965609186c_fm_i > $o,A: set_se3485332733965609186c_fm_i] :
( ! [A7: set_se3485332733965609186c_fm_i] :
( ~ ( finite7933139204641697195c_fm_i @ A7 )
=> ( P3 @ A7 ) )
=> ( ( P3 @ bot_bo145720340923748686c_fm_i )
=> ( ! [X3: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ F3 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert7698009978809854162c_fm_i @ X3 @ F3 ) ) ) ) )
=> ( P3 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_686_infinite__finite__induct,axiom,
! [P3: set_Epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ! [A7: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ A7 )
=> ( P3 @ A7 ) )
=> ( ( P3 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ~ ( member6642669606046002379c_fm_i @ X3 @ F3 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert7817948997695205106c_fm_i @ X3 @ F3 ) ) ) ) )
=> ( P3 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_687_infinite__finite__induct,axiom,
! [P3: set_Product_unit > $o,A: set_Product_unit] :
( ! [A7: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ A7 )
=> ( P3 @ A7 ) )
=> ( ( P3 @ bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ~ ( member_Product_unit @ X3 @ F3 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert_Product_unit2 @ X3 @ F3 ) ) ) ) )
=> ( P3 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_688_finite__subset__induct,axiom,
! [F2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( ord_le5389487502678872194c_fm_i @ F2 @ A )
=> ( ( P3 @ 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 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert7698009978809854162c_fm_i @ A6 @ F3 ) ) ) ) ) )
=> ( P3 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_689_finite__subset__induct,axiom,
! [F2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( ord_le3843937902494030498c_fm_i @ F2 @ A )
=> ( ( P3 @ 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 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert7817948997695205106c_fm_i @ A6 @ F3 ) ) ) ) ) )
=> ( P3 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_690_finite__subset__induct,axiom,
! [F2: set_Product_unit,A: set_Product_unit,P3: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ( ord_le3507040750410214029t_unit @ F2 @ A )
=> ( ( P3 @ bot_bo3957492148770167129t_unit )
=> ( ! [A6: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ( member_Product_unit @ A6 @ A )
=> ( ~ ( member_Product_unit @ A6 @ F3 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert_Product_unit2 @ A6 @ F3 ) ) ) ) ) )
=> ( P3 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_691_finite__subset__induct_H,axiom,
! [F2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( ord_le5389487502678872194c_fm_i @ F2 @ A )
=> ( ( P3 @ 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 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert7698009978809854162c_fm_i @ A6 @ F3 ) ) ) ) ) ) )
=> ( P3 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_692_finite__subset__induct_H,axiom,
! [F2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( ord_le3843937902494030498c_fm_i @ F2 @ A )
=> ( ( P3 @ 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 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert7817948997695205106c_fm_i @ A6 @ F3 ) ) ) ) ) ) )
=> ( P3 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_693_finite__subset__induct_H,axiom,
! [F2: set_Product_unit,A: set_Product_unit,P3: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ( ord_le3507040750410214029t_unit @ F2 @ A )
=> ( ( P3 @ bot_bo3957492148770167129t_unit )
=> ( ! [A6: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ( member_Product_unit @ A6 @ A )
=> ( ( ord_le3507040750410214029t_unit @ F3 @ A )
=> ( ~ ( member_Product_unit @ A6 @ F3 )
=> ( ( P3 @ F3 )
=> ( P3 @ ( insert_Product_unit2 @ A6 @ F3 ) ) ) ) ) ) )
=> ( P3 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_694_transitive_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [A: epistemic_fm_i > $o] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( episte2966952217288386430t_unit
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) ) ) ).
% transitive\<^sub>K\<^sub>4
thf(fact_695_soundness__AxTB4,axiom,
! [P2: epistemic_fm_i,M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P2 )
| ( epistemic_AxB_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
=> ( ( ( episte251305198166243731t_unit @ M )
& ( episte80897895827131294t_unit @ M )
& ( episte2966952217288386430t_unit @ M ) )
=> ( ( member1104366573291651755c_fm_i @ W2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 ) ) ) ) ).
% soundness_AxTB4
thf(fact_696_soundness__AxT5,axiom,
! [P2: epistemic_fm_i,M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax5_i @ P2 ) )
=> ( ( ( episte251305198166243731t_unit @ M )
& ( episte80897895827131294t_unit @ M )
& ( episte2966952217288386430t_unit @ M ) )
=> ( ( member1104366573291651755c_fm_i @ W2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 ) ) ) ) ).
% soundness_AxT5
thf(fact_697_soundness__Ax4,axiom,
! [P2: epistemic_fm_i,M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i] :
( ( epistemic_Ax4_i @ P2 )
=> ( ( episte2966952217288386430t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 ) ) ) ) ).
% soundness_Ax4
thf(fact_698_pos__introspection,axiom,
! [M: episte4846006725945195288t_unit,W2: epistemic_fm_i,I2: i,P2: epistemic_fm_i] :
( ( episte6394096064484582270t_unit @ M )
=> ( ( member6642669606046002379c_fm_i @ W2 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P2 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_699_pos__introspection,axiom,
! [M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i,I2: i,P2: epistemic_fm_i] :
( ( episte2966952217288386430t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P2 ) ) ) ) ) ) ).
% pos_introspection
thf(fact_700_soundness__AxT4,axiom,
! [P2: epistemic_fm_i,M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i] :
( ( ( epistemic_AxT_i @ P2 )
| ( epistemic_Ax4_i @ P2 ) )
=> ( ( ( episte251305198166243731t_unit @ M )
& ( episte2966952217288386430t_unit @ M ) )
=> ( ( member1104366573291651755c_fm_i @ W2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 ) ) ) ) ).
% soundness_AxT4
thf(fact_701_neg__introspection,axiom,
! [M: episte4846006725945195288t_unit,W2: epistemic_fm_i,I2: i,P2: epistemic_fm_i] :
( ( episte6568605431049406366t_unit @ M )
=> ( ( episte6394096064484582270t_unit @ M )
=> ( ( member6642669606046002379c_fm_i @ W2 @ ( episte1297369412406343275t_unit @ M ) )
=> ( episte5182706529199770816c_fm_i @ M @ W2 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ epistemic_FF_i ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_702_neg__introspection,axiom,
! [M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i,I2: i,P2: epistemic_fm_i] :
( ( episte80897895827131294t_unit @ M )
=> ( ( episte2966952217288386430t_unit @ M )
=> ( ( member1104366573291651755c_fm_i @ W2 @ ( episte2675268102452679275t_unit @ M ) )
=> ( episte2498298383968489120c_fm_i @ M @ W2 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P2 ) @ epistemic_FF_i ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_703_cofinite__bot,axiom,
( ( cofini2965452609477090530c_fm_i = bot_bo3270057444204665492c_fm_i )
= ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% cofinite_bot
thf(fact_704_cofinite__bot,axiom,
( ( cofini76534143805230541t_unit = bot_bo7697929036666283903t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% cofinite_bot
thf(fact_705_insert__subsetI,axiom,
! [X2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ X4 @ A )
=> ( ord_le5389487502678872194c_fm_i @ ( insert7698009978809854162c_fm_i @ X2 @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_706_insert__subsetI,axiom,
! [X2: epistemic_fm_i,A: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X2 @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_707_insert__subsetI,axiom,
! [X2: product_unit,A: set_Product_unit,X4: set_Product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ( ord_le3507040750410214029t_unit @ X4 @ A )
=> ( ord_le3507040750410214029t_unit @ ( insert_Product_unit2 @ X2 @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_708_prop__restrict,axiom,
! [X2: set_Epistemic_fm_i,Z4: set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ X2 @ Z4 )
=> ( ( ord_le5389487502678872194c_fm_i @ Z4
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ X4 )
& ( P3 @ X ) ) ) )
=> ( P3 @ X2 ) ) ) ).
% prop_restrict
thf(fact_709_prop__restrict,axiom,
! [X2: epistemic_fm_i,Z4: set_Epistemic_fm_i,X4: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ X2 @ Z4 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z4
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ X4 )
& ( P3 @ X ) ) ) )
=> ( P3 @ X2 ) ) ) ).
% prop_restrict
thf(fact_710_prop__restrict,axiom,
! [X2: product_unit,Z4: set_Product_unit,X4: set_Product_unit,P3: product_unit > $o] :
( ( member_Product_unit @ X2 @ Z4 )
=> ( ( ord_le3507040750410214029t_unit @ Z4
@ ( collect_Product_unit
@ ^ [X: product_unit] :
( ( member_Product_unit @ X @ X4 )
& ( P3 @ X ) ) ) )
=> ( P3 @ X2 ) ) ) ).
% prop_restrict
thf(fact_711_Collect__restrict,axiom,
! [X4: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ X4 )
& ( P3 @ X ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_712_Collect__restrict,axiom,
! [X4: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ X4 )
& ( P3 @ X ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_713_Collect__restrict,axiom,
! [X4: set_Product_unit,P3: product_unit > $o] :
( ord_le3507040750410214029t_unit
@ ( collect_Product_unit
@ ^ [X: product_unit] :
( ( member_Product_unit @ X @ X4 )
& ( P3 @ X ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_714_subset__emptyI,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [X3: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A @ bot_bo145720340923748686c_fm_i ) ) ).
% subset_emptyI
thf(fact_715_subset__emptyI,axiom,
! [A: set_Epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) ) ).
% subset_emptyI
thf(fact_716_subset__emptyI,axiom,
! [A: set_Product_unit] :
( ! [X3: product_unit] :
~ ( member_Product_unit @ X3 @ A )
=> ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit ) ) ).
% subset_emptyI
thf(fact_717_strong__completeness,axiom,
! [P3: episte5489734138418521880t_unit > $o,G5: set_Epistemic_fm_i,P2: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [M2: episte5489734138418521880t_unit] :
( ( P3 @ M2 )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G5 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ P2 ) ) ) )
=> ( ( P3
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G5 )
& ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Qs @ P2 ) ) ) ) ) ).
% strong_completeness
thf(fact_718_strong__completeness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M2 )
& ( episte80897895827131294t_unit @ M2 )
& ( episte2966952217288386430t_unit @ M2 ) )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G5 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ P2 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G5 )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_AxB_i @ P )
| ( epistemic_Ax4_i @ P ) )
@ ( epistemic_imply_i @ Qs @ P2 ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5
thf(fact_719_main_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M3 )
& ( episte80897895827131294t_unit @ M3 )
& ( episte2966952217288386430t_unit @ M3 ) )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ P2 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G5 )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_AxB_i @ P )
| ( epistemic_Ax4_i @ P ) )
@ ( epistemic_imply_i @ Qs2 @ P2 ) ) ) ) ) ).
% main\<^sub>S\<^sub>5
thf(fact_720_cone__def,axiom,
( bNF_Cardinal_cone
= ( bNF_Ca7083678892733797993t_unit @ ( insert_Product_unit2 @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ) ).
% cone_def
thf(fact_721_K__imply__weaken,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,Q: epistemic_fm_i,Ps2: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q ) )
=> ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ps ) @ ( set_Epistemic_fm_i2 @ Ps2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps2 @ Q ) ) ) ) ).
% K_imply_weaken
thf(fact_722_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_723_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_724_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_725_semantics_Osimps_I3_J,axiom,
! [M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2498298383968489120c_fm_i @ M @ W2 @ ( epistemic_Dis_i @ P2 @ Q ) )
= ( ( episte2498298383968489120c_fm_i @ M @ W2 @ P2 )
| ( episte2498298383968489120c_fm_i @ M @ W2 @ Q ) ) ) ).
% semantics.simps(3)
thf(fact_726_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_727_fm_Odistinct_I11_J,axiom,
! [X22: list_char,X31: epistemic_fm_i,X32: epistemic_fm_i] :
( ( epistemic_Pro_i @ X22 )
!= ( epistemic_Dis_i @ X31 @ X32 ) ) ).
% fm.distinct(11)
thf(fact_728_soundness__imply,axiom,
! [A: epistemic_fm_i > $o,P3: episte4846006725945195288t_unit > $o,Ps: list_Epistemic_fm_i,P2: epistemic_fm_i] :
( ! [M2: episte4846006725945195288t_unit,W5: epistemic_fm_i,P4: epistemic_fm_i] :
( ( A @ P4 )
=> ( ( P3 @ M2 )
=> ( ( member6642669606046002379c_fm_i @ W5 @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ W5 @ P4 ) ) ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ P2 ) )
=> ! [M4: episte4846006725945195288t_unit] :
( ( P3 @ M4 )
=> ! [X5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X5 @ ( episte1297369412406343275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ ( set_Epistemic_fm_i2 @ Ps ) )
=> ( episte5182706529199770816c_fm_i @ M4 @ X5 @ Xa2 ) )
=> ( episte5182706529199770816c_fm_i @ M4 @ X5 @ P2 ) ) ) ) ) ) ).
% soundness_imply
thf(fact_729_soundness__imply,axiom,
! [A: epistemic_fm_i > $o,P3: episte5489734138418521880t_unit > $o,Ps: list_Epistemic_fm_i,P2: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit,W5: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( A @ P4 )
=> ( ( P3 @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W5 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W5 @ P4 ) ) ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ P2 ) )
=> ! [M4: episte5489734138418521880t_unit] :
( ( P3 @ M4 )
=> ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ ( set_Epistemic_fm_i2 @ Ps ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ P2 ) ) ) ) ) ) ).
% soundness_imply
thf(fact_730_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_i,Qs3: list_Epistemic_fm_i,R: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ps ) @ ( set_Epistemic_fm_i2 @ Qs3 ) )
=> ! [G3: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H3 @ ( epistemic_Imp_i @ ( epistemic_imply_i @ Ps @ R ) @ ( epistemic_imply_i @ Qs3 @ R ) ) ) ) ).
% tautology_imply_superset
thf(fact_731_consistent__def,axiom,
( episte2285483198712856234tent_i
= ( ^ [A3: epistemic_fm_i > $o,S6: set_Epistemic_fm_i] :
~ ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ S6 )
& ( epistemic_AK_i @ A3 @ ( epistemic_imply_i @ Qs2 @ epistemic_FF_i ) ) ) ) ) ).
% consistent_def
thf(fact_732_strong__soundness,axiom,
! [A: epistemic_fm_i > $o,P3: episte4846006725945195288t_unit > $o,G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ! [M2: episte4846006725945195288t_unit,W5: epistemic_fm_i,P4: epistemic_fm_i] :
( ( A @ P4 )
=> ( ( P3 @ M2 )
=> ( ( member6642669606046002379c_fm_i @ W5 @ ( episte1297369412406343275t_unit @ M2 ) )
=> ( episte5182706529199770816c_fm_i @ M2 @ W5 @ P4 ) ) ) )
=> ( ? [Qs4: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs4 ) @ G5 )
& ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Qs4 @ P2 ) ) )
=> ! [M4: episte4846006725945195288t_unit] :
( ( P3 @ M4 )
=> ! [X5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X5 @ ( episte1297369412406343275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G5 )
=> ( episte5182706529199770816c_fm_i @ M4 @ X5 @ Xa2 ) )
=> ( episte5182706529199770816c_fm_i @ M4 @ X5 @ P2 ) ) ) ) ) ) ).
% strong_soundness
thf(fact_733_strong__soundness,axiom,
! [A: epistemic_fm_i > $o,P3: episte5489734138418521880t_unit > $o,G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit,W5: set_Epistemic_fm_i,P4: epistemic_fm_i] :
( ( A @ P4 )
=> ( ( P3 @ M2 )
=> ( ( member1104366573291651755c_fm_i @ W5 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ W5 @ P4 ) ) ) )
=> ( ? [Qs4: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs4 ) @ G5 )
& ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Qs4 @ P2 ) ) )
=> ! [M4: episte5489734138418521880t_unit] :
( ( P3 @ M4 )
=> ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ P2 ) ) ) ) ) ) ).
% strong_soundness
thf(fact_734_K__right__mp,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ P2 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ ( epistemic_Imp_i @ P2 @ Q ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q ) ) ) ) ).
% K_right_mp
thf(fact_735_strong__soundness_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i,P3: episte5489734138418521880t_unit > $o] :
( ? [Qs4: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs4 ) @ G5 )
& ( epistemic_AK_i
@ ^ [Uu3: epistemic_fm_i] : $false
@ ( epistemic_imply_i @ Qs4 @ P2 ) ) )
=> ! [M4: episte5489734138418521880t_unit] :
( ( P3 @ M4 )
=> ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ P2 ) ) ) ) ) ).
% strong_soundness\<^sub>K
thf(fact_736_strong__completeness_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit,X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G5 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ P2 ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G5 )
& ( epistemic_AK_i
@ ^ [Uu3: epistemic_fm_i] : $false
@ ( epistemic_imply_i @ Qs @ P2 ) ) ) ) ).
% strong_completeness\<^sub>K
thf(fact_737_main_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ P2 ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G5 )
& ( epistemic_AK_i
@ ^ [Uu3: epistemic_fm_i] : $false
@ ( epistemic_imply_i @ Qs2 @ P2 ) ) ) ) ) ).
% main\<^sub>K
thf(fact_738_strong__soundness_092_060_094sub_062T,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ? [Qs4: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs4 ) @ G5 )
& ( epistemic_AK_i @ epistemic_AxT_i @ ( epistemic_imply_i @ Qs4 @ P2 ) ) )
=> ! [M4: episte5489734138418521880t_unit] :
( ( episte251305198166243731t_unit @ M4 )
=> ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ P2 ) ) ) ) ) ).
% strong_soundness\<^sub>T
thf(fact_739_strong__soundness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ? [Qs4: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs4 ) @ G5 )
& ( epistemic_AK_i @ epistemic_Ax4_i @ ( epistemic_imply_i @ Qs4 @ P2 ) ) )
=> ! [M4: episte5489734138418521880t_unit] :
( ( episte2966952217288386430t_unit @ M4 )
=> ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ P2 ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>4
thf(fact_740_strong__soundness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ? [Qs4: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs4 ) @ G5 )
& ( epistemic_AK_i @ epistemic_AxB_i @ ( epistemic_imply_i @ Qs4 @ P2 ) ) )
=> ! [M4: episte5489734138418521880t_unit] :
( ( episte80897895827131294t_unit @ M4 )
=> ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ P2 ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>B
thf(fact_741_main_092_060_094sub_062T,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( episte251305198166243731t_unit @ M3 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ P2 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G5 )
& ( epistemic_AK_i @ epistemic_AxT_i @ ( epistemic_imply_i @ Qs2 @ P2 ) ) ) ) ) ).
% main\<^sub>T
thf(fact_742_strong__completeness_092_060_094sub_062T,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( episte251305198166243731t_unit @ M2 )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G5 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ P2 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G5 )
& ( epistemic_AK_i @ epistemic_AxT_i @ ( epistemic_imply_i @ Qs @ P2 ) ) ) ) ).
% strong_completeness\<^sub>T
thf(fact_743_main_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( episte2966952217288386430t_unit @ M3 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ P2 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G5 )
& ( epistemic_AK_i @ epistemic_Ax4_i @ ( epistemic_imply_i @ Qs2 @ P2 ) ) ) ) ) ).
% main\<^sub>K\<^sub>4
thf(fact_744_strong__completeness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( episte2966952217288386430t_unit @ M2 )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G5 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ P2 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G5 )
& ( epistemic_AK_i @ epistemic_Ax4_i @ ( epistemic_imply_i @ Qs @ P2 ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>4
thf(fact_745_strong__soundness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ? [Qs4: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs4 ) @ G5 )
& ( epistemic_AK_i @ epistemic_Ax5_i @ ( epistemic_imply_i @ Qs4 @ P2 ) ) )
=> ! [M4: episte5489734138418521880t_unit] :
( ( episte6275404236303465413t_unit @ M4 )
=> ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ P2 ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>5
thf(fact_746_main_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( episte80897895827131294t_unit @ M3 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ P2 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G5 )
& ( epistemic_AK_i @ epistemic_AxB_i @ ( epistemic_imply_i @ Qs2 @ P2 ) ) ) ) ) ).
% main\<^sub>K\<^sub>B
thf(fact_747_strong__completeness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( episte80897895827131294t_unit @ M2 )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G5 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ P2 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G5 )
& ( epistemic_AK_i @ epistemic_AxB_i @ ( epistemic_imply_i @ Qs @ P2 ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>B
thf(fact_748_S5__S5_H__assms,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G5 )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_AxB_i @ P )
| ( epistemic_Ax4_i @ P ) )
@ ( epistemic_imply_i @ Qs2 @ P2 ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G5 )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_Ax5_i @ P ) )
@ ( epistemic_imply_i @ Qs2 @ P2 ) ) ) ) ) ).
% S5_S5'_assms
thf(fact_749_main_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( episte6275404236303465413t_unit @ M3 )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ P2 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G5 )
& ( epistemic_AK_i @ epistemic_Ax5_i @ ( epistemic_imply_i @ Qs2 @ P2 ) ) ) ) ) ).
% main\<^sub>K\<^sub>5
thf(fact_750_strong__completeness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( episte6275404236303465413t_unit @ M2 )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G5 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ P2 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G5 )
& ( epistemic_AK_i @ epistemic_Ax5_i @ ( epistemic_imply_i @ Qs @ P2 ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>5
thf(fact_751_inconsistent__imply,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,G5: list_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ ( set_Epistemic_fm_i2 @ G5 ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G5 @ P2 ) ) ) ).
% inconsistent_imply
thf(fact_752_strong__soundness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ? [Qs4: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs4 ) @ G5 )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_Ax4_i @ P ) )
@ ( epistemic_imply_i @ Qs4 @ P2 ) ) )
=> ! [M4: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M4 )
& ( episte2966952217288386430t_unit @ M4 ) )
=> ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ P2 ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>4
thf(fact_753_main_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M3 )
& ( episte2966952217288386430t_unit @ M3 ) )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ P2 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G5 )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_Ax4_i @ P ) )
@ ( epistemic_imply_i @ Qs2 @ P2 ) ) ) ) ) ).
% main\<^sub>S\<^sub>4
thf(fact_754_strong__completeness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M2 )
& ( episte2966952217288386430t_unit @ M2 ) )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G5 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ P2 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G5 )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_Ax4_i @ P ) )
@ ( epistemic_imply_i @ Qs @ P2 ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>4
thf(fact_755_strong__soundness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ? [Qs4: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs4 ) @ G5 )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_Ax5_i @ P ) )
@ ( epistemic_imply_i @ Qs4 @ P2 ) ) )
=> ! [M4: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M4 )
& ( episte80897895827131294t_unit @ M4 )
& ( episte2966952217288386430t_unit @ M4 ) )
=> ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ P2 ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5'
thf(fact_756_consistent__disjuncts,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Dis_i @ P2 @ Q ) @ V2 )
=> ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P2 @ bot_bo4194595901900360558c_fm_i ) @ V2 ) )
| ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q @ bot_bo4194595901900360558c_fm_i ) @ V2 ) ) ) ) ) ).
% consistent_disjuncts
thf(fact_757_main_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( ! [M3: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M3 )
& ( episte80897895827131294t_unit @ M3 )
& ( episte2966952217288386430t_unit @ M3 ) )
=> ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ Y5 ) )
=> ( episte2498298383968489120c_fm_i @ M3 @ X @ P2 ) ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G5 )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_Ax5_i @ P ) )
@ ( epistemic_imply_i @ Qs2 @ P2 ) ) ) ) ) ).
% main\<^sub>S\<^sub>5'
thf(fact_758_strong__completeness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ! [M2: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M2 )
& ( episte80897895827131294t_unit @ M2 )
& ( episte2966952217288386430t_unit @ M2 ) )
=> ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( episte2675268102452679275t_unit @ M2 ) )
=> ( ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ G5 )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ Xa ) )
=> ( episte2498298383968489120c_fm_i @ M2 @ X3 @ P2 ) ) ) )
=> ? [Qs: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs ) @ G5 )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_Ax5_i @ P ) )
@ ( epistemic_imply_i @ Qs @ P2 ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5'
thf(fact_759_strong__soundness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ? [Qs4: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs4 ) @ G5 )
& ( epistemic_AK_i
@ ^ [P: epistemic_fm_i] :
( ( epistemic_AxT_i @ P )
| ( epistemic_AxB_i @ P )
| ( epistemic_Ax4_i @ P ) )
@ ( epistemic_imply_i @ Qs4 @ P2 ) ) )
=> ! [M4: episte5489734138418521880t_unit] :
( ( ( episte251305198166243731t_unit @ M4 )
& ( episte80897895827131294t_unit @ M4 )
& ( episte2966952217288386430t_unit @ M4 ) )
=> ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ ( episte2675268102452679275t_unit @ M4 ) )
=> ( ! [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ G5 )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ Xa2 ) )
=> ( episte2498298383968489120c_fm_i @ M4 @ X5 @ P2 ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5
thf(fact_760_fm_Oexhaust,axiom,
! [Y4: epistemic_fm_i] :
( ( Y4 != epistemic_FF_i )
=> ( ! [X23: list_char] :
( Y4
!= ( epistemic_Pro_i @ X23 ) )
=> ( ! [X312: epistemic_fm_i,X322: epistemic_fm_i] :
( Y4
!= ( epistemic_Dis_i @ X312 @ X322 ) )
=> ( ! [X412: epistemic_fm_i,X422: epistemic_fm_i] :
( Y4
!= ( epistemic_Con_i @ X412 @ X422 ) )
=> ( ! [X512: epistemic_fm_i,X522: epistemic_fm_i] :
( Y4
!= ( epistemic_Imp_i @ X512 @ X522 ) )
=> ~ ! [X612: i,X622: epistemic_fm_i] :
( Y4
!= ( epistemic_K_i @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_761_inconsistent__subset,axiom,
! [A: epistemic_fm_i > $o,V2: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V2 )
=> ( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P2 @ bot_bo4194595901900360558c_fm_i ) @ V2 ) )
=> ~ ! [V3: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ V3 ) @ V2 )
=> ~ ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ V3 ) @ epistemic_FF_i ) ) ) ) ) ).
