TPTP Problem File: SLH0138^1.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_00086_002318__6016376_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1543 ( 324 unt; 258 typ; 0 def)
% Number of atoms : 5474 ( 855 equ; 0 cnn)
% Maximal formula atoms : 21 ( 4 avg)
% Number of connectives : 15624 ( 330 ~; 61 |; 246 &;12493 @)
% ( 0 <=>;2494 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 9 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 2987 (2987 >; 0 *; 0 +; 0 <<)
% Number of symbols : 240 ( 237 usr; 19 con; 0-5 aty)
% Number of variables : 4547 ( 440 ^;3960 !; 147 ?;4547 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:11:27.084
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
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set_Epistemic_fm_a_o: $tType ).
thf(ty_n_t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
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set_Epistemic_fm_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(ty_n_t__Epistemic____Logic__Ofm_Itf__a_J,type,
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thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
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thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (237)
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
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thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
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thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
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thf(sy_c_Complete__Partial__Order_Ochain_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
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thf(sy_c_Epistemic__Logic_OAx4_001tf__a,type,
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thf(sy_c_Epistemic__Logic_OAx5_001tf__a,type,
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thf(sy_c_Epistemic__Logic_OAxB_001tf__a,type,
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thf(sy_c_Epistemic__Logic_OAxT_001tf__a,type,
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thf(sy_c_Epistemic__Logic_Ofm_OCon_001tf__a,type,
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thf(sy_c_Epistemic__Logic_Ofm_ODis_001tf__a,type,
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thf(sy_c_Epistemic__Logic_Ofm_OFF_001tf__a,type,
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thf(sy_c_Epistemic__Logic_Ofm_Opred__fm_001tf__a,type,
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thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Epistemic____Logic__Ofm_Itf__a_J_001tf__a,type,
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collec4904205152690461189c_fm_a: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J_J,type,
collec2512887859873555566fm_a_o: ( set_Epistemic_fm_a_o > $o ) > set_se3897318249199329743fm_a_o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
collec2519470961442302949c_fm_a: ( set_Epistemic_fm_a > $o ) > set_se5208064806568342746c_fm_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
collec4484617205158951365c_fm_a: ( set_se5208064806568342746c_fm_a > $o ) > set_se8609989359752948410c_fm_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
image_6282542973537860301fm_a_o: ( ( epistemic_fm_a > $o ) > epistemic_fm_a > $o ) > set_Epistemic_fm_a_o > set_Epistemic_fm_a_o ).
thf(sy_c_Set_Oimage_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
image_3065101469317257744c_fm_a: ( ( epistemic_fm_a > $o ) > set_Epistemic_fm_a ) > set_Epistemic_fm_a_o > set_se5208064806568342746c_fm_a ).
thf(sy_c_Set_Oimage_001t__Epistemic____Logic__Ofm_Itf__a_J_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
image_2381060403230254136fm_a_o: ( epistemic_fm_a > epistemic_fm_a > $o ) > set_Epistemic_fm_a > set_Epistemic_fm_a_o ).
thf(sy_c_Set_Oimage_001t__Epistemic____Logic__Ofm_Itf__a_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
image_1263732536703906021c_fm_a: ( epistemic_fm_a > set_Epistemic_fm_a ) > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J_J_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
image_3680702584007291907fm_a_o: ( set_Epistemic_fm_a_o > epistemic_fm_a > $o ) > set_se3897318249199329743fm_a_o > set_Epistemic_fm_a_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
image_6902499629341727256fm_a_o: ( set_Epistemic_fm_a > epistemic_fm_a > $o ) > set_se5208064806568342746c_fm_a > set_Epistemic_fm_a_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
image_8582343580727714565c_fm_a: ( set_Epistemic_fm_a > set_Epistemic_fm_a ) > set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
image_114381943869839653c_fm_a: ( set_se5208064806568342746c_fm_a > set_Epistemic_fm_a ) > set_se8609989359752948410c_fm_a > set_se5208064806568342746c_fm_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
image_3809950421912819996fm_a_o: ( a > epistemic_fm_a > $o ) > set_a > set_Epistemic_fm_a_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
image_3480842985853347585c_fm_a: ( a > set_Epistemic_fm_a ) > set_a > set_se5208064806568342746c_fm_a ).
thf(sy_c_Set_Oinsert_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
insert6817254513119271059fm_a_o: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a_o > set_Epistemic_fm_a_o ).
thf(sy_c_Set_Oinsert_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
insert7817948963269374698c_fm_a: epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
insert7129737658439086282c_fm_a: set_Epistemic_fm_a > set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
set_or7471044323730626962fm_a_o: ( epistemic_fm_a > $o ) > ( epistemic_fm_a > $o ) > set_Epistemic_fm_a_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
set_or1522782340569636619c_fm_a: set_Epistemic_fm_a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a ).
thf(sy_c_Wellorder__Constructions_Owo__rel_Oadm__woL_001t__Epistemic____Logic__Ofm_Itf__a_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
wellor8531372491315211020c_fm_a: set_Pr8948481099588399239c_fm_a > ( ( epistemic_fm_a > set_Epistemic_fm_a ) > epistemic_fm_a > set_Epistemic_fm_a ) > $o ).
thf(sy_c_Wellorder__Constructions_Owo__rel_OisLimOrd_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
wellor5744531135217078139c_fm_a: set_Pr8948481099588399239c_fm_a > $o ).
thf(sy_c_Wellorder__Constructions_Owo__rel_OworecZSL_001t__Epistemic____Logic__Ofm_Itf__a_J_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
wellor7234309849078405374c_fm_a: set_Pr8948481099588399239c_fm_a > set_Epistemic_fm_a > ( epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a ) > ( ( epistemic_fm_a > set_Epistemic_fm_a ) > epistemic_fm_a > set_Epistemic_fm_a ) > epistemic_fm_a > set_Epistemic_fm_a ).
thf(sy_c_member_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
member4486839677911940090fm_a_o: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a_o > $o ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
member6642669571620171971c_fm_a: epistemic_fm_a > set_Epistemic_fm_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J_J,type,
member3894352381434343216fm_a_o: set_Epistemic_fm_a_o > set_se3897318249199329743fm_a_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
member536094252920883875c_fm_a: set_Epistemic_fm_a > set_se5208064806568342746c_fm_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
member3183810400728440963c_fm_a: set_se5208064806568342746c_fm_a > set_se8609989359752948410c_fm_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: epistemic_fm_a > $o ).
thf(sy_v_i,type,
i: a ).
thf(sy_v_p,type,
p: epistemic_fm_a ).
thf(sy_v_q,type,
q: epistemic_fm_a ).
% Relevant facts (1279)
thf(fact_0__092_060open_062A_A_092_060turnstile_062_AK_Ai_Ap_A_092_060_094bold_062_092_060longrightarrow_062_AL_Ai_Aq_A_092_060_094bold_062_092_060longrightarrow_062_AL_Ai_A_Ip_A_092_060_094bold_062_092_060and_062_Aq_J_092_060close_062,axiom,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ p ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( epistemic_Imp_a @ q @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( epistemic_Imp_a @ ( epistemic_Con_a @ p @ q ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ).
% \<open>A \<turnstile> K i p \<^bold>\<longrightarrow> L i q \<^bold>\<longrightarrow> L i (p \<^bold>\<and> q)\<close>
thf(fact_1__092_060open_062A_A_092_060turnstile_062_Ap_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_A_Ip_A_092_060_094bold_062_092_060and_062_Aq_J_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_Aq_092_060close_062,axiom,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ p @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Con_a @ p @ q ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ q @ epistemic_FF_a ) ) ) ).