% inconsistent_subset
thf(fact_762_conjunct__imply,axiom,
! [G5: list_Epistemic_fm_i,P2: epistemic_fm_i,G3: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H3 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G5 ) @ P2 ) @ ( epistemic_imply_i @ G5 @ P2 ) ) ) ).
% conjunct_imply
thf(fact_763_imply__conjunct,axiom,
! [G5: list_Epistemic_fm_i,P2: epistemic_fm_i,G3: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H3 @ ( epistemic_Imp_i @ ( epistemic_imply_i @ G5 @ P2 ) @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G5 ) @ P2 ) ) ) ).
% imply_conjunct
thf(fact_764_K__conjunct__imply,axiom,
! [A: epistemic_fm_i > $o,G5: list_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G5 ) @ P2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G5 @ P2 ) ) ) ).
% K_conjunct_imply
thf(fact_765_list_Osimps_I15_J,axiom,
! [X21: epistemic_fm_i,X222: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X222 ) )
= ( insert7817948997695205106c_fm_i @ X21 @ ( set_Epistemic_fm_i2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_766_list_Osimps_I15_J,axiom,
! [X21: product_unit,X222: list_Product_unit] :
( ( set_Product_unit2 @ ( cons_Product_unit @ X21 @ X222 ) )
= ( insert_Product_unit2 @ X21 @ ( set_Product_unit2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_767_conjunct_Osimps_I2_J,axiom,
! [P2: epistemic_fm_i,Ps: list_Epistemic_fm_i] :
( ( stalnaker_conjunct_i @ ( cons_Epistemic_fm_i @ P2 @ Ps ) )
= ( epistemic_Con_i @ P2 @ ( stalnaker_conjunct_i @ Ps ) ) ) ).
% conjunct.simps(2)
thf(fact_768_set__subset__Cons,axiom,
! [Xs: list_Epistemic_fm_i,X2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_769_set__subset__Cons,axiom,
! [Xs: list_Product_unit,X2: product_unit] : ( ord_le3507040750410214029t_unit @ ( set_Product_unit2 @ Xs ) @ ( set_Product_unit2 @ ( cons_Product_unit @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_770_imply_Osimps_I2_J,axiom,
! [P2: epistemic_fm_i,Ps: list_Epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ Ps ) @ Q )
= ( epistemic_Imp_i @ P2 @ ( epistemic_imply_i @ Ps @ Q ) ) ) ).
% imply.simps(2)
thf(fact_771_K__swap,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i,G5: list_Epistemic_fm_i,R: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ ( cons_Epistemic_fm_i @ Q @ G5 ) ) @ R ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ Q @ ( cons_Epistemic_fm_i @ P2 @ G5 ) ) @ R ) ) ) ).
% K_swap
thf(fact_772_K__imply__Cons,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,Q: epistemic_fm_i,P2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ Ps ) @ Q ) ) ) ).
% K_imply_Cons
thf(fact_773_K__imply__head,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Ps: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ Ps ) @ P2 ) ) ).
% K_imply_head
thf(fact_774_K__mp,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i,G5: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ ( cons_Epistemic_fm_i @ ( epistemic_Imp_i @ P2 @ Q ) @ G5 ) ) @ Q ) ) ).
% K_mp
thf(fact_775_K__ImpI,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,G5: list_Epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ G5 ) @ Q ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G5 @ ( epistemic_Imp_i @ P2 @ Q ) ) ) ) ).
% K_ImpI
thf(fact_776_K__DisL,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Ps: list_Epistemic_fm_i,Q: epistemic_fm_i,P5: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ Ps ) @ Q ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P5 @ Ps ) @ Q ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ ( epistemic_Dis_i @ P2 @ P5 ) @ Ps ) @ Q ) ) ) ) ).
% K_DisL
thf(fact_777_K__DisE,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,G5: list_Epistemic_fm_i,R: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ G5 ) @ R ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ Q @ G5 ) @ R ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G5 @ ( epistemic_Dis_i @ P2 @ Q ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G5 @ R ) ) ) ) ) ).
% K_DisE
thf(fact_778_K__Boole,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,G5: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ G5 ) @ epistemic_FF_i ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G5 @ P2 ) ) ) ).
% K_Boole
thf(fact_779_subset__code_I1_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ B2 )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( member1104366573291651755c_fm_i @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_780_subset__code_I1_J,axiom,
! [Xs: list_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ B2 )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_781_subset__code_I1_J,axiom,
! [Xs: list_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( set_Product_unit2 @ Xs ) @ B2 )
= ( ! [X: product_unit] :
( ( member_Product_unit @ X @ ( set_Product_unit2 @ Xs ) )
=> ( member_Product_unit @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_782_K__imply__conjunct,axiom,
! [A: epistemic_fm_i > $o,G5: list_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G5 @ P2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G5 ) @ P2 ) ) ) ).
% K_imply_conjunct
thf(fact_783_kripke_Oupdate__convs_I2_J,axiom,
! [More2: product_unit > product_unit,W3: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi: set_Epistemic_fm_i > list_char > $o,More: product_unit] :
( ( episte4537596280781771625c_fm_i @ More2 @ ( episte8397520288747125024t_unit @ W3 @ K @ ( episte4955834246032687171t_unit @ Pi @ More ) ) )
= ( episte8397520288747125024t_unit @ W3 @ K @ ( episte4955834246032687171t_unit @ Pi @ ( More2 @ More ) ) ) ) ).
% kripke.update_convs(2)
thf(fact_784_Epistemic__Logic_Oextend__def,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,N: epistemic_fm_i] :
( ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ N )
= ( wellor8883914554786644734c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ S @ ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S ) @ ( maxima7895811110999683971c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) ) @ N ) ) ).
% Epistemic_Logic.extend_def
thf(fact_785_extend__under,axiom,
! [M5: epistemic_fm_i,N: epistemic_fm_i,A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ M5 @ ( order_1815523504696313095c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ N ) )
=> ( ord_le3843937902494030498c_fm_i @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ M5 ) @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ N ) ) ) ).
% extend_under
thf(fact_786_under__incr2,axiom,
! [R: set_Pr4658907567593863815c_fm_i,R3: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( ord_le3932863852503339559c_fm_i @ R @ R3 )
=> ( ord_le3843937902494030498c_fm_i @ ( order_1815523504696313095c_fm_i @ R @ A2 ) @ ( order_1815523504696313095c_fm_i @ R3 @ A2 ) ) ) ).
% under_incr2
thf(fact_787_under__incr2,axiom,
! [R: set_Pr5094982260447487303t_unit,R3: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( ord_le930853871177014503t_unit @ R @ R3 )
=> ( ord_le3507040750410214029t_unit @ ( order_6949322460238004850t_unit @ R @ A2 ) @ ( order_6949322460238004850t_unit @ R3 @ A2 ) ) ) ).
% under_incr2
thf(fact_788_under__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_1815523504696313095c_fm_i @ R @ A2 ) @ ( field_Epistemic_fm_i @ R ) ) ).
% under_Field
thf(fact_789_under__Field,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit] : ( ord_le3507040750410214029t_unit @ ( order_6949322460238004850t_unit @ R @ A2 ) @ ( field_Product_unit @ R ) ) ).
% under_Field
thf(fact_790_under__empty,axiom,
! [A2: set_Epistemic_fm_i,R: set_Pr3972831103112126087c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( order_4888947244107180775c_fm_i @ R @ A2 )
= bot_bo145720340923748686c_fm_i ) ) ).
% under_empty
thf(fact_791_under__empty,axiom,
! [A2: epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( order_1815523504696313095c_fm_i @ R @ A2 )
= bot_bo4194595901900360558c_fm_i ) ) ).
% under_empty
thf(fact_792_under__empty,axiom,
! [A2: product_unit,R: set_Pr5094982260447487303t_unit] :
( ~ ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ( order_6949322460238004850t_unit @ R @ A2 )
= bot_bo3957492148770167129t_unit ) ) ).
% under_empty
thf(fact_793_adm__woL__extendL,axiom,
wellor4631791200432093460c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( maxima7895811110999683971c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% adm_woL_extendL
thf(fact_794_set__rec,axiom,
( set_Epistemic_fm_i2
= ( rec_li5834409330748533462c_fm_i @ bot_bo4194595901900360558c_fm_i
@ ^ [X: epistemic_fm_i,Uu3: list_Epistemic_fm_i] : ( insert7817948997695205106c_fm_i @ X ) ) ) ).
% set_rec
thf(fact_795_set__rec,axiom,
( set_Product_unit2
= ( rec_li8686922504669531926t_unit @ bot_bo3957492148770167129t_unit
@ ^ [X: product_unit,Uu3: list_Product_unit] : ( insert_Product_unit2 @ X ) ) ) ).
% set_rec
thf(fact_796_fm_Orel__cases,axiom,
! [R2: i > i > $o,A2: epistemic_fm_i,B: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R2 @ A2 @ B )
=> ( ( ( A2 = epistemic_FF_i )
=> ( B != epistemic_FF_i ) )
=> ( ! [X3: list_char] :
( ( A2
= ( epistemic_Pro_i @ X3 ) )
=> ! [Y3: list_char] :
( ( B
= ( epistemic_Pro_i @ Y3 ) )
=> ( X3 != Y3 ) ) )
=> ( ! [X1: epistemic_fm_i,X2a: epistemic_fm_i] :
( ( A2
= ( epistemic_Dis_i @ X1 @ X2a ) )
=> ! [Y1: epistemic_fm_i,Y2a: epistemic_fm_i] :
( ( B
= ( epistemic_Dis_i @ Y1 @ Y2a ) )
=> ( ( epistemic_rel_fm_i_i @ R2 @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_i_i @ R2 @ X2a @ Y2a ) ) ) )
=> ( ! [X1a: epistemic_fm_i,X2b: epistemic_fm_i] :
( ( A2
= ( epistemic_Con_i @ X1a @ X2b ) )
=> ! [Y1a: epistemic_fm_i,Y2b: epistemic_fm_i] :
( ( B
= ( epistemic_Con_i @ Y1a @ Y2b ) )
=> ( ( epistemic_rel_fm_i_i @ R2 @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_i_i @ R2 @ X2b @ Y2b ) ) ) )
=> ( ! [X1b: epistemic_fm_i,X2c: epistemic_fm_i] :
( ( A2
= ( epistemic_Imp_i @ X1b @ X2c ) )
=> ! [Y1b: epistemic_fm_i,Y2c: epistemic_fm_i] :
( ( B
= ( epistemic_Imp_i @ Y1b @ Y2c ) )
=> ( ( epistemic_rel_fm_i_i @ R2 @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_i_i @ R2 @ X2c @ Y2c ) ) ) )
=> ~ ! [X1c: i,X2d: epistemic_fm_i] :
( ( A2
= ( epistemic_K_i @ X1c @ X2d ) )
=> ! [Y1c: i,Y2d: epistemic_fm_i] :
( ( B
= ( epistemic_K_i @ Y1c @ Y2d ) )
=> ( ( R2 @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_i_i @ R2 @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_797_fm_Orel__induct,axiom,
! [R2: i > i > $o,X2: epistemic_fm_i,Y4: epistemic_fm_i,Q2: epistemic_fm_i > epistemic_fm_i > $o] :
( ( epistemic_rel_fm_i_i @ R2 @ X2 @ Y4 )
=> ( ( Q2 @ epistemic_FF_i @ epistemic_FF_i )
=> ( ! [A22: list_char,B22: list_char] :
( ( A22 = B22 )
=> ( Q2 @ ( epistemic_Pro_i @ A22 ) @ ( epistemic_Pro_i @ B22 ) ) )
=> ( ! [A31: epistemic_fm_i,A32: epistemic_fm_i,B31: epistemic_fm_i,B32: epistemic_fm_i] :
( ( Q2 @ A31 @ B31 )
=> ( ( Q2 @ A32 @ B32 )
=> ( Q2 @ ( 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] :
( ( Q2 @ A41 @ B41 )
=> ( ( Q2 @ A42 @ B42 )
=> ( Q2 @ ( 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] :
( ( Q2 @ A51 @ B51 )
=> ( ( Q2 @ A52 @ B52 )
=> ( Q2 @ ( epistemic_Imp_i @ A51 @ A52 ) @ ( epistemic_Imp_i @ B51 @ B52 ) ) ) )
=> ( ! [A61: i,A62: epistemic_fm_i,B61: i,B62: epistemic_fm_i] :
( ( R2 @ A61 @ B61 )
=> ( ( Q2 @ A62 @ B62 )
=> ( Q2 @ ( epistemic_K_i @ A61 @ A62 ) @ ( epistemic_K_i @ B61 @ B62 ) ) ) )
=> ( Q2 @ X2 @ Y4 ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_798_fm_Orel__inject_I5_J,axiom,
! [R2: i > i > $o,X51: epistemic_fm_i,X52: epistemic_fm_i,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Imp_i @ X51 @ X52 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) )
= ( ( epistemic_rel_fm_i_i @ R2 @ X51 @ Y51 )
& ( epistemic_rel_fm_i_i @ R2 @ X52 @ Y52 ) ) ) ).
% fm.rel_inject(5)
thf(fact_799_fm_Orel__intros_I5_J,axiom,
! [R2: i > i > $o,X51: epistemic_fm_i,Y51: epistemic_fm_i,X52: epistemic_fm_i,Y52: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R2 @ X51 @ Y51 )
=> ( ( epistemic_rel_fm_i_i @ R2 @ X52 @ Y52 )
=> ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Imp_i @ X51 @ X52 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ) ) ).
% fm.rel_intros(5)
thf(fact_800_fm_Orel__inject_I6_J,axiom,
! [R2: i > i > $o,X61: i,X62: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_K_i @ X61 @ X62 ) @ ( epistemic_K_i @ Y61 @ Y62 ) )
= ( ( R2 @ X61 @ Y61 )
& ( epistemic_rel_fm_i_i @ R2 @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_801_fm_Orel__intros_I6_J,axiom,
! [R2: i > i > $o,X61: i,Y61: i,X62: epistemic_fm_i,Y62: epistemic_fm_i] :
( ( R2 @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_i_i @ R2 @ X62 @ Y62 )
=> ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_K_i @ X61 @ X62 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_802_fm_Octr__transfer_I1_J,axiom,
! [R2: i > i > $o] : ( epistemic_rel_fm_i_i @ R2 @ epistemic_FF_i @ epistemic_FF_i ) ).
% fm.ctr_transfer(1)
thf(fact_803_MCS__Lim__Ord_Oconsistent__hereditary,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i,S3: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( Consistent @ S )
=> ( ( ord_le3843937902494030498c_fm_i @ S3 @ S )
=> ( Consistent @ S3 ) ) ) ) ).
% MCS_Lim_Ord.consistent_hereditary
thf(fact_804_MCS__Lim__Ord_Oconsistent__hereditary,axiom,
! [R: set_Pr5094982260447487303t_unit,Consistent: set_Product_unit > $o,S: set_Product_unit,S3: set_Product_unit] :
( ( maxima7229560971830504200t_unit @ R @ Consistent )
=> ( ( Consistent @ S )
=> ( ( ord_le3507040750410214029t_unit @ S3 @ S )
=> ( Consistent @ S3 ) ) ) ) ).
% MCS_Lim_Ord.consistent_hereditary
thf(fact_805_MCS__Lim__Ord_Oconsistent__Extend,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( Consistent @ S )
=> ( Consistent @ ( maxima2580775659384275475c_fm_i @ R @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.consistent_Extend
thf(fact_806_fm_Orel__inject_I2_J,axiom,
! [R2: i > i > $o,X22: list_char,Y2: list_char] :
( ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Pro_i @ X22 ) @ ( epistemic_Pro_i @ Y2 ) )
= ( X22 = Y2 ) ) ).
% fm.rel_inject(2)
thf(fact_807_fm_Orel__intros_I2_J,axiom,
! [X22: list_char,Y2: list_char,R2: i > i > $o] :
( ( X22 = Y2 )
=> ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Pro_i @ X22 ) @ ( epistemic_Pro_i @ Y2 ) ) ) ).
% fm.rel_intros(2)
thf(fact_808_fm_Orel__inject_I4_J,axiom,
! [R2: i > i > $o,X41: epistemic_fm_i,X42: epistemic_fm_i,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Con_i @ X41 @ X42 ) @ ( epistemic_Con_i @ Y41 @ Y42 ) )
= ( ( epistemic_rel_fm_i_i @ R2 @ X41 @ Y41 )
& ( epistemic_rel_fm_i_i @ R2 @ X42 @ Y42 ) ) ) ).
% fm.rel_inject(4)
thf(fact_809_fm_Orel__intros_I4_J,axiom,
! [R2: i > i > $o,X41: epistemic_fm_i,Y41: epistemic_fm_i,X42: epistemic_fm_i,Y42: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R2 @ X41 @ Y41 )
=> ( ( epistemic_rel_fm_i_i @ R2 @ X42 @ Y42 )
=> ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Con_i @ X41 @ X42 ) @ ( epistemic_Con_i @ Y41 @ Y42 ) ) ) ) ).
% fm.rel_intros(4)
thf(fact_810_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S5 @ S )
& ( finite3304564979551393739c_fm_i @ S5 )
& ~ ( Consistent @ S5 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_811_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R: set_Pr5094982260447487303t_unit,Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7229560971830504200t_unit @ R @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ S5 @ S )
& ( finite4290736615968046902t_unit @ S5 )
& ~ ( Consistent @ S5 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_812_fm_Orel__distinct_I29_J,axiom,
! [R2: i > i > $o,X51: epistemic_fm_i,X52: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Imp_i @ X51 @ X52 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(29)
thf(fact_813_fm_Orel__distinct_I30_J,axiom,
! [R2: i > i > $o,Y61: i,Y62: epistemic_fm_i,X51: epistemic_fm_i,X52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_K_i @ Y61 @ Y62 ) @ ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.rel_distinct(30)
thf(fact_814_fm_Orel__distinct_I7_J,axiom,
! [R2: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ epistemic_FF_i @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(7)
thf(fact_815_fm_Orel__distinct_I8_J,axiom,
! [R2: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Imp_i @ Y51 @ Y52 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(8)
thf(fact_816_fm_Orel__distinct_I9_J,axiom,
! [R2: i > i > $o,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ epistemic_FF_i @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(9)
thf(fact_817_fm_Orel__distinct_I10_J,axiom,
! [R2: i > i > $o,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_K_i @ Y61 @ Y62 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(10)
thf(fact_818_fm_Orel__distinct_I25_J,axiom,
! [R2: i > i > $o,X41: epistemic_fm_i,X42: epistemic_fm_i,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Con_i @ X41 @ X42 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(25)
thf(fact_819_fm_Orel__distinct_I26_J,axiom,
! [R2: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i,X41: epistemic_fm_i,X42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Imp_i @ Y51 @ Y52 ) @ ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.rel_distinct(26)
thf(fact_820_fm_Orel__distinct_I27_J,axiom,
! [R2: i > i > $o,X41: epistemic_fm_i,X42: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Con_i @ X41 @ X42 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(27)
thf(fact_821_fm_Orel__distinct_I28_J,axiom,
! [R2: i > i > $o,Y61: i,Y62: epistemic_fm_i,X41: epistemic_fm_i,X42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_K_i @ Y61 @ Y62 ) @ ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.rel_distinct(28)
thf(fact_822_fm_Orel__distinct_I15_J,axiom,
! [R2: i > i > $o,X22: list_char,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Pro_i @ X22 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(15)
thf(fact_823_fm_Orel__distinct_I16_J,axiom,
! [R2: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i,X22: list_char] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Imp_i @ Y51 @ Y52 ) @ ( epistemic_Pro_i @ X22 ) ) ).
% fm.rel_distinct(16)
thf(fact_824_fm_Orel__distinct_I17_J,axiom,
! [R2: i > i > $o,X22: list_char,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Pro_i @ X22 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(17)
thf(fact_825_fm_Orel__distinct_I18_J,axiom,
! [R2: i > i > $o,Y61: i,Y62: epistemic_fm_i,X22: list_char] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_K_i @ Y61 @ Y62 ) @ ( epistemic_Pro_i @ X22 ) ) ).
% fm.rel_distinct(18)
thf(fact_826_fm_Orel__distinct_I5_J,axiom,
! [R2: i > i > $o,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ epistemic_FF_i @ ( epistemic_Con_i @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(5)
thf(fact_827_fm_Orel__distinct_I6_J,axiom,
! [R2: i > i > $o,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Con_i @ Y41 @ Y42 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(6)
thf(fact_828_fm_Orel__distinct_I1_J,axiom,
! [R2: i > i > $o,Y2: list_char] :
~ ( epistemic_rel_fm_i_i @ R2 @ epistemic_FF_i @ ( epistemic_Pro_i @ Y2 ) ) ).
% fm.rel_distinct(1)
thf(fact_829_fm_Orel__distinct_I2_J,axiom,
! [R2: i > i > $o,Y2: list_char] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Pro_i @ Y2 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(2)
thf(fact_830_fm_Orel__distinct_I13_J,axiom,
! [R2: i > i > $o,X22: list_char,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Pro_i @ X22 ) @ ( epistemic_Con_i @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(13)
thf(fact_831_fm_Orel__distinct_I14_J,axiom,
! [R2: i > i > $o,Y41: epistemic_fm_i,Y42: epistemic_fm_i,X22: list_char] :
~ ( epistemic_rel_fm_i_i @ R2 @ ( epistemic_Con_i @ Y41 @ Y42 ) @ ( epistemic_Pro_i @ X22 ) ) ).
% fm.rel_distinct(14)
thf(fact_832_MCS__Lim__Ord_Omaximal_H__Extend,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( maxima2621630817641868953c_fm_i @ R @ Consistent @ ( maxima2580775659384275475c_fm_i @ R @ Consistent @ S ) ) ) ).
% MCS_Lim_Ord.maximal'_Extend
thf(fact_833_MCS__Lim__Ord_Oextend__subset,axiom,
! [R: set_Pr3972831103112126087c_fm_i,Consistent: set_se3485332733965609186c_fm_i > $o,N: set_Epistemic_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( maxima9073963094320882813c_fm_i @ R @ Consistent )
=> ( ( member1104366573291651755c_fm_i @ N @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ord_le5389487502678872194c_fm_i @ S @ ( maxima3510324002270828563c_fm_i @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_subset
thf(fact_834_MCS__Lim__Ord_Oextend__subset,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,N: epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( member6642669606046002379c_fm_i @ N @ ( field_Epistemic_fm_i @ R ) )
=> ( ord_le3843937902494030498c_fm_i @ S @ ( maxima1168816371290972723c_fm_i @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_subset
thf(fact_835_MCS__Lim__Ord_Oextend__subset,axiom,
! [R: set_Pr5094982260447487303t_unit,Consistent: set_Product_unit > $o,N: product_unit,S: set_Product_unit] :
( ( maxima7229560971830504200t_unit @ R @ Consistent )
=> ( ( member_Product_unit @ N @ ( field_Product_unit @ R ) )
=> ( ord_le3507040750410214029t_unit @ S @ ( maxima7300585265216834974t_unit @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_subset
thf(fact_836_MCS__Lim__Ord_Oextend__under,axiom,
! [R: set_Pr3972831103112126087c_fm_i,Consistent: set_se3485332733965609186c_fm_i > $o,M5: set_Epistemic_fm_i,N: set_Epistemic_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( maxima9073963094320882813c_fm_i @ R @ Consistent )
=> ( ( member1104366573291651755c_fm_i @ M5 @ ( order_4888947244107180775c_fm_i @ R @ N ) )
=> ( ord_le5389487502678872194c_fm_i @ ( maxima3510324002270828563c_fm_i @ R @ Consistent @ S @ M5 ) @ ( maxima3510324002270828563c_fm_i @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_under
thf(fact_837_MCS__Lim__Ord_Oextend__under,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,M5: epistemic_fm_i,N: epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( member6642669606046002379c_fm_i @ M5 @ ( order_1815523504696313095c_fm_i @ R @ N ) )
=> ( ord_le3843937902494030498c_fm_i @ ( maxima1168816371290972723c_fm_i @ R @ Consistent @ S @ M5 ) @ ( maxima1168816371290972723c_fm_i @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_under
thf(fact_838_MCS__Lim__Ord_Oextend__under,axiom,
! [R: set_Pr5094982260447487303t_unit,Consistent: set_Product_unit > $o,M5: product_unit,N: product_unit,S: set_Product_unit] :
( ( maxima7229560971830504200t_unit @ R @ Consistent )
=> ( ( member_Product_unit @ M5 @ ( order_6949322460238004850t_unit @ R @ N ) )
=> ( ord_le3507040750410214029t_unit @ ( maxima7300585265216834974t_unit @ R @ Consistent @ S @ M5 ) @ ( maxima7300585265216834974t_unit @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_under
thf(fact_839_MCS__Lim__Ord_OExtend__subset_H,axiom,
! [R: set_Pr5094982260447487303t_unit,Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7229560971830504200t_unit @ R @ Consistent )
=> ( ( ( field_Product_unit @ R )
!= bot_bo3957492148770167129t_unit )
=> ( ord_le3507040750410214029t_unit @ S @ ( maxima3365502296767225214t_unit @ R @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_subset'
thf(fact_840_MCS__Lim__Ord_OExtend__subset_H,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( ( field_Epistemic_fm_i @ R )
!= bot_bo4194595901900360558c_fm_i )
=> ( ord_le3843937902494030498c_fm_i @ S @ ( maxima2580775659384275475c_fm_i @ R @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_subset'
thf(fact_841_MCS__Lim__Ord_OExtend__bound,axiom,
! [R: set_Pr3972831103112126087c_fm_i,Consistent: set_se3485332733965609186c_fm_i > $o,N: set_Epistemic_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( maxima9073963094320882813c_fm_i @ R @ Consistent )
=> ( ( member1104366573291651755c_fm_i @ N @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ord_le5389487502678872194c_fm_i @ ( maxima3510324002270828563c_fm_i @ R @ Consistent @ S @ N ) @ ( maxima2705192052082030579c_fm_i @ R @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_bound
thf(fact_842_MCS__Lim__Ord_OExtend__bound,axiom,
! [R: set_Pr5094982260447487303t_unit,Consistent: set_Product_unit > $o,N: product_unit,S: set_Product_unit] :
( ( maxima7229560971830504200t_unit @ R @ Consistent )
=> ( ( member_Product_unit @ N @ ( field_Product_unit @ R ) )
=> ( ord_le3507040750410214029t_unit @ ( maxima7300585265216834974t_unit @ R @ Consistent @ S @ N ) @ ( maxima3365502296767225214t_unit @ R @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_bound
thf(fact_843_MCS__Lim__Ord_OExtend__bound,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,N: epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( member6642669606046002379c_fm_i @ N @ ( field_Epistemic_fm_i @ R ) )
=> ( ord_le3843937902494030498c_fm_i @ ( maxima1168816371290972723c_fm_i @ R @ Consistent @ S @ N ) @ ( maxima2580775659384275475c_fm_i @ R @ Consistent @ S ) ) ) ) ).
% MCS_Lim_Ord.Extend_bound
thf(fact_844_MCS__Lim__Ord_OextendS__def,axiom,
! [R: 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 @ R @ 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_845_MCS__Lim__Ord_OextendS__def,axiom,
! [R: set_Pr5094982260447487303t_unit,Consistent: set_Product_unit > $o,N: product_unit,Prev: set_Product_unit,S: set_Product_unit] :
( ( maxima7229560971830504200t_unit @ R @ Consistent )
=> ( ( ( Consistent @ ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ N @ bot_bo3957492148770167129t_unit ) @ Prev ) )
=> ( ( maxima8314523877101631911t_unit @ Consistent @ S @ N @ Prev )
= ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ N @ bot_bo3957492148770167129t_unit ) @ Prev ) ) )
& ( ~ ( Consistent @ ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ N @ bot_bo3957492148770167129t_unit ) @ Prev ) )
=> ( ( maxima8314523877101631911t_unit @ Consistent @ S @ N @ Prev )
= Prev ) ) ) ) ).
% MCS_Lim_Ord.extendS_def
thf(fact_846_MCS_Omaximal_Ocong,axiom,
maxima3264069618988350929c_fm_i = maxima3264069618988350929c_fm_i ).
% MCS.maximal.cong
thf(fact_847_MCS__Lim__Ord_OExtend_Ocong,axiom,
maxima2580775659384275475c_fm_i = maxima2580775659384275475c_fm_i ).
% MCS_Lim_Ord.Extend.cong
thf(fact_848_MCS__Lim__Ord__axioms,axiom,
! [A: epistemic_fm_i > $o] : ( maxima8116622286610175645c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) ) ).
% MCS_Lim_Ord_axioms
thf(fact_849_MCS__Lim__Ord_Omaximal_H__def,axiom,
! [R: set_Pr3972831103112126087c_fm_i,Consistent: set_se3485332733965609186c_fm_i > $o,S: set_se3485332733965609186c_fm_i] :
( ( maxima9073963094320882813c_fm_i @ R @ Consistent )
=> ( ( maxima2879964782442171513c_fm_i @ R @ Consistent @ S )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( Consistent @ ( sup_su2582925890723967158c_fm_i @ ( insert7698009978809854162c_fm_i @ X @ bot_bo145720340923748686c_fm_i ) @ S ) )
=> ( member1104366573291651755c_fm_i @ X @ S ) ) ) ) ) ) ).
% MCS_Lim_Ord.maximal'_def
thf(fact_850_MCS__Lim__Ord_Omaximal_H__def,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( maxima2621630817641868953c_fm_i @ R @ Consistent @ S )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ ( field_Epistemic_fm_i @ R ) )
=> ( ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ X @ S ) ) ) ) ) ) ).
% MCS_Lim_Ord.maximal'_def
thf(fact_851_MCS__Lim__Ord_Omaximal_H__def,axiom,
! [R: set_Pr5094982260447487303t_unit,Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7229560971830504200t_unit @ R @ Consistent )
=> ( ( maxima4602283525647497476t_unit @ R @ Consistent @ S )
= ( ! [X: product_unit] :
( ( member_Product_unit @ X @ ( field_Product_unit @ R ) )
=> ( ( Consistent @ ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ X @ bot_bo3957492148770167129t_unit ) @ S ) )
=> ( member_Product_unit @ X @ S ) ) ) ) ) ) ).
% MCS_Lim_Ord.maximal'_def
thf(fact_852_MCS_Omaximal__maximal_H,axiom,
! [Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7871471802976955641t_unit @ Consistent )
=> ( ( maxima1338112684437001276t_unit @ Consistent @ S )
= ( maxima4602283525647497476t_unit @ ( bNF_Ca7083678892733797993t_unit @ top_to1996260823553986621t_unit ) @ Consistent @ S ) ) ) ).
% MCS.maximal_maximal'
thf(fact_853_MCS_Omaximal__maximal_H,axiom,
! [Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ( ( maxima3264069618988350929c_fm_i @ Consistent @ S )
= ( maxima2621630817641868953c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ Consistent @ S ) ) ) ).
% MCS.maximal_maximal'
thf(fact_854_MCS_Omaximal__Extend,axiom,
! [Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7871471802976955641t_unit @ Consistent )
=> ( maxima1338112684437001276t_unit @ Consistent @ ( maxima3365502296767225214t_unit @ ( bNF_Ca7083678892733797993t_unit @ top_to1996260823553986621t_unit ) @ Consistent @ S ) ) ) ).
% MCS.maximal_Extend
thf(fact_855_MCS_Omaximal__Extend,axiom,
! [Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ( maxima3264069618988350929c_fm_i @ Consistent @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ Consistent @ S ) ) ) ).
% MCS.maximal_Extend
thf(fact_856_MCS_OExtend__subset,axiom,
! [Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7871471802976955641t_unit @ Consistent )
=> ( ord_le3507040750410214029t_unit @ S @ ( maxima3365502296767225214t_unit @ ( bNF_Ca7083678892733797993t_unit @ top_to1996260823553986621t_unit ) @ Consistent @ S ) ) ) ).
% MCS.Extend_subset
thf(fact_857_MCS_OExtend__subset,axiom,
! [Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ( ord_le3843937902494030498c_fm_i @ S @ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ Consistent @ S ) ) ) ).
% MCS.Extend_subset
thf(fact_858_MCS__axioms,axiom,
! [A: epistemic_fm_i > $o] : ( maxima1924290493700099598c_fm_i @ ( episte2285483198712856234tent_i @ A ) ) ).
% MCS_axioms
thf(fact_859_MCS_Oinfinite__UNIV,axiom,
! [Consistent: set_Epistemic_fm_i > $o] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i ) ) ).
% MCS.infinite_UNIV
thf(fact_860_MCS_Oinfinite__UNIV,axiom,
! [Consistent: set_Product_unit > $o] :
( ( maxima7871471802976955641t_unit @ Consistent )
=> ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% MCS.infinite_UNIV
thf(fact_861_subset__CollectI,axiom,
! [B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,Q2: set_Epistemic_fm_i > $o,P3: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ A )
=> ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ B2 )
=> ( ( Q2 @ X3 )
=> ( P3 @ X3 ) ) )
=> ( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ B2 )
& ( Q2 @ X ) ) )
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P3 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_862_subset__CollectI,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,Q2: epistemic_fm_i > $o,P3: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ B2 )
=> ( ( Q2 @ X3 )
=> ( P3 @ X3 ) ) )
=> ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B2 )
& ( Q2 @ X ) ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P3 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_863_subset__CollectI,axiom,
! [B2: set_Product_unit,A: set_Product_unit,Q2: product_unit > $o,P3: product_unit > $o] :
( ( ord_le3507040750410214029t_unit @ B2 @ A )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ B2 )
=> ( ( Q2 @ X3 )
=> ( P3 @ X3 ) ) )
=> ( ord_le3507040750410214029t_unit
@ ( collect_Product_unit
@ ^ [X: product_unit] :
( ( member_Product_unit @ X @ B2 )
& ( Q2 @ X ) ) )
@ ( collect_Product_unit
@ ^ [X: product_unit] :
( ( member_Product_unit @ X @ A )
& ( P3 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_864_subset__Collect__iff,axiom,
! [B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ B2
@ ( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
& ( P3 @ X ) ) ) )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ B2 )
=> ( P3 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_865_subset__Collect__iff,axiom,
! [B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P3: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ B2
@ ( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A )
& ( P3 @ X ) ) ) )
= ( ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ B2 )
=> ( P3 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_866_subset__Collect__iff,axiom,
! [B2: set_Product_unit,A: set_Product_unit,P3: product_unit > $o] :
( ( ord_le3507040750410214029t_unit @ B2 @ A )
=> ( ( ord_le3507040750410214029t_unit @ B2
@ ( collect_Product_unit
@ ^ [X: product_unit] :
( ( member_Product_unit @ X @ A )
& ( P3 @ X ) ) ) )
= ( ! [X: product_unit] :
( ( member_Product_unit @ X @ B2 )
=> ( P3 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_867_MCS_Ointro,axiom,
! [Consistent: set_Epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
=> ( ! [S7: set_Epistemic_fm_i,S5: set_Epistemic_fm_i] :
( ( Consistent @ S7 )
=> ( ( ord_le3843937902494030498c_fm_i @ S5 @ S7 )
=> ( Consistent @ S5 ) ) )
=> ( ! [S7: set_Epistemic_fm_i] :
( ~ ( Consistent @ S7 )
=> ? [S8: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S8 @ S7 )
& ( finite3304564979551393739c_fm_i @ S8 )
& ~ ( Consistent @ S8 ) ) )
=> ( maxima1924290493700099598c_fm_i @ Consistent ) ) ) ) ).
% MCS.intro
thf(fact_868_MCS_Ointro,axiom,
! [Consistent: set_Product_unit > $o] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ! [S7: set_Product_unit,S5: set_Product_unit] :
( ( Consistent @ S7 )
=> ( ( ord_le3507040750410214029t_unit @ S5 @ S7 )
=> ( Consistent @ S5 ) ) )
=> ( ! [S7: set_Product_unit] :
( ~ ( Consistent @ S7 )
=> ? [S8: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ S8 @ S7 )
& ( finite4290736615968046902t_unit @ S8 )
& ~ ( Consistent @ S8 ) ) )
=> ( maxima7871471802976955641t_unit @ Consistent ) ) ) ) ).
% MCS.intro
thf(fact_869_MCS__def,axiom,
( maxima1924290493700099598c_fm_i
= ( ^ [Consistent2: set_Epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ top_to8364301187473967186c_fm_i )
& ! [S6: set_Epistemic_fm_i,S9: set_Epistemic_fm_i] :
( ( Consistent2 @ S6 )
=> ( ( ord_le3843937902494030498c_fm_i @ S9 @ S6 )
=> ( Consistent2 @ S9 ) ) )
& ! [S6: set_Epistemic_fm_i] :
( ~ ( Consistent2 @ S6 )
=> ? [S9: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S9 @ S6 )
& ( finite3304564979551393739c_fm_i @ S9 )
& ~ ( Consistent2 @ S9 ) ) ) ) ) ) ).
% MCS_def
thf(fact_870_MCS__def,axiom,
( maxima7871471802976955641t_unit
= ( ^ [Consistent2: set_Product_unit > $o] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ! [S6: set_Product_unit,S9: set_Product_unit] :
( ( Consistent2 @ S6 )
=> ( ( ord_le3507040750410214029t_unit @ S9 @ S6 )
=> ( Consistent2 @ S9 ) ) )
& ! [S6: set_Product_unit] :
( ~ ( Consistent2 @ S6 )
=> ? [S9: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ S9 @ S6 )
& ( finite4290736615968046902t_unit @ S9 )
& ~ ( Consistent2 @ S9 ) ) ) ) ) ) ).
% MCS_def
thf(fact_871_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_872_MCS_Omaximal__def,axiom,
! [Consistent: set_Product_unit > $o,S: set_Product_unit] :
( ( maxima7871471802976955641t_unit @ Consistent )
=> ( ( maxima1338112684437001276t_unit @ Consistent @ S )
= ( ! [P: product_unit] :
( ( Consistent @ ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ P @ bot_bo3957492148770167129t_unit ) @ S ) )
=> ( member_Product_unit @ P @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_873_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_874_kripke_Oselect__convs_I2_J,axiom,
! [W3: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi: set_Epistemic_fm_i > list_char > $o,More: product_unit] :
( ( episte484680259488342554t_unit @ ( episte8397520288747125024t_unit @ W3 @ K @ ( episte4955834246032687171t_unit @ Pi @ More ) ) )
= More ) ).
% kripke.select_convs(2)
thf(fact_875_kripke_Oselect__convs_I1_J,axiom,
! [W3: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi: set_Epistemic_fm_i > list_char > $o,More: product_unit] :
( ( episte6627496282998226676t_unit @ ( episte8397520288747125024t_unit @ W3 @ K @ ( episte4955834246032687171t_unit @ Pi @ More ) ) )
= Pi ) ).
% kripke.select_convs(1)
thf(fact_876_kripke_Oupdate__convs_I1_J,axiom,
! [Pi2: ( set_Epistemic_fm_i > list_char > $o ) > set_Epistemic_fm_i > list_char > $o,W3: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,Pi: set_Epistemic_fm_i > list_char > $o,More: product_unit] :
( ( episte8744582347493487171t_unit @ Pi2 @ ( episte8397520288747125024t_unit @ W3 @ K @ ( episte4955834246032687171t_unit @ Pi @ More ) ) )
= ( episte8397520288747125024t_unit @ W3 @ K @ ( episte4955834246032687171t_unit @ ( Pi2 @ Pi ) @ More ) ) ) ).
% kripke.update_convs(1)
thf(fact_877_Sup__fin_Oinsert,axiom,
! [A: set_Product_unit,X2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit2 @ X2 @ A ) )
= ( sup_sup_Product_unit @ X2 @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_878_Sup__fin_Osingleton,axiom,
! [X2: product_unit] :
( ( lattic5294303975357428420t_unit @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
= X2 ) ).
% Sup_fin.singleton
thf(fact_879_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_880_Sup__fin_OcoboundedI,axiom,
! [A: set_set_Product_unit,A2: set_Product_unit] :
( ( finite1772178364199683094t_unit @ A )
=> ( ( member5877623283571906838t_unit @ A2 @ A )
=> ( ord_le3507040750410214029t_unit @ A2 @ ( lattic5714169312027131812t_unit @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_881_Sup__fin_Oin__idem,axiom,
! [A: set_se3485332733965609186c_fm_i,X2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ X2 @ ( lattic3744900211830597177c_fm_i @ A ) )
= ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_882_Sup__fin_Oin__idem,axiom,
! [A: set_Product_unit,X2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X2 @ A )
=> ( ( sup_sup_Product_unit @ X2 @ ( lattic5294303975357428420t_unit @ A ) )
= ( lattic5294303975357428420t_unit @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_883_Sup__fin_Obounded__iff,axiom,
! [A: set_Product_unit,X2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X2 )
= ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A )
=> ( ord_le3221252021190050221t_unit @ X @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_884_Sup__fin_Obounded__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X2 )
= ( ! [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A )
=> ( ord_le3843937902494030498c_fm_i @ X @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_885_Sup__fin_Obounded__iff,axiom,
! [A: set_set_Product_unit,X2: set_Product_unit] :
( ( finite1772178364199683094t_unit @ A )
=> ( ( A != bot_bo6284674557190688313t_unit )
=> ( ( ord_le3507040750410214029t_unit @ ( lattic5714169312027131812t_unit @ A ) @ X2 )
= ( ! [X: set_Product_unit] :
( ( member5877623283571906838t_unit @ X @ A )
=> ( ord_le3507040750410214029t_unit @ X @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_886_Sup__fin_OboundedI,axiom,
! [A: set_Product_unit,X2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [A6: product_unit] :
( ( member_Product_unit @ A6 @ A )
=> ( ord_le3221252021190050221t_unit @ A6 @ X2 ) )
=> ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_887_Sup__fin_OboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,X2: 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 @ X2 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_888_Sup__fin_OboundedI,axiom,
! [A: set_set_Product_unit,X2: set_Product_unit] :
( ( finite1772178364199683094t_unit @ A )
=> ( ( A != bot_bo6284674557190688313t_unit )
=> ( ! [A6: set_Product_unit] :
( ( member5877623283571906838t_unit @ A6 @ A )
=> ( ord_le3507040750410214029t_unit @ A6 @ X2 ) )
=> ( ord_le3507040750410214029t_unit @ ( lattic5714169312027131812t_unit @ A ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_889_Sup__fin_OboundedE,axiom,
! [A: set_Product_unit,X2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X2 )
=> ! [A8: product_unit] :
( ( member_Product_unit @ A8 @ A )
=> ( ord_le3221252021190050221t_unit @ A8 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_890_Sup__fin_OboundedE,axiom,
! [A: set_se3485332733965609186c_fm_i,X2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X2 )
=> ! [A8: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A8 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A8 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_891_Sup__fin_OboundedE,axiom,
! [A: set_set_Product_unit,X2: set_Product_unit] :
( ( finite1772178364199683094t_unit @ A )
=> ( ( A != bot_bo6284674557190688313t_unit )
=> ( ( ord_le3507040750410214029t_unit @ ( lattic5714169312027131812t_unit @ A ) @ X2 )
=> ! [A8: set_Product_unit] :
( ( member5877623283571906838t_unit @ A8 @ A )
=> ( ord_le3507040750410214029t_unit @ A8 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_892_Sup__fin_Osubset__imp,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B2 )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_893_Sup__fin_Osubset__imp,axiom,
! [A: set_set_Product_unit,B2: set_set_Product_unit] :
( ( ord_le3674001228145610605t_unit @ A @ B2 )
=> ( ( A != bot_bo6284674557190688313t_unit )
=> ( ( finite1772178364199683094t_unit @ B2 )
=> ( ord_le3507040750410214029t_unit @ ( lattic5714169312027131812t_unit @ A ) @ ( lattic5714169312027131812t_unit @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_894_Sup__fin_Osubset__imp,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ ( lattic5294303975357428420t_unit @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_895_Sup__fin_Osubset,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( B2 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ A )
=> ( ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ B2 ) @ ( lattic5294303975357428420t_unit @ A ) )
= ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_896_Sup__fin_Oclosed,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y3 ) @ ( insert7698009978809854162c_fm_i @ X3 @ ( insert7698009978809854162c_fm_i @ Y3 @ bot_bo145720340923748686c_fm_i ) ) )
=> ( member1104366573291651755c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_897_Sup__fin_Oclosed,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,Y3: product_unit] : ( member_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y3 ) @ ( insert_Product_unit2 @ X3 @ ( insert_Product_unit2 @ Y3 @ bot_bo3957492148770167129t_unit ) ) )
=> ( member_Product_unit @ ( lattic5294303975357428420t_unit @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_898_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_se3485332733965609186c_fm_i,X2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X2 @ A ) )
= ( sup_su1936195050962291414c_fm_i @ X2 @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_899_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_Product_unit,X2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X2 @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit2 @ X2 @ A ) )
= ( sup_sup_Product_unit @ X2 @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_900_Sup__fin_Ounion,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ( B2 != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( sup_su793286257634532545t_unit @ A @ B2 ) )
= ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ A ) @ ( lattic5294303975357428420t_unit @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_901_semantics_Osimps_I2_J,axiom,
! [M: episte5489734138418521880t_unit,W2: set_Epistemic_fm_i,X2: list_char] :
( ( episte2498298383968489120c_fm_i @ M @ W2 @ ( epistemic_Pro_i @ X2 ) )
= ( episte6627496282998226676t_unit @ M @ W2 @ X2 ) ) ).
% semantics.simps(2)
thf(fact_902_kripke_Osurjective,axiom,
! [R: episte5489734138418521880t_unit] :
( R
= ( episte8397520288747125024t_unit @ ( episte2675268102452679275t_unit @ R ) @ ( episte852950631662840415t_unit @ R ) @ ( episte4955834246032687171t_unit @ ( episte6627496282998226676t_unit @ R ) @ ( episte484680259488342554t_unit @ R ) ) ) ) ).
% kripke.surjective
thf(fact_903_kripke_Oequality,axiom,
! [R: episte5489734138418521880t_unit,R3: episte5489734138418521880t_unit] :
( ( ( episte2675268102452679275t_unit @ R )
= ( episte2675268102452679275t_unit @ R3 ) )
=> ( ( ( episte852950631662840415t_unit @ R )
= ( episte852950631662840415t_unit @ R3 ) )
=> ( ( ( episte6627496282998226676t_unit @ R )
= ( episte6627496282998226676t_unit @ R3 ) )
=> ( ( ( episte484680259488342554t_unit @ R )
= ( episte484680259488342554t_unit @ R3 ) )
=> ( R = R3 ) ) ) ) ) ).
% kripke.equality
thf(fact_904_extend__underS,axiom,
! [M5: epistemic_fm_i,N: epistemic_fm_i,A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ M5 @ ( order_1288088345848551912c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ N ) )
=> ( ord_le3843937902494030498c_fm_i @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ M5 ) @ ( maxima1168816371290972723c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ A ) @ S @ N ) ) ) ).
% extend_underS
thf(fact_905_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I2: epistemic_fm_i,R2: set_Pr4658907567593863815c_fm_i,J: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ I2 @ ( order_1288088345848551912c_fm_i @ R2 @ J ) )
=> ( member6642669606046002379c_fm_i @ I2 @ ( field_Epistemic_fm_i @ R2 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_906_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I2: set_Epistemic_fm_i,R2: set_Pr3972831103112126087c_fm_i,J: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ I2 @ ( order_8439613377573682120c_fm_i @ R2 @ J ) )
=> ( member1104366573291651755c_fm_i @ I2 @ ( field_4031064140798694607c_fm_i @ R2 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_907_underS__empty,axiom,
! [A2: set_Epistemic_fm_i,R: set_Pr3972831103112126087c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( order_8439613377573682120c_fm_i @ R @ A2 )
= bot_bo145720340923748686c_fm_i ) ) ).
% underS_empty
thf(fact_908_underS__empty,axiom,
! [A2: epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( order_1288088345848551912c_fm_i @ R @ A2 )
= bot_bo4194595901900360558c_fm_i ) ) ).
% underS_empty
thf(fact_909_underS__empty,axiom,
! [A2: product_unit,R: set_Pr5094982260447487303t_unit] :
( ~ ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ( order_5809128134054296019t_unit @ R @ A2 )
= bot_bo3957492148770167129t_unit ) ) ).
% underS_empty
thf(fact_910_Order__Relation_OunderS__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_1288088345848551912c_fm_i @ R @ A2 ) @ ( field_Epistemic_fm_i @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_911_Order__Relation_OunderS__Field,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit] : ( ord_le3507040750410214029t_unit @ ( order_5809128134054296019t_unit @ R @ A2 ) @ ( field_Product_unit @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_912_underS__incr2,axiom,
! [R: set_Pr4658907567593863815c_fm_i,R3: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( ord_le3932863852503339559c_fm_i @ R @ R3 )
=> ( ord_le3843937902494030498c_fm_i @ ( order_1288088345848551912c_fm_i @ R @ A2 ) @ ( order_1288088345848551912c_fm_i @ R3 @ A2 ) ) ) ).
% underS_incr2
thf(fact_913_underS__incr2,axiom,
! [R: set_Pr5094982260447487303t_unit,R3: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( ord_le930853871177014503t_unit @ R @ R3 )
=> ( ord_le3507040750410214029t_unit @ ( order_5809128134054296019t_unit @ R @ A2 ) @ ( order_5809128134054296019t_unit @ R3 @ A2 ) ) ) ).
% underS_incr2
thf(fact_914_underS__subset__under,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_1288088345848551912c_fm_i @ R @ A2 ) @ ( order_1815523504696313095c_fm_i @ R @ A2 ) ) ).