% \<open>A \<turnstile> p \<^bold>\<longrightarrow> \<^bold>\<not> (p \<^bold>\<and> q) \<^bold>\<longrightarrow> \<^bold>\<not> q\<close>
thf(fact_2__092_060open_062A_A_092_060turnstile_062_AK_Ai_Ap_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_A_I_092_060_094bold_062_092_060not_062_A_Ip_A_092_060_094bold_062_092_060and_062_Aq_J_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_Aq_J_092_060close_062,axiom,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ p ) @ ( epistemic_K_a @ i @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Con_a @ p @ q ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ q @ epistemic_FF_a ) ) ) ) ).
% \<open>A \<turnstile> K i p \<^bold>\<longrightarrow> K i (\<^bold>\<not> (p \<^bold>\<and> q) \<^bold>\<longrightarrow> \<^bold>\<not> q)\<close>
thf(fact_3__092_060open_062A_A_092_060turnstile_062_AK_Ai_Ap_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_A_I_092_060_094bold_062_092_060not_062_A_Ip_A_092_060_094bold_062_092_060and_062_Aq_J_J_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_A_I_092_060_094bold_062_092_060not_062_Aq_J_092_060close_062,axiom,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ p ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( epistemic_Imp_a @ ( epistemic_Con_a @ p @ q ) @ epistemic_FF_a ) ) @ ( epistemic_K_a @ i @ ( epistemic_Imp_a @ q @ epistemic_FF_a ) ) ) ) ).
% \<open>A \<turnstile> K i p \<^bold>\<longrightarrow> K i (\<^bold>\<not> (p \<^bold>\<and> q)) \<^bold>\<longrightarrow> K i (\<^bold>\<not> q)\<close>
thf(fact_4__092_060open_062A_A_092_060turnstile_062_AK_Ai_A_I_092_060_094bold_062_092_060not_062_A_Ip_A_092_060_094bold_062_092_060and_062_Aq_J_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_Aq_J_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_A_I_092_060_094bold_062_092_060not_062_A_Ip_A_092_060_094bold_062_092_060and_062_Aq_J_J_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_A_I_092_060_094bold_062_092_060not_062_Aq_J_092_060close_062,axiom,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Con_a @ p @ q ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ q @ epistemic_FF_a ) ) ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( epistemic_Imp_a @ ( epistemic_Con_a @ p @ q ) @ epistemic_FF_a ) ) @ ( epistemic_K_a @ i @ ( epistemic_Imp_a @ q @ epistemic_FF_a ) ) ) ) ).
% \<open>A \<turnstile> K i (\<^bold>\<not> (p \<^bold>\<and> q) \<^bold>\<longrightarrow> \<^bold>\<not> q) \<^bold>\<longrightarrow> K i (\<^bold>\<not> (p \<^bold>\<and> q)) \<^bold>\<longrightarrow> K i (\<^bold>\<not> q)\<close>
thf(fact_5_K__imp__trans,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,R: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ Q @ R ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ R ) ) ) ) ).
% K_imp_trans
thf(fact_6_K__imp__trans_H,axiom,
! [A: epistemic_fm_a > $o,Q: epistemic_fm_a,R: epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ Q @ R ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ Q ) @ ( epistemic_Imp_a @ P @ R ) ) ) ) ).
% K_imp_trans'
thf(fact_7_K__imply__multi,axiom,
! [A: epistemic_fm_a > $o,A2: epistemic_fm_a,B: epistemic_fm_a,C: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ A2 @ B ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ A2 @ C ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ A2 @ ( epistemic_Con_a @ B @ C ) ) ) ) ) ).
% K_imply_multi
thf(fact_8_K__multi__imply,axiom,
! [A: epistemic_fm_a > $o,A2: epistemic_fm_a,B: epistemic_fm_a,C: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ A2 @ ( epistemic_Imp_a @ B @ C ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Con_a @ A2 @ B ) @ C ) ) ) ).
% K_multi_imply
thf(fact_9__092_060open_062tautology_A_Ip_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_A_Ip_A_092_060_094bold_062_092_060and_062_Aq_J_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_Aq_J_092_060close_062,axiom,
! [G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ ( epistemic_Imp_a @ p @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Con_a @ p @ q ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ q @ epistemic_FF_a ) ) ) ) ).
% \<open>tautology (p \<^bold>\<longrightarrow> \<^bold>\<not> (p \<^bold>\<and> q) \<^bold>\<longrightarrow> \<^bold>\<not> q)\<close>
thf(fact_10_A2,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ Q ) ) ) @ ( epistemic_K_a @ I @ Q ) ) ) ).
% A2
thf(fact_11_K__LK,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ).
% K_LK
thf(fact_12_K__L__dual,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ P ) ) ) ).
% K_L_dual
thf(fact_13_K__A2_H,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ Q ) ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_A2'
thf(fact_14_K__map,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,I: a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_map
thf(fact_15_fm_Oinject_I3_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( ( epistemic_Con_a @ X41 @ X42 )
= ( epistemic_Con_a @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_16_fm_Oinject_I5_J,axiom,
! [X61: a,X62: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
( ( ( epistemic_K_a @ X61 @ X62 )
= ( epistemic_K_a @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_17_fm_Oinject_I4_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( ( epistemic_Imp_a @ X51 @ X52 )
= ( epistemic_Imp_a @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_18_fm_Odistinct_I5_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_19_fm_Odistinct_I27_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Con_a @ X41 @ X42 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_20_fm_Odistinct_I25_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Con_a @ X41 @ X42 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_21_eval_Osimps_I5_J,axiom,
! [G2: list_char > $o,H2: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G2 @ H2 @ ( epistemic_Imp_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G2 @ H2 @ P )
=> ( epistemic_eval_a @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_22_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H2: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( epistemic_eval_a @ Ux @ H2 @ ( epistemic_K_a @ I @ P ) )
= ( H2 @ ( epistemic_K_a @ I @ P ) ) ) ).
% eval.simps(6)
thf(fact_23_eval_Osimps_I1_J,axiom,
! [Uu: list_char > $o,Uv: epistemic_fm_a > $o] :
~ ( epistemic_eval_a @ Uu @ Uv @ epistemic_FF_a ) ).
% eval.simps(1)
thf(fact_24_A1,axiom,
! [P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [G3: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H3 @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% A1
thf(fact_25_eval_Osimps_I4_J,axiom,
! [G2: list_char > $o,H2: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G2 @ H2 @ ( epistemic_Con_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G2 @ H2 @ P )
& ( epistemic_eval_a @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(4)
thf(fact_26_Ax,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a] :
( ( A @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% Ax
thf(fact_27_AK_Osimps,axiom,
( epistemic_AK_a
= ( ^ [A3: epistemic_fm_a > $o,A4: epistemic_fm_a] :
( ? [P2: epistemic_fm_a] :
( ( A4 = P2 )
& ! [G4: list_char > $o,H4: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H4 @ P2 ) )
| ? [I2: a,P2: epistemic_fm_a,Q2: epistemic_fm_a] :
( A4
= ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I2 @ P2 ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P2 @ Q2 ) ) ) @ ( epistemic_K_a @ I2 @ Q2 ) ) )
| ? [P2: epistemic_fm_a] :
( ( A4 = P2 )
& ( A3 @ P2 ) )
| ? [P2: epistemic_fm_a,Q2: epistemic_fm_a] :
( ( A4 = Q2 )
& ( epistemic_AK_a @ A3 @ P2 )
& ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ P2 @ Q2 ) ) )
| ? [P2: epistemic_fm_a,I2: a] :
( ( A4
= ( epistemic_K_a @ I2 @ P2 ) )
& ( epistemic_AK_a @ A3 @ P2 ) ) ) ) ) ).
% AK.simps
thf(fact_28_AK_Ocases,axiom,
! [A: epistemic_fm_a > $o,A2: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ A2 )
=> ( ~ ! [G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ A2 )
=> ( ! [I3: a,P3: epistemic_fm_a,Q3: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I3 @ P3 ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P3 @ Q3 ) ) ) @ ( epistemic_K_a @ I3 @ Q3 ) ) )
=> ( ~ ( A @ A2 )
=> ( ! [P3: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ P3 )
=> ~ ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P3 @ A2 ) ) )
=> ~ ! [P3: epistemic_fm_a] :
( ? [I3: a] :
( A2
= ( epistemic_K_a @ I3 @ P3 ) )
=> ~ ( epistemic_AK_a @ A @ P3 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_29_duality__taut,axiom,
! [I: a,P: epistemic_fm_a,Q: epistemic_fm_a,G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ).
% duality_taut
thf(fact_30_fm_Odistinct_I29_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Imp_a @ X51 @ X52 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_31_fm_Odistinct_I7_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_32_K__trans,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,R: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ Q ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ Q @ R ) @ ( epistemic_Imp_a @ P @ R ) ) ) ) ).
% K_trans
thf(fact_33_R1,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ P )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( epistemic_AK_a @ A @ Q ) ) ) ).
% R1
thf(fact_34_fm_Odistinct_I9_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_35_R2,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,I: a] :
( ( epistemic_AK_a @ A @ P )
=> ( epistemic_AK_a @ A @ ( epistemic_K_a @ I @ P ) ) ) ).
% R2
thf(fact_36_AxB_Ointros,axiom,
! [P: epistemic_fm_a,I: a] : ( epistemic_AxB_a @ ( epistemic_Imp_a @ P @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% AxB.intros
thf(fact_37_AxB_Osimps,axiom,
( epistemic_AxB_a
= ( ^ [A4: epistemic_fm_a] :
? [P2: epistemic_fm_a,I2: a] :
( A4
= ( epistemic_Imp_a @ P2 @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% AxB.simps
thf(fact_38_AxB_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_AxB_a @ A2 )
=> ~ ! [P3: epistemic_fm_a,I3: a] :
( A2
!= ( epistemic_Imp_a @ P3 @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ).
% AxB.cases
thf(fact_39_Ax5_Ointros,axiom,
! [I: a,P: epistemic_fm_a] : ( epistemic_Ax5_a @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% Ax5.intros
thf(fact_40_Ax5_Osimps,axiom,
( epistemic_Ax5_a
= ( ^ [A4: epistemic_fm_a] :
? [I2: a,P2: epistemic_fm_a] :
( A4
= ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_41_Ax5_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_Ax5_a @ A2 )
=> ~ ! [I3: a,P3: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ).
% Ax5.cases
thf(fact_42_mem__Collect__eq,axiom,
! [A2: a,P4: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A2: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ A2 @ ( collec2519470961442302949c_fm_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A2: epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ A2 @ ( collec4904205152690461189c_fm_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A: set_Epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_48_Collect__cong,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collec2519470961442302949c_fm_a @ P4 )
= ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_49_Collect__cong,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collec4904205152690461189c_fm_a @ P4 )
= ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_50_distribution,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ Q ) ) ) @ ( epistemic_K_a @ I @ Q ) ) ) ).
% distribution
thf(fact_51_Ax4_Ointros,axiom,
! [I: a,P: epistemic_fm_a] : ( epistemic_Ax4_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% Ax4.intros
thf(fact_52_Imp__intro,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( ( episte7081087998767065248c_fm_a @ M @ W @ P )
=> ( episte7081087998767065248c_fm_a @ M @ W @ Q ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ P @ Q ) ) ) ).
% Imp_intro
thf(fact_53_semantics_Osimps_I5_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ P @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M @ W @ P )
=> ( episte7081087998767065248c_fm_a @ M @ W @ Q ) ) ) ).
% semantics.simps(5)
thf(fact_54_semantics_Osimps_I1_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
~ ( episte7081087998767065248c_fm_a @ M @ W @ epistemic_FF_a ) ).
% semantics.simps(1)
thf(fact_55_semantics_Osimps_I4_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Con_a @ P @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M @ W @ P )
& ( episte7081087998767065248c_fm_a @ M @ W @ Q ) ) ) ).
% semantics.simps(4)
thf(fact_56_tautology,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [G3: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H3 @ P )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ).
% tautology
thf(fact_57_Ax4_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_Ax4_a @ A2 )
=> ~ ! [I3: a,P3: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P3 ) @ ( epistemic_K_a @ I3 @ ( epistemic_K_a @ I3 @ P3 ) ) ) ) ) ).
% Ax4.cases
thf(fact_58_Ax4_Osimps,axiom,
( epistemic_Ax4_a
= ( ^ [A4: epistemic_fm_a] :
? [I2: a,P2: epistemic_fm_a] :
( A4
= ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P2 ) @ ( epistemic_K_a @ I2 @ ( epistemic_K_a @ I2 @ P2 ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_59_KB4__5,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% KB4_5
thf(fact_60_K5__L,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% K5_L
thf(fact_61_K4__L,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% K4_L
thf(fact_62_S5_H__B,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% S5'_B
thf(fact_63_AxT_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_AxT_a @ A2 )
=> ~ ! [I3: a,P3: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P3 ) @ P3 ) ) ) ).
% AxT.cases
thf(fact_64_AxT_Osimps,axiom,
( epistemic_AxT_a
= ( ^ [A4: epistemic_fm_a] :
? [I2: a,P2: epistemic_fm_a] :
( A4
= ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P2 ) @ P2 ) ) ) ) ).
% AxT.simps
thf(fact_65_AxT_Ointros,axiom,
! [I: a,P: epistemic_fm_a] : ( epistemic_AxT_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ).
% AxT.intros
thf(fact_66_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ! [M2: episte94448284482925344t_unit,W2: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_67_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ! [M2: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_a @ W2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_68_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_69_generalization,axiom,
! [P: epistemic_fm_a,W: epistemic_fm_a,M: episte94448284482925344t_unit,I: a] :
( ! [M2: episte94448284482925344t_unit,X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X2 @ P ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_70_generalization,axiom,
! [P: epistemic_fm_a,W: a,M: episte6182337868402532512t_unit,I: a] :
( ! [M2: episte6182337868402532512t_unit,X2: a] :
( ( member_a @ X2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X2 @ P ) )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_71_generalization,axiom,
! [P: epistemic_fm_a,W: set_Epistemic_fm_a,M: episte1560738328020401952t_unit,I: a] :
( ! [M2: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_72_S5_H__4,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% S5'_4
thf(fact_73_T__L,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% T_L
thf(fact_74_predicate1I,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 ) ) ).
% predicate1I
thf(fact_75_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte4583239219080210381t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_76_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte2339904321507024205t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_77_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte2449151000174023629t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_78_order__refl,axiom,
! [X3: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X3 @ X3 ) ).
% order_refl
thf(fact_79_order__refl,axiom,
! [X3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X3 @ X3 ) ).
% order_refl
thf(fact_80_dual__order_Orefl,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_81_dual__order_Orefl,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_82_truth,axiom,
! [M: episte94448284482925344t_unit,W: epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte5633553332388074907t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ) ) ).
% truth
thf(fact_83_truth,axiom,
! [M: episte6182337868402532512t_unit,W: a,I: a,P: epistemic_fm_a] :
( ( episte8571156416534912283t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ) ) ).