% underS_subset_under
thf(fact_915_underS__subset__under,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit] : ( ord_le3507040750410214029t_unit @ ( order_5809128134054296019t_unit @ R @ A2 ) @ ( order_6949322460238004850t_unit @ R @ A2 ) ) ).
% underS_subset_under
thf(fact_916_frame_Oselect__convs_I2_J,axiom,
! [W3: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More: episte7133455005732253587t_unit] :
( ( episte852950631662840415t_unit @ ( episte8397520288747125024t_unit @ W3 @ K @ More ) )
= K ) ).
% frame.select_convs(2)
thf(fact_917_MCS__Lim__Ord_Oextend__underS,axiom,
! [R: set_Pr3972831103112126087c_fm_i,Consistent: set_se3485332733965609186c_fm_i > $o,M5: set_Epistemic_fm_i,N: set_Epistemic_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( maxima9073963094320882813c_fm_i @ R @ Consistent )
=> ( ( member1104366573291651755c_fm_i @ M5 @ ( order_8439613377573682120c_fm_i @ R @ N ) )
=> ( ord_le5389487502678872194c_fm_i @ ( maxima3510324002270828563c_fm_i @ R @ Consistent @ S @ M5 ) @ ( maxima3510324002270828563c_fm_i @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_underS
thf(fact_918_MCS__Lim__Ord_Oextend__underS,axiom,
! [R: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,M5: epistemic_fm_i,N: epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R @ Consistent )
=> ( ( member6642669606046002379c_fm_i @ M5 @ ( order_1288088345848551912c_fm_i @ R @ N ) )
=> ( ord_le3843937902494030498c_fm_i @ ( maxima1168816371290972723c_fm_i @ R @ Consistent @ S @ M5 ) @ ( maxima1168816371290972723c_fm_i @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_underS
thf(fact_919_MCS__Lim__Ord_Oextend__underS,axiom,
! [R: set_Pr5094982260447487303t_unit,Consistent: set_Product_unit > $o,M5: product_unit,N: product_unit,S: set_Product_unit] :
( ( maxima7229560971830504200t_unit @ R @ Consistent )
=> ( ( member_Product_unit @ M5 @ ( order_5809128134054296019t_unit @ R @ N ) )
=> ( ord_le3507040750410214029t_unit @ ( maxima7300585265216834974t_unit @ R @ Consistent @ S @ M5 ) @ ( maxima7300585265216834974t_unit @ R @ Consistent @ S @ N ) ) ) ) ).
% MCS_Lim_Ord.extend_underS
thf(fact_920_transitive__def,axiom,
( episte2966952217288386430t_unit
= ( ^ [M3: episte5489734138418521880t_unit] :
! [I: i,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M3 ) )
=> ! [Y5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y5 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ! [Z5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z5 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ( ( member1104366573291651755c_fm_i @ Z5 @ ( episte852950631662840415t_unit @ M3 @ I @ Y5 ) )
& ( member1104366573291651755c_fm_i @ X @ ( episte852950631662840415t_unit @ M3 @ I @ Z5 ) ) )
=> ( member1104366573291651755c_fm_i @ X @ ( episte852950631662840415t_unit @ M3 @ I @ Y5 ) ) ) ) ) ) ) ) ).
% transitive_def
thf(fact_921_reflexive__def,axiom,
( episte251305198166243731t_unit
= ( ^ [M3: episte5489734138418521880t_unit] :
! [I: i,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( member1104366573291651755c_fm_i @ X @ ( episte852950631662840415t_unit @ M3 @ I @ X ) ) ) ) ) ).
% reflexive_def
thf(fact_922_symmetric__def,axiom,
( episte80897895827131294t_unit
= ( ^ [M3: episte5489734138418521880t_unit] :
! [I: i,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M3 ) )
=> ! [Y5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y5 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ( member1104366573291651755c_fm_i @ X @ ( episte852950631662840415t_unit @ M3 @ I @ Y5 ) )
= ( member1104366573291651755c_fm_i @ Y5 @ ( episte852950631662840415t_unit @ M3 @ I @ X ) ) ) ) ) ) ) ).
% symmetric_def
thf(fact_923_Euclidean__def,axiom,
( episte6275404236303465413t_unit
= ( ^ [M3: episte5489734138418521880t_unit] :
! [I: i,X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ ( episte2675268102452679275t_unit @ M3 ) )
=> ! [Y5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y5 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ! [Z5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z5 @ ( episte2675268102452679275t_unit @ M3 ) )
=> ( ( member1104366573291651755c_fm_i @ Y5 @ ( episte852950631662840415t_unit @ M3 @ I @ X ) )
=> ( ( member1104366573291651755c_fm_i @ Z5 @ ( episte852950631662840415t_unit @ M3 @ I @ X ) )
=> ( member1104366573291651755c_fm_i @ Z5 @ ( episte852950631662840415t_unit @ M3 @ I @ Y5 ) ) ) ) ) ) ) ) ) ).
% Euclidean_def
thf(fact_924_frame_Osurjective,axiom,
! [R: episte5489734138418521880t_unit] :
( R
= ( episte8397520288747125024t_unit @ ( episte2675268102452679275t_unit @ R ) @ ( episte852950631662840415t_unit @ R ) @ ( episte7135767042998387833t_unit @ R ) ) ) ).
% frame.surjective
thf(fact_925_Refl__under__underS,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ ( field_4031064140798694607c_fm_i @ R ) @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( order_4888947244107180775c_fm_i @ R @ A2 )
= ( sup_su2582925890723967158c_fm_i @ ( order_8439613377573682120c_fm_i @ R @ A2 ) @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ).
% Refl_under_underS
thf(fact_926_Refl__under__underS,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ ( field_Epistemic_fm_i @ R ) @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( order_1815523504696313095c_fm_i @ R @ A2 )
= ( sup_su1936195050962291414c_fm_i @ ( order_1288088345848551912c_fm_i @ R @ A2 ) @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% Refl_under_underS
thf(fact_927_Refl__under__underS,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( refl_on_Product_unit @ ( field_Product_unit @ R ) @ R )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ( order_6949322460238004850t_unit @ R @ A2 )
= ( sup_su793286257634532545t_unit @ ( order_5809128134054296019t_unit @ R @ A2 ) @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% Refl_under_underS
thf(fact_928_frame_Oequality,axiom,
! [R: episte5489734138418521880t_unit,R3: episte5489734138418521880t_unit] :
( ( ( episte2675268102452679275t_unit @ R )
= ( episte2675268102452679275t_unit @ R3 ) )
=> ( ( ( episte852950631662840415t_unit @ R )
= ( episte852950631662840415t_unit @ R3 ) )
=> ( ( ( episte7135767042998387833t_unit @ R )
= ( episte7135767042998387833t_unit @ R3 ) )
=> ( R = R3 ) ) ) ) ).
% frame.equality
thf(fact_929_refl__on__empty,axiom,
refl_o2748477423337579169c_fm_i @ bot_bo4194595901900360558c_fm_i @ bot_bo3668964140382854387c_fm_i ).
% refl_on_empty
thf(fact_930_refl__on__empty,axiom,
refl_on_Product_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% refl_on_empty
thf(fact_931_Refl__under__in,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ ( field_Epistemic_fm_i @ R ) @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( member6642669606046002379c_fm_i @ A2 @ ( order_1815523504696313095c_fm_i @ R @ A2 ) ) ) ) ).
% Refl_under_in
thf(fact_932_Refl__under__in,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ ( field_4031064140798694607c_fm_i @ R ) @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( member1104366573291651755c_fm_i @ A2 @ ( order_4888947244107180775c_fm_i @ R @ A2 ) ) ) ) ).
% Refl_under_in
thf(fact_933_frame_Oselect__convs_I3_J,axiom,
! [W3: set_se3485332733965609186c_fm_i,K: i > set_Epistemic_fm_i > set_se3485332733965609186c_fm_i,More: episte7133455005732253587t_unit] :
( ( episte7135767042998387833t_unit @ ( episte8397520288747125024t_unit @ W3 @ K @ More ) )
= More ) ).
% frame.select_convs(3)
thf(fact_934_subset__singleton__iff__Uniq,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ? [A4: set_Epistemic_fm_i] : ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A4 @ bot_bo145720340923748686c_fm_i ) ) )
= ( uniq_s5891897140743797767c_fm_i
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_935_subset__singleton__iff__Uniq,axiom,
! [A: set_Epistemic_fm_i] :
( ( ? [A4: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A4 @ bot_bo4194595901900360558c_fm_i ) ) )
= ( uniq_Epistemic_fm_i
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_936_subset__singleton__iff__Uniq,axiom,
! [A: set_Product_unit] :
( ( ? [A4: product_unit] : ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ A4 @ bot_bo3957492148770167129t_unit ) ) )
= ( uniq_Product_unit
@ ^ [X: product_unit] : ( member_Product_unit @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_937_Sup__fin_Oinsert__remove,axiom,
! [A: set_Product_unit,X2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit2 @ X2 @ A ) )
= X2 ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit2 @ X2 @ A ) )
= ( sup_sup_Product_unit @ X2 @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_938_DiffI,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ C @ B2 )
=> ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_939_DiffI,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ C @ B2 )
=> ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_940_Diff__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
& ~ ( member6642669606046002379c_fm_i @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_941_Diff__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B2 ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
& ~ ( member1104366573291651755c_fm_i @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_942_Diff__cancel,axiom,
! [A: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ A )
= bot_bo4194595901900360558c_fm_i ) ).
% Diff_cancel
thf(fact_943_Diff__cancel,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ A )
= bot_bo3957492148770167129t_unit ) ).
% Diff_cancel
thf(fact_944_empty__Diff,axiom,
! [A: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ bot_bo4194595901900360558c_fm_i @ A )
= bot_bo4194595901900360558c_fm_i ) ).
% empty_Diff
thf(fact_945_empty__Diff,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ bot_bo3957492148770167129t_unit @ A )
= bot_bo3957492148770167129t_unit ) ).
% empty_Diff
thf(fact_946_Diff__empty,axiom,
! [A: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= A ) ).
% Diff_empty
thf(fact_947_Diff__empty,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ bot_bo3957492148770167129t_unit )
= A ) ).
% Diff_empty
thf(fact_948_insert__Diff1,axiom,
! [X2: product_unit,B2: set_Product_unit,A: set_Product_unit] :
( ( member_Product_unit @ X2 @ B2 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit2 @ X2 @ A ) @ B2 )
= ( minus_6452836326544984404t_unit @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_949_insert__Diff1,axiom,
! [X2: epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ B2 )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X2 @ A ) @ B2 )
= ( minus_1543973939109951465c_fm_i @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_950_insert__Diff1,axiom,
! [X2: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ B2 )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X2 @ A ) @ B2 )
= ( minus_1073301138667672009c_fm_i @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_951_Diff__insert0,axiom,
! [X2: product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ~ ( member_Product_unit @ X2 @ A )
=> ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X2 @ B2 ) )
= ( minus_6452836326544984404t_unit @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_952_Diff__insert0,axiom,
! [X2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X2 @ B2 ) )
= ( minus_1543973939109951465c_fm_i @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_953_Diff__insert0,axiom,
! [X2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X2 @ B2 ) )
= ( minus_1073301138667672009c_fm_i @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_954_Diff__UNIV,axiom,
! [A: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ top_to8364301187473967186c_fm_i )
= bot_bo4194595901900360558c_fm_i ) ).
% Diff_UNIV
thf(fact_955_Diff__UNIV,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ top_to1996260823553986621t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Diff_UNIV
thf(fact_956_Diff__eq__empty__iff,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ A @ B2 )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_957_Diff__eq__empty__iff,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ( minus_6452836326544984404t_unit @ A @ B2 )
= bot_bo3957492148770167129t_unit )
= ( ord_le3507040750410214029t_unit @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_958_insert__Diff__single,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( insert7817948997695205106c_fm_i @ A2 @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
= ( insert7817948997695205106c_fm_i @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_959_insert__Diff__single,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( insert_Product_unit2 @ A2 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( insert_Product_unit2 @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_960_finite__Diff__insert,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) ) )
= ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_961_finite__Diff__insert,axiom,
! [A: set_Product_unit,A2: product_unit,B2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ A2 @ B2 ) ) )
= ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_962_insert__Diff__if,axiom,
! [X2: product_unit,B2: set_Product_unit,A: set_Product_unit] :
( ( ( member_Product_unit @ X2 @ B2 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit2 @ X2 @ A ) @ B2 )
= ( minus_6452836326544984404t_unit @ A @ B2 ) ) )
& ( ~ ( member_Product_unit @ X2 @ B2 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit2 @ X2 @ A ) @ B2 )
= ( insert_Product_unit2 @ X2 @ ( minus_6452836326544984404t_unit @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_963_insert__Diff__if,axiom,
! [X2: epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ X2 @ B2 )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X2 @ A ) @ B2 )
= ( minus_1543973939109951465c_fm_i @ A @ B2 ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ X2 @ B2 )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X2 @ A ) @ B2 )
= ( insert7817948997695205106c_fm_i @ X2 @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_964_insert__Diff__if,axiom,
! [X2: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ X2 @ B2 )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X2 @ A ) @ B2 )
= ( minus_1073301138667672009c_fm_i @ A @ B2 ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ X2 @ B2 )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X2 @ A ) @ B2 )
= ( insert7698009978809854162c_fm_i @ X2 @ ( minus_1073301138667672009c_fm_i @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_965_DiffE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B2 ) ) ) ).
% DiffE
thf(fact_966_DiffE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B2 ) )
=> ~ ( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B2 ) ) ) ).
% DiffE
thf(fact_967_DiffD1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) )
=> ( member6642669606046002379c_fm_i @ C @ A ) ) ).
% DiffD1
thf(fact_968_DiffD1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B2 ) )
=> ( member1104366573291651755c_fm_i @ C @ A ) ) ).
% DiffD1
thf(fact_969_DiffD2,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) )
=> ~ ( member6642669606046002379c_fm_i @ C @ B2 ) ) ).
% DiffD2
thf(fact_970_DiffD2,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B2 ) )
=> ~ ( member1104366573291651755c_fm_i @ C @ B2 ) ) ).
% DiffD2
thf(fact_971_set__diff__eq,axiom,
( minus_1073301138667672009c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X @ A3 )
& ~ ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_972_set__diff__eq,axiom,
( minus_1543973939109951465c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ A3 )
& ~ ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_973_Diff__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ D2 @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) @ ( minus_1543973939109951465c_fm_i @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_974_Diff__mono,axiom,
! [A: set_Product_unit,C2: set_Product_unit,D2: set_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ C2 )
=> ( ( ord_le3507040750410214029t_unit @ D2 @ B2 )
=> ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ B2 ) @ ( minus_6452836326544984404t_unit @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_975_Diff__subset,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_976_Diff__subset,axiom,
! [A: set_Product_unit,B2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_977_double__diff,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C2 )
=> ( ( minus_1543973939109951465c_fm_i @ B2 @ ( minus_1543973939109951465c_fm_i @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_978_double__diff,axiom,
! [A: set_Product_unit,B2: set_Product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ C2 )
=> ( ( minus_6452836326544984404t_unit @ B2 @ ( minus_6452836326544984404t_unit @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_979_diff__shunt__var,axiom,
! [X2: product_unit,Y4: product_unit] :
( ( ( minus_3524152463667985524t_unit @ X2 @ Y4 )
= bot_bot_Product_unit )
= ( ord_le3221252021190050221t_unit @ X2 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_980_diff__shunt__var,axiom,
! [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ X2 @ Y4 )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ X2 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_981_diff__shunt__var,axiom,
! [X2: set_Product_unit,Y4: set_Product_unit] :
( ( ( minus_6452836326544984404t_unit @ X2 @ Y4 )
= bot_bo3957492148770167129t_unit )
= ( ord_le3507040750410214029t_unit @ X2 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_982_Diff__insert,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) )
= ( minus_1543973939109951465c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% Diff_insert
thf(fact_983_Diff__insert,axiom,
! [A: set_Product_unit,A2: product_unit,B2: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ A2 @ B2 ) )
= ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ B2 ) @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ).
% Diff_insert
thf(fact_984_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_985_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_986_insert__Diff,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ( ( insert_Product_unit2 @ A2 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= A ) ) ).
% insert_Diff
thf(fact_987_Diff__insert2,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B2 ) )
= ( minus_1543973939109951465c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_988_Diff__insert2,axiom,
! [A: set_Product_unit,A2: product_unit,B2: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ A2 @ B2 ) )
= ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_989_Diff__insert__absorb,axiom,
! [X2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X2 @ A ) @ ( insert7698009978809854162c_fm_i @ X2 @ bot_bo145720340923748686c_fm_i ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_990_Diff__insert__absorb,axiom,
! [X2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X2 @ A ) @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_991_Diff__insert__absorb,axiom,
! [X2: product_unit,A: set_Product_unit] :
( ~ ( member_Product_unit @ X2 @ A )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit2 @ X2 @ A ) @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_992_subset__Diff__insert,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,X2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B2 @ ( insert7698009978809854162c_fm_i @ X2 @ C2 ) ) )
= ( ( ord_le5389487502678872194c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B2 @ C2 ) )
& ~ ( member1104366573291651755c_fm_i @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_993_subset__Diff__insert,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,X2: epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B2 @ ( insert7817948997695205106c_fm_i @ X2 @ C2 ) ) )
= ( ( ord_le3843937902494030498c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B2 @ C2 ) )
& ~ ( member6642669606046002379c_fm_i @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_994_subset__Diff__insert,axiom,
! [A: set_Product_unit,B2: set_Product_unit,X2: product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ ( minus_6452836326544984404t_unit @ B2 @ ( insert_Product_unit2 @ X2 @ C2 ) ) )
= ( ( ord_le3507040750410214029t_unit @ A @ ( minus_6452836326544984404t_unit @ B2 @ C2 ) )
& ~ ( member_Product_unit @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_995_Diff__partition,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B2 @ A ) )
= B2 ) ) ).
% Diff_partition
thf(fact_996_Diff__partition,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ( sup_su793286257634532545t_unit @ A @ ( minus_6452836326544984404t_unit @ B2 @ A ) )
= B2 ) ) ).
% Diff_partition
thf(fact_997_Diff__subset__conv,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B2 ) @ C2 )
= ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_998_Diff__subset__conv,axiom,
! [A: set_Product_unit,B2: set_Product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ B2 ) @ C2 )
= ( ord_le3507040750410214029t_unit @ A @ ( sup_su793286257634532545t_unit @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_999_finite__empty__induct,axiom,
! [A: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( P3 @ A )
=> ( ! [A6: set_Epistemic_fm_i,A7: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A7 )
=> ( ( member1104366573291651755c_fm_i @ A6 @ A7 )
=> ( ( P3 @ A7 )
=> ( P3 @ ( minus_1073301138667672009c_fm_i @ A7 @ ( insert7698009978809854162c_fm_i @ A6 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) )
=> ( P3 @ bot_bo145720340923748686c_fm_i ) ) ) ) ).
% finite_empty_induct
thf(fact_1000_finite__empty__induct,axiom,
! [A: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( P3 @ A )
=> ( ! [A6: epistemic_fm_i,A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( member6642669606046002379c_fm_i @ A6 @ A7 )
=> ( ( P3 @ A7 )
=> ( P3 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ A6 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ( P3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% finite_empty_induct
thf(fact_1001_finite__empty__induct,axiom,
! [A: set_Product_unit,P3: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( P3 @ A )
=> ( ! [A6: product_unit,A7: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A7 )
=> ( ( member_Product_unit @ A6 @ A7 )
=> ( ( P3 @ A7 )
=> ( P3 @ ( minus_6452836326544984404t_unit @ A7 @ ( insert_Product_unit2 @ A6 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
=> ( P3 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% finite_empty_induct
thf(fact_1002_infinite__coinduct,axiom,
! [X4: set_Epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ( X4 @ A )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( X4 @ A7 )
=> ? [X5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X5 @ A7 )
& ( ( X4 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X5 @ bot_bo4194595901900360558c_fm_i ) ) )
| ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X5 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ~ ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% infinite_coinduct
thf(fact_1003_infinite__coinduct,axiom,
! [X4: set_Product_unit > $o,A: set_Product_unit] :
( ( X4 @ A )
=> ( ! [A7: set_Product_unit] :
( ( X4 @ A7 )
=> ? [X5: product_unit] :
( ( member_Product_unit @ X5 @ A7 )
& ( ( X4 @ ( minus_6452836326544984404t_unit @ A7 @ ( insert_Product_unit2 @ X5 @ bot_bo3957492148770167129t_unit ) ) )
| ~ ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A7 @ ( insert_Product_unit2 @ X5 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
=> ~ ( finite4290736615968046902t_unit @ A ) ) ) ).
% infinite_coinduct
thf(fact_1004_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_1005_infinite__remove,axiom,
! [S: set_Product_unit,A2: product_unit] :
( ~ ( finite4290736615968046902t_unit @ S )
=> ~ ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ S @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% infinite_remove
thf(fact_1006_subset__insert__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X2: set_Epistemic_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X2 @ B2 ) )
= ( ( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ord_le5389487502678872194c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X2 @ bot_bo145720340923748686c_fm_i ) ) @ B2 ) )
& ( ~ ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1007_subset__insert__iff,axiom,
! [A: set_Epistemic_fm_i,X2: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X2 @ B2 ) )
= ( ( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) @ B2 ) )
& ( ~ ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1008_subset__insert__iff,axiom,
! [A: set_Product_unit,X2: product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ X2 @ B2 ) )
= ( ( ( member_Product_unit @ X2 @ A )
=> ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) @ B2 ) )
& ( ~ ( member_Product_unit @ X2 @ A )
=> ( ord_le3507040750410214029t_unit @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1009_Diff__single__insert,axiom,
! [A: set_Epistemic_fm_i,X2: epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1010_Diff__single__insert,axiom,
! [A: set_Product_unit,X2: product_unit,B2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) @ B2 )
=> ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit2 @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1011_remove__induct,axiom,
! [P3: set_se3485332733965609186c_fm_i > $o,B2: set_se3485332733965609186c_fm_i] :
( ( P3 @ bot_bo145720340923748686c_fm_i )
=> ( ( ~ ( finite7933139204641697195c_fm_i @ B2 )
=> ( P3 @ B2 ) )
=> ( ! [A7: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A7 )
=> ( ( A7 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ A7 @ B2 )
=> ( ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ A7 )
=> ( P3 @ ( minus_1073301138667672009c_fm_i @ A7 @ ( insert7698009978809854162c_fm_i @ X5 @ bot_bo145720340923748686c_fm_i ) ) ) )
=> ( P3 @ A7 ) ) ) ) )
=> ( P3 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1012_remove__induct,axiom,
! [P3: set_Epistemic_fm_i > $o,B2: set_Epistemic_fm_i] :
( ( P3 @ bot_bo4194595901900360558c_fm_i )
=> ( ( ~ ( finite3304564979551393739c_fm_i @ B2 )
=> ( P3 @ B2 ) )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( A7 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A7 @ B2 )
=> ( ! [X5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X5 @ A7 )
=> ( P3 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X5 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P3 @ A7 ) ) ) ) )
=> ( P3 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1013_remove__induct,axiom,
! [P3: set_Product_unit > $o,B2: set_Product_unit] :
( ( P3 @ bot_bo3957492148770167129t_unit )
=> ( ( ~ ( finite4290736615968046902t_unit @ B2 )
=> ( P3 @ B2 ) )
=> ( ! [A7: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A7 )
=> ( ( A7 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ A7 @ B2 )
=> ( ! [X5: product_unit] :
( ( member_Product_unit @ X5 @ A7 )
=> ( P3 @ ( minus_6452836326544984404t_unit @ A7 @ ( insert_Product_unit2 @ X5 @ bot_bo3957492148770167129t_unit ) ) ) )
=> ( P3 @ A7 ) ) ) ) )
=> ( P3 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1014_finite__remove__induct,axiom,
! [B2: set_se3485332733965609186c_fm_i,P3: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ B2 )
=> ( ( P3 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A7: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A7 )
=> ( ( A7 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ A7 @ B2 )
=> ( ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ A7 )
=> ( P3 @ ( minus_1073301138667672009c_fm_i @ A7 @ ( insert7698009978809854162c_fm_i @ X5 @ bot_bo145720340923748686c_fm_i ) ) ) )
=> ( P3 @ A7 ) ) ) ) )
=> ( P3 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1015_finite__remove__induct,axiom,
! [B2: set_Epistemic_fm_i,P3: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ B2 )
=> ( ( P3 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( A7 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A7 @ B2 )
=> ( ! [X5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X5 @ A7 )
=> ( P3 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X5 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P3 @ A7 ) ) ) ) )
=> ( P3 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1016_finite__remove__induct,axiom,
! [B2: set_Product_unit,P3: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ( P3 @ bot_bo3957492148770167129t_unit )
=> ( ! [A7: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A7 )
=> ( ( A7 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ A7 @ B2 )
=> ( ! [X5: product_unit] :
( ( member_Product_unit @ X5 @ A7 )
=> ( P3 @ ( minus_6452836326544984404t_unit @ A7 @ ( insert_Product_unit2 @ X5 @ bot_bo3957492148770167129t_unit ) ) ) )
=> ( P3 @ A7 ) ) ) ) )
=> ( P3 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1017_Sup__fin_Oremove,axiom,
! [A: set_se3485332733965609186c_fm_i,X2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X2 @ bot_bo145720340923748686c_fm_i ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ A )
= X2 ) )
& ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X2 @ bot_bo145720340923748686c_fm_i ) )
!= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ A )
= ( sup_su1936195050962291414c_fm_i @ X2 @ ( lattic3744900211830597177c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X2 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1018_Sup__fin_Oremove,axiom,
! [A: set_Product_unit,X2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X2 @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ A )
= X2 ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ A )
= ( sup_sup_Product_unit @ X2 @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1019_underS__under,axiom,
! [I2: set_Epistemic_fm_i,R: set_Pr3972831103112126087c_fm_i] :
( ( member1104366573291651755c_fm_i @ I2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( order_8439613377573682120c_fm_i @ R @ I2 )
= ( minus_1073301138667672009c_fm_i @ ( order_4888947244107180775c_fm_i @ R @ I2 ) @ ( insert7698009978809854162c_fm_i @ I2 @ bot_bo145720340923748686c_fm_i ) ) ) ) ).