% truth
thf(fact_84_truth,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte5648423998891577755t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ) ) ).
% truth
thf(fact_85_pos__introspection,axiom,
! [M: episte94448284482925344t_unit,W: epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_86_pos__introspection,axiom,
! [M: episte6182337868402532512t_unit,W: a,I: a,P: epistemic_fm_a] :
( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_87_pos__introspection,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_88_le__funD,axiom,
! [F: epistemic_fm_a > $o,G2: epistemic_fm_a > $o,X3: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ).
% le_funD
thf(fact_89_le__funE,axiom,
! [F: epistemic_fm_a > $o,G2: epistemic_fm_a > $o,X3: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ).
% le_funE
thf(fact_90_le__funI,axiom,
! [F: epistemic_fm_a > $o,G2: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_le4043730696559282883fm_a_o @ F @ G2 ) ) ).
% le_funI
thf(fact_91_le__fun__def,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [F2: epistemic_fm_a > $o,G4: epistemic_fm_a > $o] :
! [X: epistemic_fm_a] : ( ord_less_eq_o @ ( F2 @ X ) @ ( G4 @ X ) ) ) ) ).
% le_fun_def
thf(fact_92_order__antisym__conv,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_93_order__antisym__conv,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_94_ord__le__eq__subst,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_95_ord__le__eq__subst,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_96_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_97_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_98_ord__eq__le__subst,axiom,
! [A2: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_99_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_a,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_100_ord__eq__le__subst,axiom,
! [A2: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_101_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_102_order__eq__refl,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( X3 = Y )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_103_order__eq__refl,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( X3 = Y )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_104_order__subst2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ B ) @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_105_order__subst2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_106_order__subst2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ B ) @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_107_order__subst2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_108_order__subst1,axiom,
! [A2: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_109_order__subst1,axiom,
! [A2: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_110_order__subst1,axiom,
! [A2: set_Epistemic_fm_a,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_111_order__subst1,axiom,
! [A2: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_112_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: epistemic_fm_a > $o,Z: epistemic_fm_a > $o] : ( Y3 = Z ) )
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A4 @ B2 )
& ( ord_le4043730696559282883fm_a_o @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_113_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] : ( Y3 = Z ) )
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A4 @ B2 )
& ( ord_le3275665582123262618c_fm_a @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_114_antisym,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_115_antisym,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_116_dual__order_Otrans,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ C @ B )
=> ( ord_le4043730696559282883fm_a_o @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_117_dual__order_Otrans,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ C @ B )
=> ( ord_le3275665582123262618c_fm_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_118_dual__order_Oantisym,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_119_dual__order_Oantisym,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_120_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: epistemic_fm_a > $o,Z: epistemic_fm_a > $o] : ( Y3 = Z ) )
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B2 @ A4 )
& ( ord_le4043730696559282883fm_a_o @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_121_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] : ( Y3 = Z ) )
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B2 @ A4 )
& ( ord_le3275665582123262618c_fm_a @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_122_order__trans,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ Z2 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_123_order__trans,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ Z2 )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_124_order_Otrans,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% order.trans
thf(fact_125_order_Otrans,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_126_order__antisym,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_127_order__antisym,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_128_ord__le__eq__trans,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( B = C )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_129_ord__le__eq__trans,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( B = C )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_130_ord__eq__le__trans,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( A2 = B )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_131_ord__eq__le__trans,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A2 = B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_132_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: epistemic_fm_a > $o,Z: epistemic_fm_a > $o] : ( Y3 = Z ) )
= ( ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y4 )
& ( ord_le4043730696559282883fm_a_o @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_133_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] : ( Y3 = Z ) )
= ( ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y4 )
& ( ord_le3275665582123262618c_fm_a @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_134_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5633553332388074907t_unit @ M )
& ( episte2264091934940175238t_unit @ M ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_135_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M )
& ( episte4939069199465351174t_unit @ M ) )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_136_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_137_rev__predicate1D,axiom,
! [P4: epistemic_fm_a > $o,X3: epistemic_fm_a,Q4: epistemic_fm_a > $o] :
( ( P4 @ X3 )
=> ( ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 )
=> ( Q4 @ X3 ) ) ) ).
% rev_predicate1D
thf(fact_138_predicate1D,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o,X3: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 )
=> ( ( P4 @ X3 )
=> ( Q4 @ X3 ) ) ) ).
% predicate1D
thf(fact_139_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_140_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_141_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_142_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte5633553332388074907t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_143_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte8571156416534912283t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_144_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte5648423998891577755t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_145_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5633553332388074907t_unit @ M )
& ( episte2438601301504999334t_unit @ M )
& ( episte2264091934940175238t_unit @ M ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_146_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M )
& ( episte820475350133869606t_unit @ M )
& ( episte4939069199465351174t_unit @ M ) )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_147_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M )
& ( episte5478016696552465318t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_148_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte5633553332388074907t_unit @ M )
& ( episte2438601301504999334t_unit @ M )
& ( episte2264091934940175238t_unit @ M ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_149_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M )
& ( episte820475350133869606t_unit @ M )
& ( episte4939069199465351174t_unit @ M ) )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_150_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M )
& ( episte5478016696552465318t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_151_refl__Euclid__equiv,axiom,
! [M: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M )
=> ( ( episte2449151000174023629t_unit @ M )
=> ( ( episte5648423998891577755t_unit @ M )
& ( episte5478016696552465318t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) ) ) ) ).
% refl_Euclid_equiv
thf(fact_152_neg__introspection,axiom,
! [M: episte94448284482925344t_unit,W: epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte2438601301504999334t_unit @ M )
=> ( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_153_neg__introspection,axiom,
! [M: episte6182337868402532512t_unit,W: a,I: a,P: epistemic_fm_a] :
( ( episte820475350133869606t_unit @ M )
=> ( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_154_neg__introspection,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte5478016696552465318t_unit @ M )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_155_symm__trans__Euclid,axiom,
! [M: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( episte2449151000174023629t_unit @ M ) ) ) ).
% symm_trans_Euclid
thf(fact_156_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte2438601301504999334t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_157_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte820475350133869606t_unit @ M )
=> ( ( member_a @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_158_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte5478016696552465318t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_159_Euclidean__def,axiom,
( episte2449151000174023629t_unit
= ( ^ [M3: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ! [Z3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ( member536094252920883875c_fm_a @ Y4 @ ( episte6250069432388174439t_unit @ M3 @ I2 @ X ) )
=> ( ( member536094252920883875c_fm_a @ Z3 @ ( episte6250069432388174439t_unit @ M3 @ I2 @ X ) )
=> ( member536094252920883875c_fm_a @ Z3 @ ( episte6250069432388174439t_unit @ M3 @ I2 @ Y4 ) ) ) ) ) ) ) ) ) ).
% Euclidean_def
thf(fact_160_reflexive__def,axiom,
( episte5648423998891577755t_unit
= ( ^ [M3: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M3 @ I2 @ X ) ) ) ) ) ).
% reflexive_def
thf(fact_161_transitive__def,axiom,
( episte8364071018013720454t_unit
= ( ^ [M3: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ! [Z3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ( ( member536094252920883875c_fm_a @ Z3 @ ( episte6250069432388174439t_unit @ M3 @ I2 @ Y4 ) )
& ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M3 @ I2 @ Z3 ) ) )
=> ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M3 @ I2 @ Y4 ) ) ) ) ) ) ) ) ).
% transitive_def
thf(fact_162_Greatest__equality,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,X3: epistemic_fm_a > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 ) )
=> ( ( order_253494837916242250fm_a_o @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_163_Greatest__equality,axiom,
! [P4: set_Epistemic_fm_a > $o,X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X3 ) )
=> ( ( order_4585748725732241747c_fm_a @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_164_symmetric__def,axiom,
( episte5478016696552465318t_unit
= ( ^ [M3: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M3 @ I2 @ Y4 ) )
= ( member536094252920883875c_fm_a @ Y4 @ ( episte6250069432388174439t_unit @ M3 @ I2 @ X ) ) ) ) ) ) ) ).
% symmetric_def
thf(fact_165_GreatestI2__order,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,X3: epistemic_fm_a > $o,Q4: ( epistemic_fm_a > $o ) > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 ) )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( P4 @ X2 )
=> ( ! [Y5: epistemic_fm_a > $o] :
( ( P4 @ Y5 )
=> ( ord_le4043730696559282883fm_a_o @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_253494837916242250fm_a_o @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_166_GreatestI2__order,axiom,
! [P4: set_Epistemic_fm_a > $o,X3: set_Epistemic_fm_a,Q4: set_Epistemic_fm_a > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X3 ) )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( ! [Y5: set_Epistemic_fm_a] :
( ( P4 @ Y5 )
=> ( ord_le3275665582123262618c_fm_a @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_4585748725732241747c_fm_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_167_frame_Oequality,axiom,
! [R: episte1560738328020401952t_unit,R2: episte1560738328020401952t_unit] :
( ( ( episte8072386903178013299t_unit @ R )
= ( episte8072386903178013299t_unit @ R2 ) )
=> ( ( ( episte6250069432388174439t_unit @ R )
= ( episte6250069432388174439t_unit @ R2 ) )
=> ( ( ( episte3309513806868946049t_unit @ R )
= ( episte3309513806868946049t_unit @ R2 ) )
=> ( R = R2 ) ) ) ) ).
% frame.equality
thf(fact_168_semantics_Osimps_I6_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_K_a @ I @ P ) )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( inf_in1884693029477671368c_fm_a @ ( episte8072386903178013299t_unit @ M ) @ ( episte6250069432388174439t_unit @ M @ I @ W ) ) )
=> ( episte7081087998767065248c_fm_a @ M @ X @ P ) ) ) ) ).
% semantics.simps(6)
thf(fact_169_verit__comp__simplify1_I2_J,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_170_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_171_le__rel__bool__arg__iff,axiom,
( ord_le2986116216980733324fm_a_o
= ( ^ [X4: $o > epistemic_fm_a > $o,Y6: $o > epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le4043730696559282883fm_a_o @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_172_le__rel__bool__arg__iff,axiom,
( ord_le1382570192702674727c_fm_a
= ( ^ [X4: $o > set_Epistemic_fm_a,Y6: $o > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le3275665582123262618c_fm_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_173_frame_Ounfold__congs_I1_J,axiom,
! [R: episte1560738328020401952t_unit,R2: episte1560738328020401952t_unit,V: set_se5208064806568342746c_fm_a,F: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a,F3: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a] :
( ( R = R2 )
=> ( ( ( episte8072386903178013299t_unit @ R2 )
= V )
=> ( ! [V2: set_se5208064806568342746c_fm_a] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte8842944213534508900t_unit @ F @ R )
= ( episte8842944213534508900t_unit @ F3 @ R2 ) ) ) ) ) ).
% frame.unfold_congs(1)
thf(fact_174_frame_Ofold__congs_I1_J,axiom,
! [R: episte1560738328020401952t_unit,R2: episte1560738328020401952t_unit,V: set_se5208064806568342746c_fm_a,F: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a,F3: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a] :
( ( R = R2 )
=> ( ( ( episte8072386903178013299t_unit @ R2 )
= V )
=> ( ! [V2: set_se5208064806568342746c_fm_a] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F3 @ V2 ) ) )
=> ( ( episte8842944213534508900t_unit @ F @ R )
= ( episte8842944213534508900t_unit @ F3 @ R2 ) ) ) ) ) ).
% frame.fold_congs(1)
thf(fact_175_inf_Obounded__iff,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ ( inf_in3606484609122063093fm_a_o @ B @ C ) )
= ( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
& ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_176_inf_Obounded__iff,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ ( inf_in2882328776738922472c_fm_a @ B @ C ) )
= ( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
& ( ord_le3275665582123262618c_fm_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_177_le__inf__iff,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ ( inf_in3606484609122063093fm_a_o @ Y @ Z2 ) )
= ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
& ( ord_le4043730696559282883fm_a_o @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_178_le__inf__iff,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ ( inf_in2882328776738922472c_fm_a @ Y @ Z2 ) )
= ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
& ( ord_le3275665582123262618c_fm_a @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_179_inf__sup__ord_I2_J,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_180_inf__sup__ord_I2_J,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_181_inf__sup__ord_I1_J,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_182_inf__sup__ord_I1_J,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_183_inf__le1,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_184_inf__le1,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_185_inf__le2,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_186_inf__le2,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_187_le__infE,axiom,
! [X3: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) )
=> ~ ( ( ord_le4043730696559282883fm_a_o @ X3 @ A2 )
=> ~ ( ord_le4043730696559282883fm_a_o @ X3 @ B ) ) ) ).
% le_infE
thf(fact_188_le__infE,axiom,
! [X3: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ X3 @ A2 )
=> ~ ( ord_le3275665582123262618c_fm_a @ X3 @ B ) ) ) ).
% le_infE
thf(fact_189_le__infI,axiom,
! [X3: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ B )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_190_le__infI,axiom,
! [X3: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ B )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_191_inf__mono,axiom,
! [A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o,B: epistemic_fm_a > $o,D: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ C )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ D )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) @ ( inf_in3606484609122063093fm_a_o @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_192_inf__mono,axiom,
! [A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a,B: set_Epistemic_fm_a,D: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ C )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ D )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) @ ( inf_in2882328776738922472c_fm_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_193_inf_OcoboundedI2,axiom,
! [B: epistemic_fm_a > $o,C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_194_inf_OcoboundedI2,axiom,
! [B: set_Epistemic_fm_a,C: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_195_inf_OcoboundedI1,axiom,
! [A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ C )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_196_inf_OcoboundedI1,axiom,
! [A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ C )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_197_inf_Oabsorb__iff2,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [B2: epistemic_fm_a > $o,A4: epistemic_fm_a > $o] :
( ( inf_in3606484609122063093fm_a_o @ A4 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_198_inf_Oabsorb__iff2,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [B2: set_Epistemic_fm_a,A4: set_Epistemic_fm_a] :
( ( inf_in2882328776738922472c_fm_a @ A4 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_199_inf_Oabsorb__iff1,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( inf_in3606484609122063093fm_a_o @ A4 @ B2 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_200_inf_Oabsorb__iff1,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( inf_in2882328776738922472c_fm_a @ A4 @ B2 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_201_inf_Ocobounded2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_202_inf_Ocobounded2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_203_inf_Ocobounded1,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_204_inf_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_205_inf_Oorder__iff,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( A4
= ( inf_in3606484609122063093fm_a_o @ A4 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_206_inf_Oorder__iff,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( A4
= ( inf_in2882328776738922472c_fm_a @ A4 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_207_inf__greatest,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Z2 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( inf_in3606484609122063093fm_a_o @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_208_inf__greatest,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Z2 )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( inf_in2882328776738922472c_fm_a @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_209_inf_OboundedI,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ A2 @ C )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( inf_in3606484609122063093fm_a_o @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_210_inf_OboundedI,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ A2 @ C )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( inf_in2882328776738922472c_fm_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_211_inf_OboundedE,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ ( inf_in3606484609122063093fm_a_o @ B @ C ) )
=> ~ ( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ~ ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_212_inf_OboundedE,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ ( inf_in2882328776738922472c_fm_a @ B @ C ) )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ~ ( ord_le3275665582123262618c_fm_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_213_inf__absorb2,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( ( inf_in3606484609122063093fm_a_o @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_214_inf__absorb2,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( ( inf_in2882328776738922472c_fm_a @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_215_inf__absorb1,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( inf_in3606484609122063093fm_a_o @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_216_inf__absorb1,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( inf_in2882328776738922472c_fm_a @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_217_inf_Oabsorb2,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( inf_in3606484609122063093fm_a_o @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_218_inf_Oabsorb2,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( inf_in2882328776738922472c_fm_a @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_219_inf_Oabsorb1,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( inf_in3606484609122063093fm_a_o @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_220_inf_Oabsorb1,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( inf_in2882328776738922472c_fm_a @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_221_le__iff__inf,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] :
( ( inf_in3606484609122063093fm_a_o @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_222_le__iff__inf,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( inf_in2882328776738922472c_fm_a @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_223_inf__unique,axiom,
! [F: ( epistemic_fm_a > $o ) > ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o,Z4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ( ord_le4043730696559282883fm_a_o @ X2 @ Z4 )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_in3606484609122063093fm_a_o @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_224_inf__unique,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ Z4 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_in2882328776738922472c_fm_a @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_225_inf_OorderI,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( A2
= ( inf_in3606484609122063093fm_a_o @ A2 @ B ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ B ) ) ).