% underS_under
thf(fact_1020_underS__under,axiom,
! [I2: epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ( member6642669606046002379c_fm_i @ I2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( order_1288088345848551912c_fm_i @ R @ I2 )
= ( minus_1543973939109951465c_fm_i @ ( order_1815523504696313095c_fm_i @ R @ I2 ) @ ( insert7817948997695205106c_fm_i @ I2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% underS_under
thf(fact_1021_underS__under,axiom,
! [I2: product_unit,R: set_Pr5094982260447487303t_unit] :
( ( member_Product_unit @ I2 @ ( field_Product_unit @ R ) )
=> ( ( order_5809128134054296019t_unit @ R @ I2 )
= ( minus_6452836326544984404t_unit @ ( order_6949322460238004850t_unit @ R @ I2 ) @ ( insert_Product_unit2 @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% underS_under
thf(fact_1022_set__removeAll,axiom,
! [X2: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( remove817376155714978094c_fm_i @ X2 @ Xs ) )
= ( minus_1543973939109951465c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% set_removeAll
thf(fact_1023_set__removeAll,axiom,
! [X2: product_unit,Xs: list_Product_unit] :
( ( set_Product_unit2 @ ( remove8762936195178819609t_unit @ X2 @ Xs ) )
= ( minus_6452836326544984404t_unit @ ( set_Product_unit2 @ Xs ) @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ).
% set_removeAll
thf(fact_1024_minus__set__def,axiom,
( minus_1073301138667672009c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ( minus_5671197887077578836fm_i_o
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ A3 )
@ ^ [X: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1025_minus__set__def,axiom,
( minus_1543973939109951465c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( minus_6519238007984220724fm_i_o
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ A3 )
@ ^ [X: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1026_remove__def,axiom,
( remove5552923401211475591c_fm_i
= ( ^ [X: epistemic_fm_i,A3: set_Epistemic_fm_i] : ( minus_1543973939109951465c_fm_i @ A3 @ ( insert7817948997695205106c_fm_i @ X @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% remove_def
thf(fact_1027_remove__def,axiom,
( remove_Product_unit
= ( ^ [X: product_unit,A3: set_Product_unit] : ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit2 @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% remove_def
thf(fact_1028_member__remove,axiom,
! [X2: epistemic_fm_i,Y4: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( remove5552923401211475591c_fm_i @ Y4 @ A ) )
= ( ( member6642669606046002379c_fm_i @ X2 @ A )
& ( X2 != Y4 ) ) ) ).
% member_remove
thf(fact_1029_member__remove,axiom,
! [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( remove1392734035364174951c_fm_i @ Y4 @ A ) )
= ( ( member1104366573291651755c_fm_i @ X2 @ A )
& ( X2 != Y4 ) ) ) ).
% member_remove
thf(fact_1030_Func__emp2,axiom,
! [A: set_Epistemic_fm_i] :
( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ( bNF_We489590214134045204c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= bot_bo1633126455607734703c_fm_i ) ) ).
% Func_emp2
thf(fact_1031_Func__emp2,axiom,
! [A: set_Epistemic_fm_i] :
( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ( bNF_We5948249774593435647t_unit @ A @ bot_bo3957492148770167129t_unit )
= bot_bo5160044010258911514t_unit ) ) ).
% Func_emp2
thf(fact_1032_Func__emp2,axiom,
! [A: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( bNF_We667984284430983401c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= bot_bo7615558272139493508c_fm_i ) ) ).
% Func_emp2
thf(fact_1033_Func__emp2,axiom,
! [A: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( bNF_We2086683926456529748t_unit @ A @ bot_bo3957492148770167129t_unit )
= bot_bo290337425153025135t_unit ) ) ).
% Func_emp2
thf(fact_1034_Func__is__emp,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( bNF_We489590214134045204c_fm_i @ A @ B2 )
= bot_bo1633126455607734703c_fm_i )
= ( ( A != bot_bo4194595901900360558c_fm_i )
& ( B2 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% Func_is_emp
thf(fact_1035_Func__is__emp,axiom,
! [A: set_Epistemic_fm_i,B2: set_Product_unit] :
( ( ( bNF_We5948249774593435647t_unit @ A @ B2 )
= bot_bo5160044010258911514t_unit )
= ( ( A != bot_bo4194595901900360558c_fm_i )
& ( B2 = bot_bo3957492148770167129t_unit ) ) ) ).
% Func_is_emp
thf(fact_1036_Func__is__emp,axiom,
! [A: set_Product_unit,B2: set_Epistemic_fm_i] :
( ( ( bNF_We667984284430983401c_fm_i @ A @ B2 )
= bot_bo7615558272139493508c_fm_i )
= ( ( A != bot_bo3957492148770167129t_unit )
& ( B2 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% Func_is_emp
thf(fact_1037_Func__is__emp,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( ( bNF_We2086683926456529748t_unit @ A @ B2 )
= bot_bo290337425153025135t_unit )
= ( ( A != bot_bo3957492148770167129t_unit )
& ( B2 = bot_bo3957492148770167129t_unit ) ) ) ).
% Func_is_emp
thf(fact_1038_List_Oset__insert,axiom,
! [X2: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( insert177310195918387262c_fm_i @ X2 @ Xs ) )
= ( insert7817948997695205106c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1039_List_Oset__insert,axiom,
! [X2: product_unit,Xs: list_Product_unit] :
( ( set_Product_unit2 @ ( insert_Product_unit @ X2 @ Xs ) )
= ( insert_Product_unit2 @ X2 @ ( set_Product_unit2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1040_empty__in__Func,axiom,
! [B2: set_Epistemic_fm_i] :
( ( B2 != bot_bo4194595901900360558c_fm_i )
=> ( member8097212860227194492c_fm_i
@ ^ [X: epistemic_fm_i] : undefi6158949293579472778c_fm_i
@ ( bNF_We489590214134045204c_fm_i @ bot_bo4194595901900360558c_fm_i @ B2 ) ) ) ).
% empty_in_Func
thf(fact_1041_empty__in__Func,axiom,
! [B2: set_Epistemic_fm_i] :
( ( B2 != bot_bo4194595901900360558c_fm_i )
=> ( member3099810079864775633c_fm_i
@ ^ [X: product_unit] : undefi6158949293579472778c_fm_i
@ ( bNF_We667984284430983401c_fm_i @ bot_bo3957492148770167129t_unit @ B2 ) ) ) ).
% empty_in_Func
thf(fact_1042_empty__in__Func,axiom,
! [B2: set_Product_unit] :
( ( B2 != bot_bo3957492148770167129t_unit )
=> ( member4534395898142662503t_unit
@ ^ [X: epistemic_fm_i] : undefi2974082611360736373t_unit
@ ( bNF_We5948249774593435647t_unit @ bot_bo4194595901900360558c_fm_i @ B2 ) ) ) ).
% empty_in_Func
thf(fact_1043_empty__in__Func,axiom,
! [B2: set_Product_unit] :
( ( B2 != bot_bo3957492148770167129t_unit )
=> ( member3685380118729357628t_unit
@ ^ [X: product_unit] : undefi2974082611360736373t_unit
@ ( bNF_We2086683926456529748t_unit @ bot_bo3957492148770167129t_unit @ B2 ) ) ) ).
% empty_in_Func
thf(fact_1044_Field__insert,axiom,
! [A2: epistemic_fm_i,B: epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ( field_Epistemic_fm_i @ ( insert347525590040402807c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ B ) @ R ) )
= ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) ) @ ( field_Epistemic_fm_i @ R ) ) ) ).
% Field_insert
thf(fact_1045_Field__insert,axiom,
! [A2: product_unit,B: product_unit,R: set_Pr5094982260447487303t_unit] :
( ( field_Product_unit @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R ) )
= ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ A2 @ ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R ) ) ) ).
% Field_insert
thf(fact_1046_Set_Ois__empty__def,axiom,
( is_emp7643256682846039796c_fm_i
= ( ^ [A3: set_Epistemic_fm_i] : ( A3 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% Set.is_empty_def
thf(fact_1047_Set_Ois__empty__def,axiom,
( is_emp5529367068699929823t_unit
= ( ^ [A3: set_Product_unit] : ( A3 = bot_bo3957492148770167129t_unit ) ) ) ).
% Set.is_empty_def
thf(fact_1048_wo__rel_Ofinite__underS__bound,axiom,
! [R: set_Pr3972831103112126087c_fm_i,X4: set_se3485332733965609186c_fm_i,N: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( finite7933139204641697195c_fm_i @ X4 )
=> ( ( ord_le5389487502678872194c_fm_i @ X4 @ ( order_8439613377573682120c_fm_i @ R @ N ) )
=> ( ( X4 != bot_bo145720340923748686c_fm_i )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ X4 )
& ! [Xa: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa @ X4 )
=> ( member1104366573291651755c_fm_i @ Xa @ ( order_4888947244107180775c_fm_i @ R @ X3 ) ) ) ) ) ) ) ) ).
% wo_rel.finite_underS_bound
thf(fact_1049_wo__rel_Ofinite__underS__bound,axiom,
! [R: set_Pr4658907567593863815c_fm_i,X4: set_Epistemic_fm_i,N: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( finite3304564979551393739c_fm_i @ X4 )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ ( order_1288088345848551912c_fm_i @ R @ N ) )
=> ( ( X4 != bot_bo4194595901900360558c_fm_i )
=> ? [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ X4 )
& ! [Xa: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa @ X4 )
=> ( member6642669606046002379c_fm_i @ Xa @ ( order_1815523504696313095c_fm_i @ R @ X3 ) ) ) ) ) ) ) ) ).
% wo_rel.finite_underS_bound
thf(fact_1050_wo__rel_Ofinite__underS__bound,axiom,
! [R: set_Pr5094982260447487303t_unit,X4: set_Product_unit,N: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( finite4290736615968046902t_unit @ X4 )
=> ( ( ord_le3507040750410214029t_unit @ X4 @ ( order_5809128134054296019t_unit @ R @ N ) )
=> ( ( X4 != bot_bo3957492148770167129t_unit )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ X4 )
& ! [Xa: product_unit] :
( ( member_Product_unit @ Xa @ X4 )
=> ( member_Product_unit @ Xa @ ( order_6949322460238004850t_unit @ R @ X3 ) ) ) ) ) ) ) ) ).
% wo_rel.finite_underS_bound
thf(fact_1051_under__def,axiom,
( order_4888947244107180775c_fm_i
= ( ^ [R4: set_Pr3972831103112126087c_fm_i,A4: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [B5: set_Epistemic_fm_i] : ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ B5 @ A4 ) @ R4 ) ) ) ) ).
% under_def
thf(fact_1052_under__def,axiom,
( order_1815523504696313095c_fm_i
= ( ^ [R4: set_Pr4658907567593863815c_fm_i,A4: epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [B5: epistemic_fm_i] : ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B5 @ A4 ) @ R4 ) ) ) ) ).
% under_def
thf(fact_1053_well__order__induct__imp,axiom,
! [R: set_Pr4658907567593863815c_fm_i,P3: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ! [X3: epistemic_fm_i] :
( ! [Y6: epistemic_fm_i] :
( ( ( Y6 != X3 )
& ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ Y6 @ X3 ) @ R ) )
=> ( ( member6642669606046002379c_fm_i @ Y6 @ ( field_Epistemic_fm_i @ R ) )
=> ( P3 @ Y6 ) ) )
=> ( ( member6642669606046002379c_fm_i @ X3 @ ( field_Epistemic_fm_i @ R ) )
=> ( P3 @ X3 ) ) )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( P3 @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1054_well__order__induct__imp,axiom,
! [R: set_Pr3972831103112126087c_fm_i,P3: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ! [X3: set_Epistemic_fm_i] :
( ! [Y6: set_Epistemic_fm_i] :
( ( ( Y6 != X3 )
& ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ Y6 @ X3 ) @ R ) )
=> ( ( member1104366573291651755c_fm_i @ Y6 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( P3 @ Y6 ) ) )
=> ( ( member1104366573291651755c_fm_i @ X3 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( P3 @ X3 ) ) )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( P3 @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1055_FieldI1,axiom,
! [I2: epistemic_fm_i,J: epistemic_fm_i,R2: set_Pr4658907567593863815c_fm_i] :
( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ I2 @ J ) @ R2 )
=> ( member6642669606046002379c_fm_i @ I2 @ ( field_Epistemic_fm_i @ R2 ) ) ) ).
% FieldI1
thf(fact_1056_FieldI1,axiom,
! [I2: set_Epistemic_fm_i,J: set_Epistemic_fm_i,R2: set_Pr3972831103112126087c_fm_i] :
( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ I2 @ J ) @ R2 )
=> ( member1104366573291651755c_fm_i @ I2 @ ( field_4031064140798694607c_fm_i @ R2 ) ) ) ).
% FieldI1
thf(fact_1057_FieldI2,axiom,
! [I2: epistemic_fm_i,J: epistemic_fm_i,R2: set_Pr4658907567593863815c_fm_i] :
( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ I2 @ J ) @ R2 )
=> ( member6642669606046002379c_fm_i @ J @ ( field_Epistemic_fm_i @ R2 ) ) ) ).
% FieldI2
thf(fact_1058_FieldI2,axiom,
! [I2: set_Epistemic_fm_i,J: set_Epistemic_fm_i,R2: set_Pr3972831103112126087c_fm_i] :
( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ I2 @ J ) @ R2 )
=> ( member1104366573291651755c_fm_i @ J @ ( field_4031064140798694607c_fm_i @ R2 ) ) ) ).
% FieldI2
thf(fact_1059_refl__onD,axiom,
! [A: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ A @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1060_refl__onD,axiom,
! [A: set_se3485332733965609186c_fm_i,R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ A @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A2 @ A2 ) @ R ) ) ) ).
% refl_onD
thf(fact_1061_refl__onD1,axiom,
! [A: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i,X2: epistemic_fm_i,Y4: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ A @ R )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ X2 @ Y4 ) @ R )
=> ( member6642669606046002379c_fm_i @ X2 @ A ) ) ) ).
% refl_onD1
thf(fact_1062_refl__onD1,axiom,
! [A: set_se3485332733965609186c_fm_i,R: set_Pr3972831103112126087c_fm_i,X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ A @ R )
=> ( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ X2 @ Y4 ) @ R )
=> ( member1104366573291651755c_fm_i @ X2 @ A ) ) ) ).
% refl_onD1
thf(fact_1063_refl__onD2,axiom,
! [A: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i,X2: epistemic_fm_i,Y4: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ A @ R )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ X2 @ Y4 ) @ R )
=> ( member6642669606046002379c_fm_i @ Y4 @ A ) ) ) ).
% refl_onD2
thf(fact_1064_refl__onD2,axiom,
! [A: set_se3485332733965609186c_fm_i,R: set_Pr3972831103112126087c_fm_i,X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ A @ R )
=> ( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ X2 @ Y4 ) @ R )
=> ( member1104366573291651755c_fm_i @ Y4 @ A ) ) ) ).
% refl_onD2
thf(fact_1065_underS__def,axiom,
( order_8439613377573682120c_fm_i
= ( ^ [R4: set_Pr3972831103112126087c_fm_i,A4: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [B5: set_Epistemic_fm_i] :
( ( B5 != A4 )
& ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ B5 @ A4 ) @ R4 ) ) ) ) ) ).
% underS_def
thf(fact_1066_underS__def,axiom,
( order_1288088345848551912c_fm_i
= ( ^ [R4: set_Pr4658907567593863815c_fm_i,A4: epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [B5: epistemic_fm_i] :
( ( B5 != A4 )
& ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B5 @ A4 ) @ R4 ) ) ) ) ) ).
% underS_def
thf(fact_1067_reflI,axiom,
! [R: set_Pr4658907567593863815c_fm_i] :
( ! [X3: epistemic_fm_i] : ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ X3 @ X3 ) @ R )
=> ( refl_o2748477423337579169c_fm_i @ top_to8364301187473967186c_fm_i @ R ) ) ).
% reflI
thf(fact_1068_reflI,axiom,
! [R: set_Pr5094982260447487303t_unit] :
( ! [X3: product_unit] : ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X3 @ X3 ) @ R )
=> ( refl_on_Product_unit @ top_to1996260823553986621t_unit @ R ) ) ).
% reflI
thf(fact_1069_reflD,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ top_to8364301187473967186c_fm_i @ R )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ A2 ) @ R ) ) ).
% reflD
thf(fact_1070_reflD,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( refl_on_Product_unit @ top_to1996260823553986621t_unit @ R )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ A2 ) @ R ) ) ).
% reflD
thf(fact_1071_wo__rel__r,axiom,
bNF_We8255954865191566094c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) ).
% wo_rel_r
thf(fact_1072_refl__on__singleton,axiom,
! [X2: epistemic_fm_i] : ( refl_o2748477423337579169c_fm_i @ ( insert7817948997695205106c_fm_i @ X2 @ bot_bo4194595901900360558c_fm_i ) @ ( insert347525590040402807c_fm_i @ ( produc4253568889897832087c_fm_i @ X2 @ X2 ) @ bot_bo3668964140382854387c_fm_i ) ) ).
% refl_on_singleton
thf(fact_1073_refl__on__singleton,axiom,
! [X2: product_unit] : ( refl_on_Product_unit @ ( insert_Product_unit2 @ X2 @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X2 @ X2 ) @ bot_bo1654022762305692595t_unit ) ) ).
% refl_on_singleton
thf(fact_1074_wo__rel_Ocases__Total,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i,B: epistemic_fm_i,Phi: epistemic_fm_i > epistemic_fm_i > $o] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) ) @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ B ) @ R )
=> ( Phi @ A2 @ B ) )
=> ( ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B @ A2 ) @ R )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1075_wo__rel_Ocases__Total,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,Phi: product_unit > product_unit > $o] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit2 @ A2 @ ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R ) )
=> ( ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R )
=> ( Phi @ A2 @ B ) )
=> ( ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ R )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_1076_wo__rel_Oin__notinI,axiom,
! [R: set_Pr4658907567593863815c_fm_i,J: epistemic_fm_i,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ~ ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ J @ I2 ) @ R )
| ( J = I2 ) )
=> ( ( member6642669606046002379c_fm_i @ I2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ J @ ( field_Epistemic_fm_i @ R ) )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ I2 @ J ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1077_wo__rel_Oin__notinI,axiom,
! [R: set_Pr3972831103112126087c_fm_i,J: set_Epistemic_fm_i,I2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ~ ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ J @ I2 ) @ R )
| ( J = I2 ) )
=> ( ( member1104366573291651755c_fm_i @ I2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ J @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ I2 @ J ) @ R ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_1078_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( B2 != bot_bo4194595901900360558c_fm_i )
=> ? [X_1: epistemic_fm_i] : ( bNF_We541765464805561304c_fm_i @ R @ B2 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_1079_wo__rel_OWell__order__isMinim__exists,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( B2 != bot_bo3957492148770167129t_unit )
=> ? [X_1: product_unit] : ( bNF_We909621271159000003t_unit @ R @ B2 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_1080_wo__rel_Ounder__zero,axiom,
! [R: set_Pr4658907567593863815c_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( field_Epistemic_fm_i @ R )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( order_1815523504696313095c_fm_i @ R @ ( wellor854379368678715862c_fm_i @ R ) )
= ( insert7817948997695205106c_fm_i @ ( wellor854379368678715862c_fm_i @ R ) @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% wo_rel.under_zero
thf(fact_1081_wo__rel_Ounder__zero,axiom,
! [R: set_Pr5094982260447487303t_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( field_Product_unit @ R )
!= bot_bo3957492148770167129t_unit )
=> ( ( order_6949322460238004850t_unit @ R @ ( wellor829852009790670017t_unit @ R ) )
= ( insert_Product_unit2 @ ( wellor829852009790670017t_unit @ R ) @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% wo_rel.under_zero
thf(fact_1082_insert__Chain,axiom,
! [R: set_Pr5094982260447487303t_unit,C2: set_Product_unit,I2: product_unit] :
( ( refl_on_Product_unit @ ( field_Product_unit @ R ) @ R )
=> ( ( member5877623283571906838t_unit @ C2 @ ( chains_Product_unit @ R ) )
=> ( ( member_Product_unit @ I2 @ ( field_Product_unit @ R ) )
=> ( ! [J2: product_unit] :
( ( member_Product_unit @ J2 @ C2 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ J2 @ I2 ) @ R )
| ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J2 ) @ R ) ) )
=> ( member5877623283571906838t_unit @ ( insert_Product_unit2 @ I2 @ C2 ) @ ( chains_Product_unit @ R ) ) ) ) ) ) ).
% insert_Chain
thf(fact_1083_insert__Chain,axiom,
! [R: set_Pr3972831103112126087c_fm_i,C2: set_se3485332733965609186c_fm_i,I2: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ ( field_4031064140798694607c_fm_i @ R ) @ R )
=> ( ( member1461078328125707403c_fm_i @ C2 @ ( chains7915783162359121842c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ I2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ! [J2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ J2 @ C2 )
=> ( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ J2 @ I2 ) @ R )
| ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ I2 @ J2 ) @ R ) ) )
=> ( member1461078328125707403c_fm_i @ ( insert7698009978809854162c_fm_i @ I2 @ C2 ) @ ( chains7915783162359121842c_fm_i @ R ) ) ) ) ) ) ).
% insert_Chain
thf(fact_1084_insert__Chain,axiom,
! [R: set_Pr4658907567593863815c_fm_i,C2: set_Epistemic_fm_i,I2: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ ( field_Epistemic_fm_i @ R ) @ R )
=> ( ( member1104366573291651755c_fm_i @ C2 @ ( chains2242491043985051090c_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ I2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ! [J2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ J2 @ C2 )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ J2 @ I2 ) @ R )
| ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ I2 @ J2 ) @ R ) ) )
=> ( member1104366573291651755c_fm_i @ ( insert7817948997695205106c_fm_i @ I2 @ C2 ) @ ( chains2242491043985051090c_fm_i @ R ) ) ) ) ) ) ).
% insert_Chain
thf(fact_1085_wo__rel_Ozero__smallest,axiom,
! [R: set_Pr4658907567593863815c_fm_i,J: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ J @ ( field_Epistemic_fm_i @ R ) )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ ( wellor854379368678715862c_fm_i @ R ) @ J ) @ R ) ) ) ).
% wo_rel.zero_smallest
thf(fact_1086_wo__rel_Ozero__smallest,axiom,
! [R: set_Pr3972831103112126087c_fm_i,J: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ J @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ ( wellor6421275854403804086c_fm_i @ R ) @ J ) @ R ) ) ) ).
% wo_rel.zero_smallest
thf(fact_1087_wo__rel_Ozero__in__Field,axiom,
! [R: set_Pr3972831103112126087c_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ( field_4031064140798694607c_fm_i @ R )
!= bot_bo145720340923748686c_fm_i )
=> ( member1104366573291651755c_fm_i @ ( wellor6421275854403804086c_fm_i @ R ) @ ( field_4031064140798694607c_fm_i @ R ) ) ) ) ).
% wo_rel.zero_in_Field
thf(fact_1088_wo__rel_Ozero__in__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( field_Epistemic_fm_i @ R )
!= bot_bo4194595901900360558c_fm_i )
=> ( member6642669606046002379c_fm_i @ ( wellor854379368678715862c_fm_i @ R ) @ ( field_Epistemic_fm_i @ R ) ) ) ) ).
% wo_rel.zero_in_Field
thf(fact_1089_wo__rel_Ozero__in__Field,axiom,
! [R: set_Pr5094982260447487303t_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( field_Product_unit @ R )
!= bot_bo3957492148770167129t_unit )
=> ( member_Product_unit @ ( wellor829852009790670017t_unit @ R ) @ ( field_Product_unit @ R ) ) ) ) ).
% wo_rel.zero_in_Field
thf(fact_1090_wo__rel_OunderS__zero,axiom,
! [R: set_Pr4658907567593863815c_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( order_1288088345848551912c_fm_i @ R @ ( wellor854379368678715862c_fm_i @ R ) )
= bot_bo4194595901900360558c_fm_i ) ) ).
% wo_rel.underS_zero
thf(fact_1091_wo__rel_OunderS__zero,axiom,
! [R: set_Pr5094982260447487303t_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( order_5809128134054296019t_unit @ R @ ( wellor829852009790670017t_unit @ R ) )
= bot_bo3957492148770167129t_unit ) ) ).