% inf.orderI
thf(fact_226_inf_OorderI,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( A2
= ( inf_in2882328776738922472c_fm_a @ A2 @ B ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ B ) ) ).
% inf.orderI
thf(fact_227_inf_OorderE,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( A2
= ( inf_in3606484609122063093fm_a_o @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_228_inf_OorderE,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( A2
= ( inf_in2882328776738922472c_fm_a @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_229_le__infI2,axiom,
! [B: epistemic_fm_a > $o,X3: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ X3 )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_230_le__infI2,axiom,
! [B: set_Epistemic_fm_a,X3: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ X3 )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_231_le__infI1,axiom,
! [A2: epistemic_fm_a > $o,X3: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ X3 )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_232_le__infI1,axiom,
! [A2: set_Epistemic_fm_a,X3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ X3 )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_233_frame_Osurjective,axiom,
! [R: episte1560738328020401952t_unit] :
( R
= ( episte2888590659910966568t_unit @ ( episte8072386903178013299t_unit @ R ) @ ( episte6250069432388174439t_unit @ R ) @ ( episte3309513806868946049t_unit @ R ) ) ) ).
% frame.surjective
thf(fact_234_kripke_Oequality,axiom,
! [R: episte1560738328020401952t_unit,R2: episte1560738328020401952t_unit] :
( ( ( episte8072386903178013299t_unit @ R )
= ( episte8072386903178013299t_unit @ R2 ) )
=> ( ( ( episte6250069432388174439t_unit @ R )
= ( episte6250069432388174439t_unit @ R2 ) )
=> ( ( ( episte2398645135750866164t_unit @ R )
= ( episte2398645135750866164t_unit @ R2 ) )
=> ( ( ( episte5479201149095757850t_unit @ R )
= ( episte5479201149095757850t_unit @ R2 ) )
=> ( R = R2 ) ) ) ) ) ).
% kripke.equality
thf(fact_235_fm_Opred__mono,axiom,
! [P4: a > $o,Pa: a > $o] :
( ( ord_less_eq_a_o @ P4 @ Pa )
=> ( ord_le4043730696559282883fm_a_o @ ( epistemic_pred_fm_a @ P4 ) @ ( epistemic_pred_fm_a @ Pa ) ) ) ).
% fm.pred_mono
thf(fact_236_fm_Opred__mono,axiom,
! [P4: epistemic_fm_a > $o,Pa: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ P4 @ Pa )
=> ( ord_le7240545311762076327fm_a_o @ ( episte3114992851182798545c_fm_a @ P4 ) @ ( episte3114992851182798545c_fm_a @ Pa ) ) ) ).
% fm.pred_mono
thf(fact_237_strong__soundness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M4 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>5
thf(fact_238_strong__soundness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M4 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>B
thf(fact_239_strong__soundness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M4 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>4
thf(fact_240_strong__soundness_092_060_094sub_062T,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M4 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>T
thf(fact_241_frame_Oext__inject,axiom,
! [W3: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit,W4: set_se5208064806568342746c_fm_a,K2: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More2: episte1193835314949844379t_unit] :
( ( ( episte2888590659910966568t_unit @ W3 @ K @ More )
= ( episte2888590659910966568t_unit @ W4 @ K2 @ More2 ) )
= ( ( W3 = W4 )
& ( K = K2 )
& ( More = More2 ) ) ) ).
% frame.ext_inject
thf(fact_242_fm_Opred__inject_I5_J,axiom,
! [P4: a > $o,A2: epistemic_fm_a,Aa: epistemic_fm_a] :
( ( epistemic_pred_fm_a @ P4 @ ( epistemic_Imp_a @ A2 @ Aa ) )
= ( ( epistemic_pred_fm_a @ P4 @ A2 )
& ( epistemic_pred_fm_a @ P4 @ Aa ) ) ) ).
% fm.pred_inject(5)
thf(fact_243_fm_Opred__inject_I6_J,axiom,
! [P4: a > $o,A2: a,Aa: epistemic_fm_a] :
( ( epistemic_pred_fm_a @ P4 @ ( epistemic_K_a @ A2 @ Aa ) )
= ( ( P4 @ A2 )
& ( epistemic_pred_fm_a @ P4 @ Aa ) ) ) ).
% fm.pred_inject(6)
thf(fact_244_fm_Opred__inject_I4_J,axiom,
! [P4: a > $o,A2: epistemic_fm_a,Aa: epistemic_fm_a] :
( ( epistemic_pred_fm_a @ P4 @ ( epistemic_Con_a @ A2 @ Aa ) )
= ( ( epistemic_pred_fm_a @ P4 @ A2 )
& ( epistemic_pred_fm_a @ P4 @ Aa ) ) ) ).
% fm.pred_inject(4)
thf(fact_245_frame_Oupdate__convs_I3_J,axiom,
! [More2: episte1193835314949844379t_unit > episte1193835314949844379t_unit,W3: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte7964699311158398698c_fm_a @ More2 @ ( episte2888590659910966568t_unit @ W3 @ K @ More ) )
= ( episte2888590659910966568t_unit @ W3 @ K @ ( More2 @ More ) ) ) ).
% frame.update_convs(3)
thf(fact_246_frame_Ocases__scheme,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W5: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More3: episte1193835314949844379t_unit] :
( R
!= ( episte2888590659910966568t_unit @ W5 @ K3 @ More3 ) ) ).
% frame.cases_scheme
thf(fact_247_K__imply__weaken,axiom,
! [A: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a,Ps2: list_Epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ Q ) )
=> ( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Ps ) @ ( set_Epistemic_fm_a2 @ Ps2 ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps2 @ Q ) ) ) ) ).
% K_imply_weaken
thf(fact_248_strong__completeness_092_060_094sub_062T,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs2 @ P ) ) ) ) ).
% strong_completeness\<^sub>T
thf(fact_249_main_092_060_094sub_062T,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M3: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M3 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>T
thf(fact_250_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_a,Qs4: list_Epistemic_fm_a,R: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Ps ) @ ( set_Epistemic_fm_a2 @ Qs4 ) )
=> ! [G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ ( epistemic_Imp_a @ ( epistemic_imply_a @ Ps @ R ) @ ( epistemic_imply_a @ Qs4 @ R ) ) ) ) ).
% tautology_imply_superset
thf(fact_251_strong__completeness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs2 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>4
thf(fact_252_main_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M3: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M3 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>4
thf(fact_253_strong__completeness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs2 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>5
thf(fact_254_main_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M3: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M3 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>5
thf(fact_255_strong__completeness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs2 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>B
thf(fact_256_main_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M3: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M3 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>B
thf(fact_257_fm_Opred__inject_I1_J,axiom,
! [P4: a > $o] : ( epistemic_pred_fm_a @ P4 @ epistemic_FF_a ) ).
% fm.pred_inject(1)
thf(fact_258_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte94448284482925344t_unit,W2: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte94448284482925344t_unit] :
( ( P4 @ M4 )
=> ! [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ ( episte6390737319716712051t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte5333283044364550848c_fm_a @ M4 @ X5 @ Xa ) )
=> ( episte5333283044364550848c_fm_a @ M4 @ X5 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_259_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_a @ W2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte6182337868402532512t_unit] :
( ( P4 @ M4 )
=> ! [X5: a] :
( ( member_a @ X5 @ ( episte6926715892928323059t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte295617885132580260cs_a_a @ M4 @ X5 @ Xa ) )
=> ( episte295617885132580260cs_a_a @ M4 @ X5 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_260_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte1560738328020401952t_unit] :
( ( P4 @ M4 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_261_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte94448284482925344t_unit,W2: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M4: episte94448284482925344t_unit] :
( ( P4 @ M4 )
=> ! [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ ( episte6390737319716712051t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte5333283044364550848c_fm_a @ M4 @ X5 @ Xa ) )
=> ( episte5333283044364550848c_fm_a @ M4 @ X5 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_262_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_a @ W2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M4: episte6182337868402532512t_unit] :
( ( P4 @ M4 )
=> ! [X5: a] :
( ( member_a @ X5 @ ( episte6926715892928323059t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte295617885132580260cs_a_a @ M4 @ X5 @ Xa ) )
=> ( episte295617885132580260cs_a_a @ M4 @ X5 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_263_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M4: episte1560738328020401952t_unit] :
( ( P4 @ M4 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_264_frame_Oselect__convs_I1_J,axiom,
! [W3: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte8072386903178013299t_unit @ ( episte2888590659910966568t_unit @ W3 @ K @ More ) )
= W3 ) ).
% frame.select_convs(1)
thf(fact_265_frame_Oselect__convs_I2_J,axiom,
! [W3: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte6250069432388174439t_unit @ ( episte2888590659910966568t_unit @ W3 @ K @ More ) )
= K ) ).
% frame.select_convs(2)
thf(fact_266_K__right__mp,axiom,
! [A: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ ( epistemic_Imp_a @ P @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ Q ) ) ) ) ).
% K_right_mp
thf(fact_267_frame_Oselect__convs_I3_J,axiom,
! [W3: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte3309513806868946049t_unit @ ( episte2888590659910966568t_unit @ W3 @ K @ More ) )
= More ) ).
% frame.select_convs(3)
thf(fact_268_frame_Oupdate__convs_I2_J,axiom,
! [K2: ( a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a ) > a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,W3: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte7087520964475219152t_unit @ K2 @ ( episte2888590659910966568t_unit @ W3 @ K @ More ) )
= ( episte2888590659910966568t_unit @ W3 @ ( K2 @ K ) @ More ) ) ).
% frame.update_convs(2)
thf(fact_269_frame_Oupdate__convs_I1_J,axiom,
! [W4: set_se5208064806568342746c_fm_a > set_se5208064806568342746c_fm_a,W3: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte8842944213534508900t_unit @ W4 @ ( episte2888590659910966568t_unit @ W3 @ K @ More ) )
= ( episte2888590659910966568t_unit @ ( W4 @ W3 ) @ K @ More ) ) ).
% frame.update_convs(1)
thf(fact_270_kripke_Osurjective,axiom,
! [R: episte1560738328020401952t_unit] :
( R
= ( episte2888590659910966568t_unit @ ( episte8072386903178013299t_unit @ R ) @ ( episte6250069432388174439t_unit @ R ) @ ( episte8239586592105053771t_unit @ ( episte2398645135750866164t_unit @ R ) @ ( episte5479201149095757850t_unit @ R ) ) ) ) ).
% kripke.surjective
thf(fact_271_strong__soundness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte5478016696552465318t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5
thf(fact_272_strong__soundness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte5478016696552465318t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5'
thf(fact_273_strong__soundness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>4
thf(fact_274_consistent__def,axiom,
( episte2285483198712856226tent_a
= ( ^ [A3: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
~ ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ S )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs3 @ epistemic_FF_a ) ) ) ) ) ).
% consistent_def
thf(fact_275_K__Boole,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G5: list_Epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ G5 ) @ epistemic_FF_a ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ P ) ) ) ).
% K_Boole
thf(fact_276_S5__S5_H__assms,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% S5_S5'_assms
thf(fact_277_kripke_Oext__inject,axiom,
! [Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit,Pi2: set_Epistemic_fm_a > list_char > $o,More2: product_unit] :
( ( ( episte8239586592105053771t_unit @ Pi @ More )
= ( episte8239586592105053771t_unit @ Pi2 @ More2 ) )
= ( ( Pi = Pi2 )
& ( More = More2 ) ) ) ).
% kripke.ext_inject
thf(fact_278_kripke_Oupdate__convs_I2_J,axiom,
! [More2: product_unit > product_unit,W3: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte9120385895580347753c_fm_a @ More2 @ ( episte2888590659910966568t_unit @ W3 @ K @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= ( episte2888590659910966568t_unit @ W3 @ K @ ( episte8239586592105053771t_unit @ Pi @ ( More2 @ More ) ) ) ) ).
% kripke.update_convs(2)
thf(fact_279_kripke_Ocases__scheme,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W5: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi3: set_Epistemic_fm_a > list_char > $o,More3: product_unit] :
( R
!= ( episte2888590659910966568t_unit @ W5 @ K3 @ ( episte8239586592105053771t_unit @ Pi3 @ More3 ) ) ) ).
% kripke.cases_scheme
thf(fact_280_fm_Opred__True,axiom,
( ( epistemic_pred_fm_a
@ ^ [Uu2: a] : $true )
= ( ^ [Uu2: epistemic_fm_a] : $true ) ) ).
% fm.pred_True
thf(fact_281_kripke_Oext__induct,axiom,
! [P4: episte1193835314949844379t_unit > $o,R: episte1193835314949844379t_unit] :
( ! [Pi3: set_Epistemic_fm_a > list_char > $o,More3: product_unit] : ( P4 @ ( episte8239586592105053771t_unit @ Pi3 @ More3 ) )
=> ( P4 @ R ) ) ).
% kripke.ext_induct
thf(fact_282_consistent__hereditary,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a,S3: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S2 )
=> ( ( ord_le3275665582123262618c_fm_a @ S3 @ S2 )
=> ( episte2285483198712856226tent_a @ A @ S3 ) ) ) ).
% consistent_hereditary
thf(fact_283_main_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M3: episte1560738328020401952t_unit,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>K
thf(fact_284_strong__completeness_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) )
=> ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G5 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs2 @ P ) ) ) ) ).