% wo_rel.underS_zero
thf(fact_1092_wo__rel_Oleq__zero,axiom,
! [R: set_Pr4658907567593863815c_fm_i,X2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( field_Epistemic_fm_i @ R )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ X2 @ ( wellor854379368678715862c_fm_i @ R ) ) @ R )
= ( X2
= ( wellor854379368678715862c_fm_i @ R ) ) ) ) ) ).
% wo_rel.leq_zero
thf(fact_1093_wo__rel_Oleq__zero,axiom,
! [R: set_Pr5094982260447487303t_unit,X2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( field_Product_unit @ R )
!= bot_bo3957492148770167129t_unit )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X2 @ ( wellor829852009790670017t_unit @ R ) ) @ R )
= ( X2
= ( wellor829852009790670017t_unit @ R ) ) ) ) ) ).
% wo_rel.leq_zero
thf(fact_1094_wo__rel_Ominim__isMinim,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( B2 != bot_bo4194595901900360558c_fm_i )
=> ( bNF_We541765464805561304c_fm_i @ R @ B2 @ ( bNF_We6247850573693329646c_fm_i @ R @ B2 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_1095_wo__rel_Ominim__isMinim,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( B2 != bot_bo3957492148770167129t_unit )
=> ( bNF_We909621271159000003t_unit @ R @ B2 @ ( bNF_We7185707985517244377t_unit @ R @ B2 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_1096_wo__rel_Ominim__least,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ B @ B2 )
=> ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ ( bNF_We5992065776426773198c_fm_i @ R @ B2 ) @ B ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1097_wo__rel_Ominim__least,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i,B: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ B @ B2 )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ ( bNF_We6247850573693329646c_fm_i @ R @ B2 ) @ B ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1098_wo__rel_Ominim__least,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit,B: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( member_Product_unit @ B @ B2 )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ ( bNF_We7185707985517244377t_unit @ R @ B2 ) @ B ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_1099_wo__rel_Oequals__minim,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ A2 @ B2 )
=> ( ! [B7: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B7 @ B2 )
=> ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A2 @ B7 ) @ R ) )
=> ( A2
= ( bNF_We5992065776426773198c_fm_i @ R @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1100_wo__rel_Oequals__minim,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ A2 @ B2 )
=> ( ! [B7: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B7 @ B2 )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ B7 ) @ R ) )
=> ( A2
= ( bNF_We6247850573693329646c_fm_i @ R @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1101_wo__rel_Oequals__minim,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit,A2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( member_Product_unit @ A2 @ B2 )
=> ( ! [B7: product_unit] :
( ( member_Product_unit @ B7 @ B2 )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B7 ) @ R ) )
=> ( A2
= ( bNF_We7185707985517244377t_unit @ R @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1102_wo__rel_Ominim__in,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( B2 != bot_bo145720340923748686c_fm_i )
=> ( member1104366573291651755c_fm_i @ ( bNF_We5992065776426773198c_fm_i @ R @ B2 ) @ B2 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1103_wo__rel_Ominim__in,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( B2 != bot_bo4194595901900360558c_fm_i )
=> ( member6642669606046002379c_fm_i @ ( bNF_We6247850573693329646c_fm_i @ R @ B2 ) @ B2 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1104_wo__rel_Ominim__in,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( B2 != bot_bo3957492148770167129t_unit )
=> ( member_Product_unit @ ( bNF_We7185707985517244377t_unit @ R @ B2 ) @ B2 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_1105_wo__rel_Ominim__inField,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( B2 != bot_bo145720340923748686c_fm_i )
=> ( member1104366573291651755c_fm_i @ ( bNF_We5992065776426773198c_fm_i @ R @ B2 ) @ ( field_4031064140798694607c_fm_i @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1106_wo__rel_Ominim__inField,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( B2 != bot_bo4194595901900360558c_fm_i )
=> ( member6642669606046002379c_fm_i @ ( bNF_We6247850573693329646c_fm_i @ R @ B2 ) @ ( field_Epistemic_fm_i @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1107_wo__rel_Ominim__inField,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( B2 != bot_bo3957492148770167129t_unit )
=> ( member_Product_unit @ ( bNF_We7185707985517244377t_unit @ R @ B2 ) @ ( field_Product_unit @ R ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_1108_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ B @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A2 @ ( bNF_We796651220403541840c_fm_i @ R @ A2 @ B ) ) @ R )
& ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ B @ ( bNF_We796651220403541840c_fm_i @ R @ A2 @ B ) ) @ R )
& ( member1104366573291651755c_fm_i @ ( bNF_We796651220403541840c_fm_i @ R @ A2 @ B ) @ ( insert7698009978809854162c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1109_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i,B: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ B @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ ( bNF_We2513106701737039216c_fm_i @ R @ A2 @ B ) ) @ R )
& ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B @ ( bNF_We2513106701737039216c_fm_i @ R @ A2 @ B ) ) @ R )
& ( member6642669606046002379c_fm_i @ ( bNF_We2513106701737039216c_fm_i @ R @ A2 @ B ) @ ( insert7817948997695205106c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1110_wo__rel_Omax2__greater__among,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ( member_Product_unit @ B @ ( field_Product_unit @ R ) )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ ( bNF_We7204996500647184731t_unit @ R @ A2 @ B ) ) @ R )
& ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ ( bNF_We7204996500647184731t_unit @ R @ A2 @ B ) ) @ R )
& ( member_Product_unit @ ( bNF_We7204996500647184731t_unit @ R @ A2 @ B ) @ ( insert_Product_unit2 @ A2 @ ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1111_Chains__subset_H,axiom,
! [R: set_Pr4658907567593863815c_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ top_to8364301187473967186c_fm_i @ R )
=> ( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ( pred_c1742013745311679565c_fm_i @ top_to8364301187473967186c_fm_i
@ ^ [X: epistemic_fm_i,Y5: epistemic_fm_i] : ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ X @ Y5 ) @ R ) ) )
@ ( chains2242491043985051090c_fm_i @ R ) ) ) ).
% Chains_subset'
thf(fact_1112_Chains__subset_H,axiom,
! [R: set_Pr5094982260447487303t_unit] :
( ( refl_on_Product_unit @ top_to1996260823553986621t_unit @ R )
=> ( ord_le3674001228145610605t_unit
@ ( collec7787489603430924120t_unit
@ ( pred_c6915491785556283576t_unit @ top_to1996260823553986621t_unit
@ ^ [X: product_unit,Y5: product_unit] : ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y5 ) @ R ) ) )
@ ( chains_Product_unit @ R ) ) ) ).
% Chains_subset'
thf(fact_1113_wo__rel_OunderS__suc,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ B @ ( order_8439613377573682120c_fm_i @ R @ ( bNF_We1910513081875451107c_fm_i @ R @ A ) ) )
=> ( ( ord_le5389487502678872194c_fm_i @ A @ ( field_4031064140798694607c_fm_i @ R ) )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( member1104366573291651755c_fm_i @ B @ ( order_4888947244107180775c_fm_i @ R @ X3 ) ) ) ) ) ) ).
% wo_rel.underS_suc
thf(fact_1114_wo__rel_OunderS__suc,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ B @ ( order_1288088345848551912c_fm_i @ R @ ( bNF_We6750397660357449475c_fm_i @ R @ A ) ) )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ ( field_Epistemic_fm_i @ R ) )
=> ? [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
& ( member6642669606046002379c_fm_i @ B @ ( order_1815523504696313095c_fm_i @ R @ X3 ) ) ) ) ) ) ).
% wo_rel.underS_suc
thf(fact_1115_wo__rel_OunderS__suc,axiom,
! [R: set_Pr5094982260447487303t_unit,B: product_unit,A: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( member_Product_unit @ B @ ( order_5809128134054296019t_unit @ R @ ( bNF_We4101066952194689646t_unit @ R @ A ) ) )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( field_Product_unit @ R ) )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
& ( member_Product_unit @ B @ ( order_6949322460238004850t_unit @ R @ X3 ) ) ) ) ) ) ).
% wo_rel.underS_suc
thf(fact_1116_pred__on_Ochain__empty,axiom,
! [A: set_Epistemic_fm_i,P3: epistemic_fm_i > epistemic_fm_i > $o] : ( pred_c1742013745311679565c_fm_i @ A @ P3 @ bot_bo4194595901900360558c_fm_i ) ).
% pred_on.chain_empty
thf(fact_1117_pred__on_Ochain__empty,axiom,
! [A: set_Product_unit,P3: product_unit > product_unit > $o] : ( pred_c6915491785556283576t_unit @ A @ P3 @ bot_bo3957492148770167129t_unit ) ).
% pred_on.chain_empty
thf(fact_1118_pred__on_Ochain__total,axiom,
! [A: set_Epistemic_fm_i,P3: epistemic_fm_i > epistemic_fm_i > $o,C2: set_Epistemic_fm_i,X2: epistemic_fm_i,Y4: epistemic_fm_i] :
( ( pred_c1742013745311679565c_fm_i @ A @ P3 @ C2 )
=> ( ( member6642669606046002379c_fm_i @ X2 @ C2 )
=> ( ( member6642669606046002379c_fm_i @ Y4 @ C2 )
=> ( ( sup_su4282625464227558722fm_i_o @ P3
@ ^ [Y: epistemic_fm_i,Z: epistemic_fm_i] : ( Y = Z )
@ X2
@ Y4 )
| ( sup_su4282625464227558722fm_i_o @ P3
@ ^ [Y: epistemic_fm_i,Z: epistemic_fm_i] : ( Y = Z )
@ Y4
@ X2 ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_1119_pred__on_Ochain__total,axiom,
! [A: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > set_Epistemic_fm_i > $o,C2: set_se3485332733965609186c_fm_i,X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( pred_c4586345430894039597c_fm_i @ A @ P3 @ C2 )
=> ( ( member1104366573291651755c_fm_i @ X2 @ C2 )
=> ( ( member1104366573291651755c_fm_i @ Y4 @ C2 )
=> ( ( sup_su7764009942809480514fm_i_o @ P3
@ ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z )
@ X2
@ Y4 )
| ( sup_su7764009942809480514fm_i_o @ P3
@ ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z )
@ Y4
@ X2 ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_1120_pred__on_Ochain__def,axiom,
( pred_c1742013745311679565c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,P6: epistemic_fm_i > epistemic_fm_i > $o,C3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C3 @ A3 )
& ! [X: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X @ C3 )
=> ! [Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y5 @ C3 )
=> ( ( sup_su4282625464227558722fm_i_o @ P6
@ ^ [Y: epistemic_fm_i,Z: epistemic_fm_i] : ( Y = Z )
@ X
@ Y5 )
| ( sup_su4282625464227558722fm_i_o @ P6
@ ^ [Y: epistemic_fm_i,Z: epistemic_fm_i] : ( Y = Z )
@ Y5
@ X ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_1121_pred__on_Ochain__def,axiom,
( pred_c6915491785556283576t_unit
= ( ^ [A3: set_Product_unit,P6: product_unit > product_unit > $o,C3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ C3 @ A3 )
& ! [X: product_unit] :
( ( member_Product_unit @ X @ C3 )
=> ! [Y5: product_unit] :
( ( member_Product_unit @ Y5 @ C3 )
=> ( ( sup_su3411595859437804162unit_o @ P6
@ ^ [Y: product_unit,Z: product_unit] : ( Y = Z )
@ X
@ Y5 )
| ( sup_su3411595859437804162unit_o @ P6
@ ^ [Y: product_unit,Z: product_unit] : ( Y = Z )
@ Y5
@ X ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_1122_pred__on_OchainI,axiom,
! [C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ C2 @ A )
=> ( ! [X3: set_Epistemic_fm_i,Y3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ C2 )
=> ( ( member1104366573291651755c_fm_i @ Y3 @ C2 )
=> ( ( sup_su7764009942809480514fm_i_o @ P3
@ ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z )
@ X3
@ Y3 )
| ( sup_su7764009942809480514fm_i_o @ P3
@ ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z )
@ Y3
@ X3 ) ) ) )
=> ( pred_c4586345430894039597c_fm_i @ A @ P3 @ C2 ) ) ) ).
% pred_on.chainI
thf(fact_1123_pred__on_OchainI,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P3: epistemic_fm_i > epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ A )
=> ( ! [X3: epistemic_fm_i,Y3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ C2 )
=> ( ( member6642669606046002379c_fm_i @ Y3 @ C2 )
=> ( ( sup_su4282625464227558722fm_i_o @ P3
@ ^ [Y: epistemic_fm_i,Z: epistemic_fm_i] : ( Y = Z )
@ X3
@ Y3 )
| ( sup_su4282625464227558722fm_i_o @ P3
@ ^ [Y: epistemic_fm_i,Z: epistemic_fm_i] : ( Y = Z )
@ Y3
@ X3 ) ) ) )
=> ( pred_c1742013745311679565c_fm_i @ A @ P3 @ C2 ) ) ) ).
% pred_on.chainI
thf(fact_1124_pred__on_OchainI,axiom,
! [C2: set_Product_unit,A: set_Product_unit,P3: product_unit > product_unit > $o] :
( ( ord_le3507040750410214029t_unit @ C2 @ A )
=> ( ! [X3: product_unit,Y3: product_unit] :
( ( member_Product_unit @ X3 @ C2 )
=> ( ( member_Product_unit @ Y3 @ C2 )
=> ( ( sup_su3411595859437804162unit_o @ P3
@ ^ [Y: product_unit,Z: product_unit] : ( Y = Z )
@ X3
@ Y3 )
| ( sup_su3411595859437804162unit_o @ P3
@ ^ [Y: product_unit,Z: product_unit] : ( Y = Z )
@ Y3
@ X3 ) ) ) )
=> ( pred_c6915491785556283576t_unit @ A @ P3 @ C2 ) ) ) ).
% pred_on.chainI
thf(fact_1125_BNF__Wellorder__Relation_Owo__rel_Osuc__underS,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( A2
= ( bNF_We6750397660357449475c_fm_i @ R @ ( order_1288088345848551912c_fm_i @ R @ A2 ) ) ) ) ) ).
% BNF_Wellorder_Relation.wo_rel.suc_underS
thf(fact_1126_BNF__Wellorder__Relation_Owo__rel_Osuc__underS,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( A2
= ( bNF_We1910513081875451107c_fm_i @ R @ ( order_8439613377573682120c_fm_i @ R @ A2 ) ) ) ) ) ).
% BNF_Wellorder_Relation.wo_rel.suc_underS
thf(fact_1127_wo__rel_Omax2__greater,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i,B: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ B @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ ( bNF_We2513106701737039216c_fm_i @ R @ A2 @ B ) ) @ R )
& ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B @ ( bNF_We2513106701737039216c_fm_i @ R @ A2 @ B ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1128_wo__rel_Omax2__greater,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ B @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A2 @ ( bNF_We796651220403541840c_fm_i @ R @ A2 @ B ) ) @ R )
& ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ B @ ( bNF_We796651220403541840c_fm_i @ R @ A2 @ B ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_1129_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i,B: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ B @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ( bNF_We2513106701737039216c_fm_i @ R @ A2 @ B )
= B )
= ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ B ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1130_wo__rel_Omax2__equals2,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ B @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( ( bNF_We796651220403541840c_fm_i @ R @ A2 @ B )
= B )
= ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A2 @ B ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_1131_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i,B: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ B @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ( bNF_We2513106701737039216c_fm_i @ R @ A2 @ B )
= A2 )
= ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1132_wo__rel_Omax2__equals1,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ B @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( ( bNF_We796651220403541840c_fm_i @ R @ A2 @ B )
= A2 )
= ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ B @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_1133_pred__on_Ochain__extend,axiom,
! [A: set_se3485332733965609186c_fm_i,P3: set_Epistemic_fm_i > set_Epistemic_fm_i > $o,C2: set_se3485332733965609186c_fm_i,Z2: set_Epistemic_fm_i] :
( ( pred_c4586345430894039597c_fm_i @ A @ P3 @ C2 )
=> ( ( member1104366573291651755c_fm_i @ Z2 @ A )
=> ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ C2 )
=> ( sup_su7764009942809480514fm_i_o @ P3
@ ^ [Y: set_Epistemic_fm_i,Z: set_Epistemic_fm_i] : ( Y = Z )
@ X3
@ Z2 ) )
=> ( pred_c4586345430894039597c_fm_i @ A @ P3 @ ( sup_su2582925890723967158c_fm_i @ ( insert7698009978809854162c_fm_i @ Z2 @ bot_bo145720340923748686c_fm_i ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1134_pred__on_Ochain__extend,axiom,
! [A: set_Epistemic_fm_i,P3: epistemic_fm_i > epistemic_fm_i > $o,C2: set_Epistemic_fm_i,Z2: epistemic_fm_i] :
( ( pred_c1742013745311679565c_fm_i @ A @ P3 @ C2 )
=> ( ( member6642669606046002379c_fm_i @ Z2 @ A )
=> ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ C2 )
=> ( sup_su4282625464227558722fm_i_o @ P3
@ ^ [Y: epistemic_fm_i,Z: epistemic_fm_i] : ( Y = Z )
@ X3
@ Z2 ) )
=> ( pred_c1742013745311679565c_fm_i @ A @ P3 @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Z2 @ bot_bo4194595901900360558c_fm_i ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1135_pred__on_Ochain__extend,axiom,
! [A: set_Product_unit,P3: product_unit > product_unit > $o,C2: set_Product_unit,Z2: product_unit] :
( ( pred_c6915491785556283576t_unit @ A @ P3 @ C2 )
=> ( ( member_Product_unit @ Z2 @ A )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ C2 )
=> ( sup_su3411595859437804162unit_o @ P3
@ ^ [Y: product_unit,Z: product_unit] : ( Y = Z )
@ X3
@ Z2 ) )
=> ( pred_c6915491785556283576t_unit @ A @ P3 @ ( sup_su793286257634532545t_unit @ ( insert_Product_unit2 @ Z2 @ bot_bo3957492148770167129t_unit ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1136_Chains__alt__def,axiom,
! [R: set_Pr4658907567593863815c_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ top_to8364301187473967186c_fm_i @ R )
=> ( ( chains2242491043985051090c_fm_i @ R )
= ( collec3087743281813070829c_fm_i
@ ( pred_c1742013745311679565c_fm_i @ top_to8364301187473967186c_fm_i
@ ^ [X: epistemic_fm_i,Y5: epistemic_fm_i] : ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ X @ Y5 ) @ R ) ) ) ) ) ).
% Chains_alt_def
thf(fact_1137_Chains__alt__def,axiom,
! [R: set_Pr5094982260447487303t_unit] :
( ( refl_on_Product_unit @ top_to1996260823553986621t_unit @ R )
=> ( ( chains_Product_unit @ R )
= ( collec7787489603430924120t_unit
@ ( pred_c6915491785556283576t_unit @ top_to1996260823553986621t_unit
@ ^ [X: product_unit,Y5: product_unit] : ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y5 ) @ R ) ) ) ) ) ).
% Chains_alt_def
thf(fact_1138_Chains__subset,axiom,
! [R: set_Pr4658907567593863815c_fm_i] :
( ord_le5389487502678872194c_fm_i @ ( chains2242491043985051090c_fm_i @ R )
@ ( collec3087743281813070829c_fm_i
@ ( pred_c1742013745311679565c_fm_i @ top_to8364301187473967186c_fm_i
@ ^ [X: epistemic_fm_i,Y5: epistemic_fm_i] : ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ X @ Y5 ) @ R ) ) ) ) ).
% Chains_subset
thf(fact_1139_Chains__subset,axiom,
! [R: set_Pr5094982260447487303t_unit] :
( ord_le3674001228145610605t_unit @ ( chains_Product_unit @ R )
@ ( collec7787489603430924120t_unit
@ ( pred_c6915491785556283576t_unit @ top_to1996260823553986621t_unit
@ ^ [X: product_unit,Y5: product_unit] : ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y5 ) @ R ) ) ) ) ).
% Chains_subset
thf(fact_1140_wo__rel_Omax2__among,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ B @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( member1104366573291651755c_fm_i @ ( bNF_We796651220403541840c_fm_i @ R @ A2 @ B ) @ ( insert7698009978809854162c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_1141_wo__rel_Omax2__among,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i,B: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ B @ ( field_Epistemic_fm_i @ R ) )
=> ( member6642669606046002379c_fm_i @ ( bNF_We2513106701737039216c_fm_i @ R @ A2 @ B ) @ ( insert7817948997695205106c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_1142_wo__rel_Omax2__among,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ( member_Product_unit @ B @ ( field_Product_unit @ R ) )
=> ( member_Product_unit @ ( bNF_We7204996500647184731t_unit @ R @ A2 @ B ) @ ( insert_Product_unit2 @ A2 @ ( insert_Product_unit2 @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_1143_wo__rel_Oequals__suc,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ! [B7: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B7 @ B2 )
=> ( ( A2 != B7 )
& ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ B7 @ A2 ) @ R ) ) )
=> ( ! [A9: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A9 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ! [B8: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B8 @ B2 )
=> ( ( A9 != B8 )
& ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ B8 @ A9 ) @ R ) ) )
=> ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A2 @ A9 ) @ R ) ) )
=> ( A2
= ( bNF_We1910513081875451107c_fm_i @ R @ B2 ) ) ) ) ) ) ) ).
% wo_rel.equals_suc
thf(fact_1144_wo__rel_Oequals__suc,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ! [B7: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B7 @ B2 )
=> ( ( A2 != B7 )
& ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B7 @ A2 ) @ R ) ) )
=> ( ! [A9: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A9 @ ( field_Epistemic_fm_i @ R ) )
=> ( ! [B8: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B8 @ B2 )
=> ( ( A9 != B8 )
& ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B8 @ A9 ) @ R ) ) )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ A9 ) @ R ) ) )
=> ( A2
= ( bNF_We6750397660357449475c_fm_i @ R @ B2 ) ) ) ) ) ) ) ).
% wo_rel.equals_suc
thf(fact_1145_wo__rel_Oequals__suc,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit,A2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ! [B7: product_unit] :
( ( member_Product_unit @ B7 @ B2 )
=> ( ( A2 != B7 )
& ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B7 @ A2 ) @ R ) ) )
=> ( ! [A9: product_unit] :
( ( member_Product_unit @ A9 @ ( field_Product_unit @ R ) )
=> ( ! [B8: product_unit] :
( ( member_Product_unit @ B8 @ B2 )
=> ( ( A9 != B8 )
& ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B8 @ A9 ) @ R ) ) )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ A9 ) @ R ) ) )
=> ( A2
= ( bNF_We4101066952194689646t_unit @ R @ B2 ) ) ) ) ) ) ) ).
% wo_rel.equals_suc
thf(fact_1146_wo__rel_Osuc__least,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ! [B7: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B7 @ B2 )
=> ( ( A2 != B7 )
& ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ B7 @ A2 ) @ R ) ) )
=> ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ ( bNF_We1910513081875451107c_fm_i @ R @ B2 ) @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.suc_least
thf(fact_1147_wo__rel_Osuc__least,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ! [B7: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B7 @ B2 )
=> ( ( A2 != B7 )
& ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B7 @ A2 ) @ R ) ) )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ ( bNF_We6750397660357449475c_fm_i @ R @ B2 ) @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.suc_least
thf(fact_1148_wo__rel_Osuc__least,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit,A2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ! [B7: product_unit] :
( ( member_Product_unit @ B7 @ B2 )
=> ( ( A2 != B7 )
& ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B7 @ A2 ) @ R ) ) )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ ( bNF_We4101066952194689646t_unit @ R @ B2 ) @ A2 ) @ R ) ) ) ) ) ).
% wo_rel.suc_least
thf(fact_1149_wo__rel_Omax2__iff,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i,B: epistemic_fm_i,C: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ B @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ ( bNF_We2513106701737039216c_fm_i @ R @ A2 @ B ) @ C ) @ R )
= ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ C ) @ R )
& ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B @ C ) @ R ) ) ) ) ) ) ).
% wo_rel.max2_iff
thf(fact_1150_wo__rel_Omax2__iff,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ B @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ ( bNF_We796651220403541840c_fm_i @ R @ A2 @ B ) @ C ) @ R )
= ( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A2 @ C ) @ R )
& ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ B @ C ) @ R ) ) ) ) ) ) ).
% wo_rel.max2_iff
thf(fact_1151_wo__rel_Ominim__UnderS__underS,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A: set_Epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ ( field_Epistemic_fm_i @ R ) )
=> ( ( order_2700047633941854664c_fm_i @ R @ A )
= ( order_1288088345848551912c_fm_i @ R @ ( bNF_We6247850573693329646c_fm_i @ R @ A ) ) ) ) ) ) ).
% wo_rel.minim_UnderS_underS
thf(fact_1152_wo__rel_Ominim__UnderS__underS,axiom,
! [R: set_Pr5094982260447487303t_unit,A: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( field_Product_unit @ R ) )
=> ( ( order_1874045165604686259t_unit @ R @ A )
= ( order_5809128134054296019t_unit @ R @ ( bNF_We7185707985517244377t_unit @ R @ A ) ) ) ) ) ) ).
% wo_rel.minim_UnderS_underS
thf(fact_1153_wo__rel_Ominim__Under__under,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A: set_Epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ ( field_Epistemic_fm_i @ R ) )
=> ( ( order_2859517305408744743c_fm_i @ R @ A )
= ( order_1815523504696313095c_fm_i @ R @ ( bNF_We6247850573693329646c_fm_i @ R @ A ) ) ) ) ) ) ).
% wo_rel.minim_Under_under
thf(fact_1154_wo__rel_Ominim__Under__under,axiom,
! [R: set_Pr5094982260447487303t_unit,A: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( field_Product_unit @ R ) )
=> ( ( order_1637587770817919634t_unit @ R @ A )
= ( order_6949322460238004850t_unit @ R @ ( bNF_We7185707985517244377t_unit @ R @ A ) ) ) ) ) ) ).
% wo_rel.minim_Under_under
thf(fact_1155_Under__decr,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ ( order_2859517305408744743c_fm_i @ R @ B2 ) @ ( order_2859517305408744743c_fm_i @ R @ A ) ) ) ).
% Under_decr
thf(fact_1156_Under__decr,axiom,
! [A: set_Product_unit,B2: set_Product_unit,R: set_Pr5094982260447487303t_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ord_le3507040750410214029t_unit @ ( order_1637587770817919634t_unit @ R @ B2 ) @ ( order_1637587770817919634t_unit @ R @ A ) ) ) ).
% Under_decr
thf(fact_1157_UnderS__subset__Under,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_2700047633941854664c_fm_i @ R @ A ) @ ( order_2859517305408744743c_fm_i @ R @ A ) ) ).
% UnderS_subset_Under
thf(fact_1158_UnderS__subset__Under,axiom,
! [R: set_Pr5094982260447487303t_unit,A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ ( order_1874045165604686259t_unit @ R @ A ) @ ( order_1637587770817919634t_unit @ R @ A ) ) ).
% UnderS_subset_Under
thf(fact_1159_UnderS__decr,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ ( order_2700047633941854664c_fm_i @ R @ B2 ) @ ( order_2700047633941854664c_fm_i @ R @ A ) ) ) ).
% UnderS_decr
thf(fact_1160_UnderS__decr,axiom,
! [A: set_Product_unit,B2: set_Product_unit,R: set_Pr5094982260447487303t_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ord_le3507040750410214029t_unit @ ( order_1874045165604686259t_unit @ R @ B2 ) @ ( order_1874045165604686259t_unit @ R @ A ) ) ) ).
% UnderS_decr
thf(fact_1161_Under__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_2859517305408744743c_fm_i @ R @ A ) @ ( field_Epistemic_fm_i @ R ) ) ).
% Under_Field
thf(fact_1162_Under__Field,axiom,
! [R: set_Pr5094982260447487303t_unit,A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ ( order_1637587770817919634t_unit @ R @ A ) @ ( field_Product_unit @ R ) ) ).
% Under_Field
thf(fact_1163_UnderS__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_2700047633941854664c_fm_i @ R @ A ) @ ( field_Epistemic_fm_i @ R ) ) ).
% UnderS_Field
thf(fact_1164_UnderS__Field,axiom,
! [R: set_Pr5094982260447487303t_unit,A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ ( order_1874045165604686259t_unit @ R @ A ) @ ( field_Product_unit @ R ) ) ).
% UnderS_Field
thf(fact_1165_under__Under__singl,axiom,
( order_1815523504696313095c_fm_i
= ( ^ [R4: set_Pr4658907567593863815c_fm_i,A4: epistemic_fm_i] : ( order_2859517305408744743c_fm_i @ R4 @ ( insert7817948997695205106c_fm_i @ A4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% under_Under_singl
thf(fact_1166_under__Under__singl,axiom,
( order_6949322460238004850t_unit
= ( ^ [R4: set_Pr5094982260447487303t_unit,A4: product_unit] : ( order_1637587770817919634t_unit @ R4 @ ( insert_Product_unit2 @ A4 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% under_Under_singl
thf(fact_1167_underS__UnderS__singl,axiom,
( order_1288088345848551912c_fm_i
= ( ^ [R4: set_Pr4658907567593863815c_fm_i,A4: epistemic_fm_i] : ( order_2700047633941854664c_fm_i @ R4 @ ( insert7817948997695205106c_fm_i @ A4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% underS_UnderS_singl
thf(fact_1168_underS__UnderS__singl,axiom,
( order_5809128134054296019t_unit
= ( ^ [R4: set_Pr5094982260447487303t_unit,A4: product_unit] : ( order_1874045165604686259t_unit @ R4 @ ( insert_Product_unit2 @ A4 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% underS_UnderS_singl
thf(fact_1169_wo__rel_OTrans__Under__regressive,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A: set_Epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ ( field_Epistemic_fm_i @ R ) )
=> ( ord_le3843937902494030498c_fm_i @ ( order_2859517305408744743c_fm_i @ R @ ( order_2859517305408744743c_fm_i @ R @ A ) ) @ ( order_2859517305408744743c_fm_i @ R @ A ) ) ) ) ) ).
% wo_rel.Trans_Under_regressive
thf(fact_1170_wo__rel_OTrans__Under__regressive,axiom,
! [R: set_Pr5094982260447487303t_unit,A: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( field_Product_unit @ R ) )
=> ( ord_le3507040750410214029t_unit @ ( order_1637587770817919634t_unit @ R @ ( order_1637587770817919634t_unit @ R @ A ) ) @ ( order_1637587770817919634t_unit @ R @ A ) ) ) ) ) ).
% wo_rel.Trans_Under_regressive
thf(fact_1171_wo__rel_Oequals__minim__Under,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ A2 @ B2 )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( order_3026938356532400903c_fm_i @ R @ B2 ) )
=> ( A2
= ( bNF_We5992065776426773198c_fm_i @ R @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim_Under
thf(fact_1172_wo__rel_Oequals__minim__Under,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ A2 @ B2 )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( order_2859517305408744743c_fm_i @ R @ B2 ) )
=> ( A2
= ( bNF_We6247850573693329646c_fm_i @ R @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim_Under
thf(fact_1173_wo__rel_Oequals__minim__Under,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit,A2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( member_Product_unit @ A2 @ B2 )
=> ( ( member_Product_unit @ A2 @ ( order_1637587770817919634t_unit @ R @ B2 ) )
=> ( A2
= ( bNF_We7185707985517244377t_unit @ R @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim_Under
thf(fact_1174_wo__rel_Ominim__iff__In__Under,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( B2 != bot_bo145720340923748686c_fm_i )
=> ( ( A2
= ( bNF_We5992065776426773198c_fm_i @ R @ B2 ) )
= ( ( member1104366573291651755c_fm_i @ A2 @ B2 )
& ( member1104366573291651755c_fm_i @ A2 @ ( order_3026938356532400903c_fm_i @ R @ B2 ) ) ) ) ) ) ) ).
% wo_rel.minim_iff_In_Under
thf(fact_1175_wo__rel_Ominim__iff__In__Under,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( B2 != bot_bo4194595901900360558c_fm_i )
=> ( ( A2
= ( bNF_We6247850573693329646c_fm_i @ R @ B2 ) )
= ( ( member6642669606046002379c_fm_i @ A2 @ B2 )
& ( member6642669606046002379c_fm_i @ A2 @ ( order_2859517305408744743c_fm_i @ R @ B2 ) ) ) ) ) ) ) ).
% wo_rel.minim_iff_In_Under
thf(fact_1176_wo__rel_Ominim__iff__In__Under,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit,A2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( B2 != bot_bo3957492148770167129t_unit )
=> ( ( A2
= ( bNF_We7185707985517244377t_unit @ R @ B2 ) )
= ( ( member_Product_unit @ A2 @ B2 )
& ( member_Product_unit @ A2 @ ( order_1637587770817919634t_unit @ R @ B2 ) ) ) ) ) ) ) ).
% wo_rel.minim_iff_In_Under
thf(fact_1177_wo__rel_Ominim__Under,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( B2 != bot_bo145720340923748686c_fm_i )
=> ( member1104366573291651755c_fm_i @ ( bNF_We5992065776426773198c_fm_i @ R @ B2 ) @ ( order_3026938356532400903c_fm_i @ R @ B2 ) ) ) ) ) ).
% wo_rel.minim_Under
thf(fact_1178_wo__rel_Ominim__Under,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( B2 != bot_bo4194595901900360558c_fm_i )
=> ( member6642669606046002379c_fm_i @ ( bNF_We6247850573693329646c_fm_i @ R @ B2 ) @ ( order_2859517305408744743c_fm_i @ R @ B2 ) ) ) ) ) ).
% wo_rel.minim_Under
thf(fact_1179_wo__rel_Ominim__Under,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( B2 != bot_bo3957492148770167129t_unit )
=> ( member_Product_unit @ ( bNF_We7185707985517244377t_unit @ R @ B2 ) @ ( order_1637587770817919634t_unit @ R @ B2 ) ) ) ) ) ).
% wo_rel.minim_Under
thf(fact_1180_wo__rel_Ounder__underS__suc,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( ( order_8048284402201309037c_fm_i @ R @ A2 )
!= bot_bo145720340923748686c_fm_i )
=> ( ( order_8439613377573682120c_fm_i @ R @ ( bNF_We1910513081875451107c_fm_i @ R @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
= ( order_4888947244107180775c_fm_i @ R @ A2 ) ) ) ) ) ).
% wo_rel.under_underS_suc
thf(fact_1181_wo__rel_Ounder__underS__suc,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ A2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( order_1288088345848551912c_fm_i @ R @ ( bNF_We6750397660357449475c_fm_i @ R @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
= ( order_1815523504696313095c_fm_i @ R @ A2 ) ) ) ) ) ).
% wo_rel.under_underS_suc
thf(fact_1182_wo__rel_Ounder__underS__suc,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ( ( order_1808334460385152504t_unit @ R @ A2 )
!= bot_bo3957492148770167129t_unit )
=> ( ( order_5809128134054296019t_unit @ R @ ( bNF_We4101066952194689646t_unit @ R @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( order_6949322460238004850t_unit @ R @ A2 ) ) ) ) ) ).
% wo_rel.under_underS_suc
thf(fact_1183_Wellorder__Constructions_Owo__rel_Osuc__underS,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( ( order_7243152452012511053c_fm_i @ R @ B2 )
!= bot_bo145720340923748686c_fm_i )
=> ( ( member1104366573291651755c_fm_i @ B @ B2 )
=> ( member1104366573291651755c_fm_i @ B @ ( order_8439613377573682120c_fm_i @ R @ ( bNF_We1910513081875451107c_fm_i @ R @ B2 ) ) ) ) ) ) ) ).
% Wellorder_Constructions.wo_rel.suc_underS
thf(fact_1184_Wellorder__Constructions_Owo__rel_Osuc__underS,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i,B: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ( order_225095690189193581c_fm_i @ R @ B2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( member6642669606046002379c_fm_i @ B @ B2 )
=> ( member6642669606046002379c_fm_i @ B @ ( order_1288088345848551912c_fm_i @ R @ ( bNF_We6750397660357449475c_fm_i @ R @ B2 ) ) ) ) ) ) ) ).
% Wellorder_Constructions.wo_rel.suc_underS
thf(fact_1185_Wellorder__Constructions_Owo__rel_Osuc__underS,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit,B: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( ( order_7096623528790318552t_unit @ R @ B2 )
!= bot_bo3957492148770167129t_unit )
=> ( ( member_Product_unit @ B @ B2 )
=> ( member_Product_unit @ B @ ( order_5809128134054296019t_unit @ R @ ( bNF_We4101066952194689646t_unit @ R @ B2 ) ) ) ) ) ) ) ).
% Wellorder_Constructions.wo_rel.suc_underS
thf(fact_1186_aboveS__incr2,axiom,
! [R: set_Pr4658907567593863815c_fm_i,R3: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( ord_le3932863852503339559c_fm_i @ R @ R3 )
=> ( ord_le3843937902494030498c_fm_i @ ( order_8036508438950666637c_fm_i @ R @ A2 ) @ ( order_8036508438950666637c_fm_i @ R3 @ A2 ) ) ) ).
% aboveS_incr2
thf(fact_1187_aboveS__incr2,axiom,
! [R: set_Pr5094982260447487303t_unit,R3: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( ord_le930853871177014503t_unit @ R @ R3 )
=> ( ord_le3507040750410214029t_unit @ ( order_1808334460385152504t_unit @ R @ A2 ) @ ( order_1808334460385152504t_unit @ R3 @ A2 ) ) ) ).
% aboveS_incr2
thf(fact_1188_aboveS__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_8036508438950666637c_fm_i @ R @ A2 ) @ ( field_Epistemic_fm_i @ R ) ) ).
% aboveS_Field
thf(fact_1189_aboveS__Field,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit] : ( ord_le3507040750410214029t_unit @ ( order_1808334460385152504t_unit @ R @ A2 ) @ ( field_Product_unit @ R ) ) ).
% aboveS_Field
thf(fact_1190_aboveS__AboveS__singl,axiom,
( order_8036508438950666637c_fm_i
= ( ^ [R4: set_Pr4658907567593863815c_fm_i,A4: epistemic_fm_i] : ( order_225095690189193581c_fm_i @ R4 @ ( insert7817948997695205106c_fm_i @ A4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% aboveS_AboveS_singl
thf(fact_1191_aboveS__AboveS__singl,axiom,
( order_1808334460385152504t_unit
= ( ^ [R4: set_Pr5094982260447487303t_unit,A4: product_unit] : ( order_7096623528790318552t_unit @ R4 @ ( insert_Product_unit2 @ A4 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% aboveS_AboveS_singl
thf(fact_1192_AboveS__decr,axiom,
! [A: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ ( order_225095690189193581c_fm_i @ R @ B2 ) @ ( order_225095690189193581c_fm_i @ R @ A ) ) ) ).
% AboveS_decr
thf(fact_1193_AboveS__decr,axiom,
! [A: set_Product_unit,B2: set_Product_unit,R: set_Pr5094982260447487303t_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ord_le3507040750410214029t_unit @ ( order_7096623528790318552t_unit @ R @ B2 ) @ ( order_7096623528790318552t_unit @ R @ A ) ) ) ).
% AboveS_decr
thf(fact_1194_aboveS__def,axiom,
( order_8048284402201309037c_fm_i
= ( ^ [R4: set_Pr3972831103112126087c_fm_i,A4: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [B5: set_Epistemic_fm_i] :
( ( B5 != A4 )
& ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A4 @ B5 ) @ R4 ) ) ) ) ) ).
% aboveS_def
thf(fact_1195_aboveS__def,axiom,
( order_8036508438950666637c_fm_i
= ( ^ [R4: set_Pr4658907567593863815c_fm_i,A4: epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [B5: epistemic_fm_i] :
( ( B5 != A4 )
& ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A4 @ B5 ) @ R4 ) ) ) ) ) ).
% aboveS_def
thf(fact_1196_in__UnderS__aboveS,axiom,
! [A2: epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( member6642669606046002379c_fm_i @ A2 @ ( order_2700047633941854664c_fm_i @ R @ ( order_8036508438950666637c_fm_i @ R @ A2 ) ) ) ) ).
% in_UnderS_aboveS
thf(fact_1197_in__UnderS__aboveS,axiom,
! [A2: set_Epistemic_fm_i,R: set_Pr3972831103112126087c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( member1104366573291651755c_fm_i @ A2 @ ( order_7634481427384884136c_fm_i @ R @ ( order_8048284402201309037c_fm_i @ R @ A2 ) ) ) ) ).
% in_UnderS_aboveS
thf(fact_1198_AboveS__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_225095690189193581c_fm_i @ R @ A ) @ ( field_Epistemic_fm_i @ R ) ) ).
% AboveS_Field
thf(fact_1199_AboveS__Field,axiom,
! [R: set_Pr5094982260447487303t_unit,A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ ( order_7096623528790318552t_unit @ R @ A ) @ ( field_Product_unit @ R ) ) ).
% AboveS_Field
thf(fact_1200_in__AboveS__underS,axiom,
! [A2: epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( member6642669606046002379c_fm_i @ A2 @ ( order_225095690189193581c_fm_i @ R @ ( order_1288088345848551912c_fm_i @ R @ A2 ) ) ) ) ).
% in_AboveS_underS
thf(fact_1201_in__AboveS__underS,axiom,
! [A2: set_Epistemic_fm_i,R: set_Pr3972831103112126087c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( member1104366573291651755c_fm_i @ A2 @ ( order_7243152452012511053c_fm_i @ R @ ( order_8439613377573682120c_fm_i @ R @ A2 ) ) ) ) ).
% in_AboveS_underS
thf(fact_1202_UnderS__AboveS__Galois,axiom,
! [B2: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ord_le3843937902494030498c_fm_i @ C2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( order_225095690189193581c_fm_i @ R @ C2 ) )
= ( ord_le3843937902494030498c_fm_i @ C2 @ ( order_2700047633941854664c_fm_i @ R @ B2 ) ) ) ) ) ).
% UnderS_AboveS_Galois
thf(fact_1203_UnderS__AboveS__Galois,axiom,
! [B2: set_Product_unit,R: set_Pr5094982260447487303t_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( ord_le3507040750410214029t_unit @ C2 @ ( field_Product_unit @ R ) )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( order_7096623528790318552t_unit @ R @ C2 ) )
= ( ord_le3507040750410214029t_unit @ C2 @ ( order_1874045165604686259t_unit @ R @ B2 ) ) ) ) ) ).
% UnderS_AboveS_Galois
thf(fact_1204_subset__AboveS__UnderS,axiom,
! [B2: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ord_le3843937902494030498c_fm_i @ B2 @ ( order_225095690189193581c_fm_i @ R @ ( order_2700047633941854664c_fm_i @ R @ B2 ) ) ) ) ).
% subset_AboveS_UnderS
thf(fact_1205_subset__AboveS__UnderS,axiom,
! [B2: set_Product_unit,R: set_Pr5094982260447487303t_unit] :
( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ord_le3507040750410214029t_unit @ B2 @ ( order_7096623528790318552t_unit @ R @ ( order_1874045165604686259t_unit @ R @ B2 ) ) ) ) ).
% subset_AboveS_UnderS
thf(fact_1206_subset__UnderS__AboveS,axiom,
! [B2: set_Epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ord_le3843937902494030498c_fm_i @ B2 @ ( order_2700047633941854664c_fm_i @ R @ ( order_225095690189193581c_fm_i @ R @ B2 ) ) ) ) ).
% subset_UnderS_AboveS
thf(fact_1207_subset__UnderS__AboveS,axiom,
! [B2: set_Product_unit,R: set_Pr5094982260447487303t_unit] :
( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ord_le3507040750410214029t_unit @ B2 @ ( order_1874045165604686259t_unit @ R @ ( order_7096623528790318552t_unit @ R @ B2 ) ) ) ) ).
% subset_UnderS_AboveS
thf(fact_1208_wo__rel_Oequals__suc__AboveS,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( order_7243152452012511053c_fm_i @ R @ B2 ) )
=> ( ! [A9: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A9 @ ( order_7243152452012511053c_fm_i @ R @ B2 ) )
=> ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A2 @ A9 ) @ R ) )
=> ( A2
= ( bNF_We1910513081875451107c_fm_i @ R @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_1209_wo__rel_Oequals__suc__AboveS,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( order_225095690189193581c_fm_i @ R @ B2 ) )
=> ( ! [A9: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A9 @ ( order_225095690189193581c_fm_i @ R @ B2 ) )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A2 @ A9 ) @ R ) )
=> ( A2
= ( bNF_We6750397660357449475c_fm_i @ R @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_1210_wo__rel_Oequals__suc__AboveS,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit,A2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( member_Product_unit @ A2 @ ( order_7096623528790318552t_unit @ R @ B2 ) )
=> ( ! [A9: product_unit] :
( ( member_Product_unit @ A9 @ ( order_7096623528790318552t_unit @ R @ B2 ) )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ A9 ) @ R ) )
=> ( A2
= ( bNF_We4101066952194689646t_unit @ R @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_1211_wo__rel_Osuc__AboveS,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( ( order_7243152452012511053c_fm_i @ R @ B2 )
!= bot_bo145720340923748686c_fm_i )
=> ( member1104366573291651755c_fm_i @ ( bNF_We1910513081875451107c_fm_i @ R @ B2 ) @ ( order_7243152452012511053c_fm_i @ R @ B2 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_1212_wo__rel_Osuc__AboveS,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ( order_225095690189193581c_fm_i @ R @ B2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( member6642669606046002379c_fm_i @ ( bNF_We6750397660357449475c_fm_i @ R @ B2 ) @ ( order_225095690189193581c_fm_i @ R @ B2 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_1213_wo__rel_Osuc__AboveS,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( ( order_7096623528790318552t_unit @ R @ B2 )
!= bot_bo3957492148770167129t_unit )
=> ( member_Product_unit @ ( bNF_We4101066952194689646t_unit @ R @ B2 ) @ ( order_7096623528790318552t_unit @ R @ B2 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_1214_wo__rel_Osuc__inField,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( ( order_7243152452012511053c_fm_i @ R @ B2 )
!= bot_bo145720340923748686c_fm_i )
=> ( member1104366573291651755c_fm_i @ ( bNF_We1910513081875451107c_fm_i @ R @ B2 ) @ ( field_4031064140798694607c_fm_i @ R ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_1215_wo__rel_Osuc__inField,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ( order_225095690189193581c_fm_i @ R @ B2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( member6642669606046002379c_fm_i @ ( bNF_We6750397660357449475c_fm_i @ R @ B2 ) @ ( field_Epistemic_fm_i @ R ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_1216_wo__rel_Osuc__inField,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( ( order_7096623528790318552t_unit @ R @ B2 )
!= bot_bo3957492148770167129t_unit )
=> ( member_Product_unit @ ( bNF_We4101066952194689646t_unit @ R @ B2 ) @ ( field_Product_unit @ R ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_1217_wo__rel_Osuc__singl__pred,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i,A10: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( ( order_8048284402201309037c_fm_i @ R @ A2 )
!= bot_bo145720340923748686c_fm_i )
=> ( ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A10 @ ( bNF_We1910513081875451107c_fm_i @ R @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ) @ R )
=> ( ( A10
!= ( bNF_We1910513081875451107c_fm_i @ R @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
=> ( ( A10 = A2 )
| ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A10 @ A2 ) @ R ) ) ) ) ) ) ) ).
% wo_rel.suc_singl_pred
thf(fact_1218_wo__rel_Osuc__singl__pred,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i,A10: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ A2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A10 @ ( bNF_We6750397660357449475c_fm_i @ R @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) @ R )
=> ( ( A10
!= ( bNF_We6750397660357449475c_fm_i @ R @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
=> ( ( A10 = A2 )
| ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A10 @ A2 ) @ R ) ) ) ) ) ) ) ).
% wo_rel.suc_singl_pred
thf(fact_1219_wo__rel_Osuc__singl__pred,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit,A10: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ( ( order_1808334460385152504t_unit @ R @ A2 )
!= bot_bo3957492148770167129t_unit )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A10 @ ( bNF_We4101066952194689646t_unit @ R @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ) @ R )
=> ( ( A10
!= ( bNF_We4101066952194689646t_unit @ R @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) )
=> ( ( A10 = A2 )
| ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A10 @ A2 ) @ R ) ) ) ) ) ) ) ).
% wo_rel.suc_singl_pred
thf(fact_1220_wo__rel_Osuc__greater,axiom,
! [R: set_Pr3972831103112126087c_fm_i,B2: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( ( order_7243152452012511053c_fm_i @ R @ B2 )
!= bot_bo145720340923748686c_fm_i )
=> ( ( member1104366573291651755c_fm_i @ B @ B2 )
=> ( ( ( bNF_We1910513081875451107c_fm_i @ R @ B2 )
!= B )
& ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ B @ ( bNF_We1910513081875451107c_fm_i @ R @ B2 ) ) @ R ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_1221_wo__rel_Osuc__greater,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i,B: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ( order_225095690189193581c_fm_i @ R @ B2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( member6642669606046002379c_fm_i @ B @ B2 )
=> ( ( ( bNF_We6750397660357449475c_fm_i @ R @ B2 )
!= B )
& ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ B @ ( bNF_We6750397660357449475c_fm_i @ R @ B2 ) ) @ R ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_1222_wo__rel_Osuc__greater,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit,B: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( ( order_7096623528790318552t_unit @ R @ B2 )
!= bot_bo3957492148770167129t_unit )
=> ( ( member_Product_unit @ B @ B2 )
=> ( ( ( bNF_We4101066952194689646t_unit @ R @ B2 )
!= B )
& ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ ( bNF_We4101066952194689646t_unit @ R @ B2 ) ) @ R ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_1223_wo__rel_Osuc__above__AboveS,axiom,
! [R: set_Pr4658907567593863815c_fm_i,B2: set_Epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ( order_225095690189193581c_fm_i @ R @ B2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( order_225095690189193581c_fm_i @ R @ B2 )
= ( order_1596567953341465634c_fm_i @ R @ ( bNF_We6750397660357449475c_fm_i @ R @ B2 ) ) ) ) ) ) ).
% wo_rel.suc_above_AboveS
thf(fact_1224_wo__rel_Osuc__above__AboveS,axiom,
! [R: set_Pr5094982260447487303t_unit,B2: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( field_Product_unit @ R ) )
=> ( ( ( order_7096623528790318552t_unit @ R @ B2 )
!= bot_bo3957492148770167129t_unit )
=> ( ( order_7096623528790318552t_unit @ R @ B2 )
= ( order_5924129906778625293t_unit @ R @ ( bNF_We4101066952194689646t_unit @ R @ B2 ) ) ) ) ) ) ).
% wo_rel.suc_above_AboveS
thf(fact_1225_Refl__above__aboveS,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ ( field_4031064140798694607c_fm_i @ R ) @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( order_3037068032131546626c_fm_i @ R @ A2 )
= ( sup_su2582925890723967158c_fm_i @ ( order_8048284402201309037c_fm_i @ R @ A2 ) @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ).
% Refl_above_aboveS
thf(fact_1226_Refl__above__aboveS,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ ( field_Epistemic_fm_i @ R ) @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( order_1596567953341465634c_fm_i @ R @ A2 )
= ( sup_su1936195050962291414c_fm_i @ ( order_8036508438950666637c_fm_i @ R @ A2 ) @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% Refl_above_aboveS
thf(fact_1227_Refl__above__aboveS,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( refl_on_Product_unit @ ( field_Product_unit @ R ) @ R )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R ) )
=> ( ( order_5924129906778625293t_unit @ R @ A2 )
= ( sup_su793286257634532545t_unit @ ( order_1808334460385152504t_unit @ R @ A2 ) @ ( insert_Product_unit2 @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% Refl_above_aboveS
thf(fact_1228_above__def,axiom,
( order_3037068032131546626c_fm_i
= ( ^ [R4: set_Pr3972831103112126087c_fm_i,A4: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [B5: set_Epistemic_fm_i] : ( member7895148549828227280c_fm_i @ ( produc1860896601521578647c_fm_i @ A4 @ B5 ) @ R4 ) ) ) ) ).
% above_def
thf(fact_1229_above__def,axiom,
( order_1596567953341465634c_fm_i
= ( ^ [R4: set_Pr4658907567593863815c_fm_i,A4: epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [B5: epistemic_fm_i] : ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ A4 @ B5 ) @ R4 ) ) ) ) ).
% above_def
thf(fact_1230_above__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_1596567953341465634c_fm_i @ R @ A2 ) @ ( field_Epistemic_fm_i @ R ) ) ).
% above_Field
thf(fact_1231_above__Field,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit] : ( ord_le3507040750410214029t_unit @ ( order_5924129906778625293t_unit @ R @ A2 ) @ ( field_Product_unit @ R ) ) ).
% above_Field
thf(fact_1232_above__incr2,axiom,
! [R: set_Pr4658907567593863815c_fm_i,R3: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( ord_le3932863852503339559c_fm_i @ R @ R3 )
=> ( ord_le3843937902494030498c_fm_i @ ( order_1596567953341465634c_fm_i @ R @ A2 ) @ ( order_1596567953341465634c_fm_i @ R3 @ A2 ) ) ) ).
% above_incr2
thf(fact_1233_above__incr2,axiom,
! [R: set_Pr5094982260447487303t_unit,R3: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( ord_le930853871177014503t_unit @ R @ R3 )
=> ( ord_le3507040750410214029t_unit @ ( order_5924129906778625293t_unit @ R @ A2 ) @ ( order_5924129906778625293t_unit @ R3 @ A2 ) ) ) ).
% above_incr2
thf(fact_1234_Refl__above__in,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] :
( ( refl_o2748477423337579169c_fm_i @ ( field_Epistemic_fm_i @ R ) @ R )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( member6642669606046002379c_fm_i @ A2 @ ( order_1596567953341465634c_fm_i @ R @ A2 ) ) ) ) ).
% Refl_above_in
thf(fact_1235_Refl__above__in,axiom,
! [R: set_Pr3972831103112126087c_fm_i,A2: set_Epistemic_fm_i] :
( ( refl_o4322866692276002945c_fm_i @ ( field_4031064140798694607c_fm_i @ R ) @ R )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( member1104366573291651755c_fm_i @ A2 @ ( order_3037068032131546626c_fm_i @ R @ A2 ) ) ) ) ).
% Refl_above_in
thf(fact_1236_aboveS__subset__above,axiom,
! [R: set_Pr4658907567593863815c_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( order_8036508438950666637c_fm_i @ R @ A2 ) @ ( order_1596567953341465634c_fm_i @ R @ A2 ) ) ).
% aboveS_subset_above
thf(fact_1237_aboveS__subset__above,axiom,
! [R: set_Pr5094982260447487303t_unit,A2: product_unit] : ( ord_le3507040750410214029t_unit @ ( order_1808334460385152504t_unit @ R @ A2 ) @ ( order_5924129906778625293t_unit @ R @ A2 ) ) ).
% aboveS_subset_above
thf(fact_1238_in__Under__above,axiom,
! [A2: epistemic_fm_i,R: set_Pr4658907567593863815c_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( field_Epistemic_fm_i @ R ) )
=> ( member6642669606046002379c_fm_i @ A2 @ ( order_2859517305408744743c_fm_i @ R @ ( order_1596567953341465634c_fm_i @ R @ A2 ) ) ) ) ).
% in_Under_above
thf(fact_1239_in__Under__above,axiom,
! [A2: set_Epistemic_fm_i,R: set_Pr3972831103112126087c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( member1104366573291651755c_fm_i @ A2 @ ( order_3026938356532400903c_fm_i @ R @ ( order_3037068032131546626c_fm_i @ R @ A2 ) ) ) ) ).
% in_Under_above
thf(fact_1240_wo__rel_OisLimOrd__aboveS,axiom,
! [R: set_Pr3972831103112126087c_fm_i,I2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( wellor2121115805534388579c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ I2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( order_8048284402201309037c_fm_i @ R @ I2 )
!= bot_bo145720340923748686c_fm_i ) ) ) ) ).
% wo_rel.isLimOrd_aboveS
thf(fact_1241_wo__rel_OisLimOrd__aboveS,axiom,
! [R: set_Pr4658907567593863815c_fm_i,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( wellor5744531169642908547c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ I2 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( order_8036508438950666637c_fm_i @ R @ I2 )
!= bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% wo_rel.isLimOrd_aboveS
thf(fact_1242_wo__rel_OisLimOrd__aboveS,axiom,
! [R: set_Pr5094982260447487303t_unit,I2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( wellor2192093370025869038t_unit @ R )
=> ( ( member_Product_unit @ I2 @ ( field_Product_unit @ R ) )
=> ( ( order_1808334460385152504t_unit @ R @ I2 )
!= bot_bo3957492148770167129t_unit ) ) ) ) ).
% wo_rel.isLimOrd_aboveS
thf(fact_1243_wo__rel_OisLimOrd__succ,axiom,
! [R: set_Pr4658907567593863815c_fm_i,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( wellor5744531169642908547c_fm_i @ R )
=> ( ( member6642669606046002379c_fm_i @ I2 @ ( field_Epistemic_fm_i @ R ) )
=> ( member6642669606046002379c_fm_i @ ( wellor2624885639821348924c_fm_i @ R @ I2 ) @ ( field_Epistemic_fm_i @ R ) ) ) ) ) ).
% wo_rel.isLimOrd_succ
thf(fact_1244_wo__rel_OisLimOrd__succ,axiom,
! [R: set_Pr3972831103112126087c_fm_i,I2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( wellor2121115805534388579c_fm_i @ R )
=> ( ( member1104366573291651755c_fm_i @ I2 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( member1104366573291651755c_fm_i @ ( wellor6171438539502936604c_fm_i @ R @ I2 ) @ ( field_4031064140798694607c_fm_i @ R ) ) ) ) ) ).
% wo_rel.isLimOrd_succ
thf(fact_1245_wo__rel_Osucc__aboveS__isLimOrd,axiom,
! [R: set_Pr3972831103112126087c_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( field_4031064140798694607c_fm_i @ R ) )
=> ( ( ( order_8048284402201309037c_fm_i @ R @ X3 )
!= bot_bo145720340923748686c_fm_i )
& ( member1104366573291651755c_fm_i @ ( wellor6171438539502936604c_fm_i @ R @ X3 ) @ ( field_4031064140798694607c_fm_i @ R ) ) ) )
=> ( wellor2121115805534388579c_fm_i @ R ) ) ) ).
% wo_rel.succ_aboveS_isLimOrd
thf(fact_1246_wo__rel_Osucc__aboveS__isLimOrd,axiom,
! [R: set_Pr4658907567593863815c_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( field_Epistemic_fm_i @ R ) )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ X3 )
!= bot_bo4194595901900360558c_fm_i )
& ( member6642669606046002379c_fm_i @ ( wellor2624885639821348924c_fm_i @ R @ X3 ) @ ( field_Epistemic_fm_i @ R ) ) ) )
=> ( wellor5744531169642908547c_fm_i @ R ) ) ) ).
% wo_rel.succ_aboveS_isLimOrd
thf(fact_1247_wo__rel_Osucc__aboveS__isLimOrd,axiom,
! [R: set_Pr5094982260447487303t_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( field_Product_unit @ R ) )
=> ( ( ( order_1808334460385152504t_unit @ R @ X3 )
!= bot_bo3957492148770167129t_unit )
& ( member_Product_unit @ ( wellor508239122659082791t_unit @ R @ X3 ) @ ( field_Product_unit @ R ) ) ) )
=> ( wellor2192093370025869038t_unit @ R ) ) ) ).
% wo_rel.succ_aboveS_isLimOrd
thf(fact_1248_wo__rel_Osucc__diff,axiom,
! [R: set_Pr4658907567593863815c_fm_i,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ I2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( wellor2624885639821348924c_fm_i @ R @ I2 )
!= I2 ) ) ) ).
% wo_rel.succ_diff
thf(fact_1249_wo__rel_Osucc__diff,axiom,
! [R: set_Pr5094982260447487303t_unit,I2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( order_1808334460385152504t_unit @ R @ I2 )
!= bot_bo3957492148770167129t_unit )
=> ( ( wellor508239122659082791t_unit @ R @ I2 )
!= I2 ) ) ) ).
% wo_rel.succ_diff
thf(fact_1250_wo__rel_Osucc__inj,axiom,
! [R: set_Pr4658907567593863815c_fm_i,I2: epistemic_fm_i,J: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ I2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ J )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( ( wellor2624885639821348924c_fm_i @ R @ I2 )
= ( wellor2624885639821348924c_fm_i @ R @ J ) )
= ( I2 = J ) ) ) ) ) ).
% wo_rel.succ_inj
thf(fact_1251_wo__rel_Osucc__inj,axiom,
! [R: set_Pr5094982260447487303t_unit,I2: product_unit,J: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( order_1808334460385152504t_unit @ R @ I2 )
!= bot_bo3957492148770167129t_unit )
=> ( ( ( order_1808334460385152504t_unit @ R @ J )
!= bot_bo3957492148770167129t_unit )
=> ( ( ( wellor508239122659082791t_unit @ R @ I2 )
= ( wellor508239122659082791t_unit @ R @ J ) )
= ( I2 = J ) ) ) ) ) ).
% wo_rel.succ_inj
thf(fact_1252_isLimOrd__r,axiom,
wellor5744531169642908547c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) ).
% isLimOrd_r
thf(fact_1253_wo__rel_Osucc__in__diff,axiom,
! [R: set_Pr4658907567593863815c_fm_i,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ I2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ I2 @ ( wellor2624885639821348924c_fm_i @ R @ I2 ) ) @ R )
& ( ( wellor2624885639821348924c_fm_i @ R @ I2 )
!= I2 ) ) ) ) ).
% wo_rel.succ_in_diff
thf(fact_1254_wo__rel_Osucc__in__diff,axiom,
! [R: set_Pr5094982260447487303t_unit,I2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( order_1808334460385152504t_unit @ R @ I2 )
!= bot_bo3957492148770167129t_unit )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ ( wellor508239122659082791t_unit @ R @ I2 ) ) @ R )
& ( ( wellor508239122659082791t_unit @ R @ I2 )
!= I2 ) ) ) ) ).
% wo_rel.succ_in_diff
thf(fact_1255_wo__rel_Osucc__not__in,axiom,
! [R: set_Pr4658907567593863815c_fm_i,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ I2 )
!= bot_bo4194595901900360558c_fm_i )
=> ~ ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ ( wellor2624885639821348924c_fm_i @ R @ I2 ) @ I2 ) @ R ) ) ) ).
% wo_rel.succ_not_in
thf(fact_1256_wo__rel_Osucc__not__in,axiom,
! [R: set_Pr5094982260447487303t_unit,I2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( order_1808334460385152504t_unit @ R @ I2 )
!= bot_bo3957492148770167129t_unit )
=> ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ ( wellor508239122659082791t_unit @ R @ I2 ) @ I2 ) @ R ) ) ) ).
% wo_rel.succ_not_in
thf(fact_1257_wo__rel_Osucc__mono,axiom,
! [R: set_Pr4658907567593863815c_fm_i,J: epistemic_fm_i,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ J )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ I2 @ J ) @ R )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ ( wellor2624885639821348924c_fm_i @ R @ I2 ) @ ( wellor2624885639821348924c_fm_i @ R @ J ) ) @ R ) ) ) ) ).
% wo_rel.succ_mono
thf(fact_1258_wo__rel_Osucc__mono,axiom,
! [R: set_Pr5094982260447487303t_unit,J: product_unit,I2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( order_1808334460385152504t_unit @ R @ J )
!= bot_bo3957492148770167129t_unit )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ ( wellor508239122659082791t_unit @ R @ I2 ) @ ( wellor508239122659082791t_unit @ R @ J ) ) @ R ) ) ) ) ).
% wo_rel.succ_mono
thf(fact_1259_wo__rel_Oless__succ,axiom,
! [R: set_Pr4658907567593863815c_fm_i,I2: epistemic_fm_i,J: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ I2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ J @ ( wellor2624885639821348924c_fm_i @ R @ I2 ) ) @ R )
= ( ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ J @ I2 ) @ R )
| ( J
= ( wellor2624885639821348924c_fm_i @ R @ I2 ) ) ) ) ) ) ).
% wo_rel.less_succ
thf(fact_1260_wo__rel_Oless__succ,axiom,
! [R: set_Pr5094982260447487303t_unit,I2: product_unit,J: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( order_1808334460385152504t_unit @ R @ I2 )
!= bot_bo3957492148770167129t_unit )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ J @ ( wellor508239122659082791t_unit @ R @ I2 ) ) @ R )
= ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ J @ I2 ) @ R )
| ( J
= ( wellor508239122659082791t_unit @ R @ I2 ) ) ) ) ) ) ).
% wo_rel.less_succ
thf(fact_1261_wo__rel_Osucc__in,axiom,
! [R: set_Pr4658907567593863815c_fm_i,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ I2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( member1268636611877661904c_fm_i @ ( produc4253568889897832087c_fm_i @ I2 @ ( wellor2624885639821348924c_fm_i @ R @ I2 ) ) @ R ) ) ) ).
% wo_rel.succ_in
thf(fact_1262_wo__rel_Osucc__in,axiom,
! [R: set_Pr5094982260447487303t_unit,I2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( order_1808334460385152504t_unit @ R @ I2 )
!= bot_bo3957492148770167129t_unit )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ ( wellor508239122659082791t_unit @ R @ I2 ) ) @ R ) ) ) ).
% wo_rel.succ_in
thf(fact_1263_wo__rel_Osucc__in__Field,axiom,
! [R: set_Pr3972831103112126087c_fm_i,I2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( ( order_8048284402201309037c_fm_i @ R @ I2 )
!= bot_bo145720340923748686c_fm_i )
=> ( member1104366573291651755c_fm_i @ ( wellor6171438539502936604c_fm_i @ R @ I2 ) @ ( field_4031064140798694607c_fm_i @ R ) ) ) ) ).
% wo_rel.succ_in_Field
thf(fact_1264_wo__rel_Osucc__in__Field,axiom,
! [R: set_Pr4658907567593863815c_fm_i,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ I2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( member6642669606046002379c_fm_i @ ( wellor2624885639821348924c_fm_i @ R @ I2 ) @ ( field_Epistemic_fm_i @ R ) ) ) ) ).
% wo_rel.succ_in_Field
thf(fact_1265_wo__rel_Osucc__in__Field,axiom,
! [R: set_Pr5094982260447487303t_unit,I2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( order_1808334460385152504t_unit @ R @ I2 )
!= bot_bo3957492148770167129t_unit )
=> ( member_Product_unit @ ( wellor508239122659082791t_unit @ R @ I2 ) @ ( field_Product_unit @ R ) ) ) ) ).
% wo_rel.succ_in_Field
thf(fact_1266_wo__rel_Osucc__def,axiom,
! [R: set_Pr4658907567593863815c_fm_i,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( wellor2624885639821348924c_fm_i @ R @ I2 )
= ( bNF_We6750397660357449475c_fm_i @ R @ ( insert7817948997695205106c_fm_i @ I2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% wo_rel.succ_def
thf(fact_1267_wo__rel_Osucc__def,axiom,
! [R: set_Pr5094982260447487303t_unit,I2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( wellor508239122659082791t_unit @ R @ I2 )
= ( bNF_We4101066952194689646t_unit @ R @ ( insert_Product_unit2 @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% wo_rel.succ_def
thf(fact_1268_wo__rel_Ounder__succ,axiom,
! [R: set_Pr4658907567593863815c_fm_i,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ I2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( order_1815523504696313095c_fm_i @ R @ ( wellor2624885639821348924c_fm_i @ R @ I2 ) )
= ( insert7817948997695205106c_fm_i @ ( wellor2624885639821348924c_fm_i @ R @ I2 ) @ ( order_1815523504696313095c_fm_i @ R @ I2 ) ) ) ) ) ).
% wo_rel.under_succ
thf(fact_1269_wo__rel_Ounder__succ,axiom,
! [R: set_Pr5094982260447487303t_unit,I2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( order_1808334460385152504t_unit @ R @ I2 )
!= bot_bo3957492148770167129t_unit )
=> ( ( order_6949322460238004850t_unit @ R @ ( wellor508239122659082791t_unit @ R @ I2 ) )
= ( insert_Product_unit2 @ ( wellor508239122659082791t_unit @ R @ I2 ) @ ( order_6949322460238004850t_unit @ R @ I2 ) ) ) ) ) ).
% wo_rel.under_succ
thf(fact_1270_wo__rel_OunderS__succ,axiom,
! [R: set_Pr4658907567593863815c_fm_i,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( ( order_8036508438950666637c_fm_i @ R @ I2 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( order_1288088345848551912c_fm_i @ R @ ( wellor2624885639821348924c_fm_i @ R @ I2 ) )
= ( order_1815523504696313095c_fm_i @ R @ I2 ) ) ) ) ).
% wo_rel.underS_succ
thf(fact_1271_wo__rel_OunderS__succ,axiom,
! [R: set_Pr5094982260447487303t_unit,I2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( ( order_1808334460385152504t_unit @ R @ I2 )
!= bot_bo3957492148770167129t_unit )
=> ( ( order_5809128134054296019t_unit @ R @ ( wellor508239122659082791t_unit @ R @ I2 ) )
= ( order_6949322460238004850t_unit @ R @ I2 ) ) ) ) ).
% wo_rel.underS_succ
thf(fact_1272_wo__rel_Owell__order__inductZSL,axiom,
! [R: set_Pr3972831103112126087c_fm_i,P3: set_Epistemic_fm_i > $o,I2: set_Epistemic_fm_i] :
( ( bNF_We224940875319969006c_fm_i @ R )
=> ( ( P3 @ ( wellor6421275854403804086c_fm_i @ R ) )
=> ( ! [I3: set_Epistemic_fm_i] :
( ( ( order_8048284402201309037c_fm_i @ R @ I3 )
!= bot_bo145720340923748686c_fm_i )
=> ( ( P3 @ I3 )
=> ( P3 @ ( wellor6171438539502936604c_fm_i @ R @ I3 ) ) ) )
=> ( ! [I3: set_Epistemic_fm_i] :
( ( wellor2139422199430923550c_fm_i @ R @ I3 )
=> ( ( I3
!= ( wellor6421275854403804086c_fm_i @ R ) )
=> ( ! [J3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ J3 @ ( order_8439613377573682120c_fm_i @ R @ I3 ) )
=> ( P3 @ J3 ) )
=> ( P3 @ I3 ) ) ) )
=> ( P3 @ I2 ) ) ) ) ) ).
% wo_rel.well_order_inductZSL
thf(fact_1273_wo__rel_Owell__order__inductZSL,axiom,
! [R: set_Pr4658907567593863815c_fm_i,P3: epistemic_fm_i > $o,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ( P3 @ ( wellor854379368678715862c_fm_i @ R ) )
=> ( ! [I3: epistemic_fm_i] :
( ( ( order_8036508438950666637c_fm_i @ R @ I3 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( P3 @ I3 )
=> ( P3 @ ( wellor2624885639821348924c_fm_i @ R @ I3 ) ) ) )
=> ( ! [I3: epistemic_fm_i] :
( ( wellor4766384553491184446c_fm_i @ R @ I3 )
=> ( ( I3
!= ( wellor854379368678715862c_fm_i @ R ) )
=> ( ! [J3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ J3 @ ( order_1288088345848551912c_fm_i @ R @ I3 ) )
=> ( P3 @ J3 ) )
=> ( P3 @ I3 ) ) ) )
=> ( P3 @ I2 ) ) ) ) ) ).
% wo_rel.well_order_inductZSL
thf(fact_1274_wo__rel_Owell__order__inductZSL,axiom,
! [R: set_Pr5094982260447487303t_unit,P3: product_unit > $o,I2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ( P3 @ ( wellor829852009790670017t_unit @ R ) )
=> ( ! [I3: product_unit] :
( ( ( order_1808334460385152504t_unit @ R @ I3 )
!= bot_bo3957492148770167129t_unit )
=> ( ( P3 @ I3 )
=> ( P3 @ ( wellor508239122659082791t_unit @ R @ I3 ) ) ) )
=> ( ! [I3: product_unit] :
( ( wellor1931483356828029481t_unit @ R @ I3 )
=> ( ( I3
!= ( wellor829852009790670017t_unit @ R ) )
=> ( ! [J3: product_unit] :
( ( member_Product_unit @ J3 @ ( order_5809128134054296019t_unit @ R @ I3 ) )
=> ( P3 @ J3 ) )
=> ( P3 @ I3 ) ) ) )
=> ( P3 @ I2 ) ) ) ) ) ).
% wo_rel.well_order_inductZSL
thf(fact_1275_wo__rel_Owell__order__inductSL,axiom,
! [R: set_Pr4658907567593863815c_fm_i,P3: epistemic_fm_i > $o,I2: epistemic_fm_i] :
( ( bNF_We8255954865191566094c_fm_i @ R )
=> ( ! [I3: epistemic_fm_i] :
( ( ( order_8036508438950666637c_fm_i @ R @ I3 )
!= bot_bo4194595901900360558c_fm_i )
=> ( ( P3 @ I3 )
=> ( P3 @ ( wellor2624885639821348924c_fm_i @ R @ I3 ) ) ) )
=> ( ! [I3: epistemic_fm_i] :
( ( wellor4766384553491184446c_fm_i @ R @ I3 )
=> ( ! [J3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ J3 @ ( order_1288088345848551912c_fm_i @ R @ I3 ) )
=> ( P3 @ J3 ) )
=> ( P3 @ I3 ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% wo_rel.well_order_inductSL
thf(fact_1276_wo__rel_Owell__order__inductSL,axiom,
! [R: set_Pr5094982260447487303t_unit,P3: product_unit > $o,I2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R )
=> ( ! [I3: product_unit] :
( ( ( order_1808334460385152504t_unit @ R @ I3 )
!= bot_bo3957492148770167129t_unit )
=> ( ( P3 @ I3 )
=> ( P3 @ ( wellor508239122659082791t_unit @ R @ I3 ) ) ) )
=> ( ! [I3: product_unit] :
( ( wellor1931483356828029481t_unit @ R @ I3 )
=> ( ! [J3: product_unit] :
( ( member_Product_unit @ J3 @ ( order_5809128134054296019t_unit @ R @ I3 ) )
=> ( P3 @ J3 ) )
=> ( P3 @ I3 ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% wo_rel.well_order_inductSL
% Conjectures (1)
thf(conj_0,conjecture,
( episte2498298383968489120c_fm_i
@ ( episte8397520288747125024t_unit
@ ( collec3087743281813070829c_fm_i
@ ^ [W: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ a @ W )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ W ) ) )
@ ^ [I: i,V: set_Epistemic_fm_i] :
( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I @ P ) @ V ) ) ) )
@ ( episte4955834246032687171t_unit
@ ^ [V: set_Epistemic_fm_i,X: list_char] : ( member6642669606046002379c_fm_i @ ( epistemic_Pro_i @ X ) @ V )
@ product_Unity ) )
@ ( maxima2580775659384275475c_fm_i @ ( bNF_Ca1305897194302070654c_fm_i @ top_to8364301187473967186c_fm_i ) @ ( episte2285483198712856234tent_i @ a ) @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ i2 @ ( epistemic_Con_i @ p @ q ) ) @ ( epistemic_Con_i @ ( epistemic_K_i @ i2 @ p ) @ ( epistemic_K_i @ i2 @ q ) ) ) @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) )
@ ( epistemic_Con_i @ ( epistemic_K_i @ i2 @ ( epistemic_Con_i @ p @ q ) ) @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ i2 @ p ) @ ( epistemic_K_i @ i2 @ q ) ) @ epistemic_FF_i ) ) ) ).
%------------------------------------------------------------------------------