% strong_completeness\<^sub>K
thf(fact_285_imply_Osimps_I2_J,axiom,
! [P: epistemic_fm_a,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ Q )
= ( epistemic_Imp_a @ P @ ( epistemic_imply_a @ Ps @ Q ) ) ) ).
% imply.simps(2)
thf(fact_286_K__imply__head,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Ps: list_Epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ P ) ) ).
% K_imply_head
thf(fact_287_K__imply__Cons,axiom,
! [A: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ Q ) ) ) ).
% K_imply_Cons
thf(fact_288_K__swap,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G5: list_Epistemic_fm_a,R: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ ( cons_Epistemic_fm_a @ Q @ G5 ) ) @ R ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ ( cons_Epistemic_fm_a @ P @ G5 ) ) @ R ) ) ) ).
% K_swap
thf(fact_289_kripke_Oselect__convs_I1_J,axiom,
! [W3: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte2398645135750866164t_unit @ ( episte2888590659910966568t_unit @ W3 @ K @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= Pi ) ).
% kripke.select_convs(1)
thf(fact_290_kripke_Oselect__convs_I2_J,axiom,
! [W3: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte5479201149095757850t_unit @ ( episte2888590659910966568t_unit @ W3 @ K @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= More ) ).
% kripke.select_convs(2)
thf(fact_291_S5__S5_H,axiom,
! [P: epistemic_fm_a] :
( ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ P )
=> ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ P ) ) ).
% S5_S5'
thf(fact_292_S5_H__S5,axiom,
! [P: epistemic_fm_a] :
( ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ P )
=> ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ P ) ) ).
% S5'_S5
thf(fact_293_strong__soundness_092_060_094sub_062K,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a,P4: episte1560738328020401952t_unit > $o] :
( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G5 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs @ P ) ) )
=> ! [M4: episte1560738328020401952t_unit] :
( ( P4 @ M4 )
=> ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Xa: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa @ G5 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ Xa ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X5 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K
thf(fact_294_strong__completeness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs2 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>4
thf(fact_295_main_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>4
thf(fact_296_strong__completeness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte5478016696552465318t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs2 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5'
thf(fact_297_main_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>5'
thf(fact_298_K__ImpI,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G5: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G5 ) @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ ( epistemic_Imp_a @ P @ Q ) ) ) ) ).
% K_ImpI
thf(fact_299_K__mp,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G5: list_Epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P @ Q ) @ G5 ) ) @ Q ) ) ).
% K_mp
thf(fact_300_main_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>5
thf(fact_301_strong__completeness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G5: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte5478016696552465318t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G5 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs2 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5
thf(fact_302_pred__subset__eq,axiom,
! [R3: set_se5208064806568342746c_fm_a,S2: set_se5208064806568342746c_fm_a] :
( ( ord_le1568123225352354019fm_a_o
@ ^ [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ R3 )
@ ^ [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ S2 ) )
= ( ord_le7112219575281605754c_fm_a @ R3 @ S2 ) ) ).
% pred_subset_eq
thf(fact_303_pred__subset__eq,axiom,
! [R3: set_a,S2: set_a] :
( ( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ R3 )
@ ^ [X: a] : ( member_a @ X @ S2 ) )
= ( ord_less_eq_set_a @ R3 @ S2 ) ) ).
% pred_subset_eq
thf(fact_304_pred__subset__eq,axiom,
! [R3: set_Epistemic_fm_a,S2: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ R3 )
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ S2 ) )
= ( ord_le3275665582123262618c_fm_a @ R3 @ S2 ) ) ).
% pred_subset_eq
thf(fact_305_less__eq__set__def,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,B3: set_se5208064806568342746c_fm_a] :
( ord_le1568123225352354019fm_a_o
@ ^ [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ A3 )
@ ^ [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_306_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A3 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_307_less__eq__set__def,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ A3 )
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_308_AxB__symmetric_H,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a,W6: set_Epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V3 )
=> ( ( episte2285483198712856226tent_a @ A @ W6 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W6 )
=> ( ( member536094252920883875c_fm_a @ W6
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V3 ) ) ) ) )
=> ( member536094252920883875c_fm_a @ V3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ W6 ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_309_Ax5__Euclidean,axiom,
! [A: epistemic_fm_a > $o,U: set_Epistemic_fm_a,V3: set_Epistemic_fm_a,W6: set_Epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ U )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ U )
=> ( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V3 )
=> ( ( episte2285483198712856226tent_a @ A @ W6 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W6 )
=> ( ( member536094252920883875c_fm_a @ V3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ U ) ) ) ) )
=> ( ( member536094252920883875c_fm_a @ W6
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ U ) ) ) ) )
=> ( member536094252920883875c_fm_a @ W6
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_310_Ax4__transitive,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a,W6: set_Epistemic_fm_a,I: a,U: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V3 )
=> ( ( member536094252920883875c_fm_a @ W6
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V3 ) ) ) ) )
=> ( ( member536094252920883875c_fm_a @ U
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ W6 ) ) ) ) )
=> ( member536094252920883875c_fm_a @ U
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V3 ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_311_AxT__reflexive,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V3 )
=> ( member536094252920883875c_fm_a @ V3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V3 ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_312_kripke_Oupdate__convs_I1_J,axiom,
! [Pi2: ( set_Epistemic_fm_a > list_char > $o ) > set_Epistemic_fm_a > list_char > $o,W3: set_se5208064806568342746c_fm_a,K: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte1857908288096881731t_unit @ Pi2 @ ( episte2888590659910966568t_unit @ W3 @ K @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= ( episte2888590659910966568t_unit @ W3 @ K @ ( episte8239586592105053771t_unit @ ( Pi2 @ Pi ) @ More ) ) ) ).
% kripke.update_convs(1)
thf(fact_313_K__DisL,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a,P5: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ Q ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P5 @ Ps ) @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ ( epistemic_Dis_a @ P @ P5 ) @ Ps ) @ Q ) ) ) ) ).
% K_DisL
thf(fact_314_fm_Oinject_I2_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( ( epistemic_Dis_a @ X31 @ X32 )
= ( epistemic_Dis_a @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fm.inject(2)
thf(fact_315_fm_Opred__inject_I3_J,axiom,
! [P4: a > $o,A2: epistemic_fm_a,Aa: epistemic_fm_a] :
( ( epistemic_pred_fm_a @ P4 @ ( epistemic_Dis_a @ A2 @ Aa ) )
= ( ( epistemic_pred_fm_a @ P4 @ A2 )
& ( epistemic_pred_fm_a @ P4 @ Aa ) ) ) ).
% fm.pred_inject(3)
thf(fact_316_fm_Odistinct_I21_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Dis_a @ X31 @ X32 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(21)
thf(fact_317_fm_Odistinct_I23_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Dis_a @ X31 @ X32 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(23)
thf(fact_318_fm_Odistinct_I3_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.distinct(3)
thf(fact_319_semantics_Osimps_I3_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Dis_a @ P @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M @ W @ P )
| ( episte7081087998767065248c_fm_a @ M @ W @ Q ) ) ) ).
% semantics.simps(3)
thf(fact_320_fm_Odistinct_I19_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( epistemic_Dis_a @ X31 @ X32 )
!= ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.distinct(19)
thf(fact_321_ax__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V3 )
=> ( ( A @ P )
=> ( member6642669571620171971c_fm_a @ P @ V3 ) ) ) ) ).
% ax_in_maximal
thf(fact_322_eval_Osimps_I3_J,axiom,
! [G2: list_char > $o,H2: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G2 @ H2 @ ( epistemic_Dis_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G2 @ H2 @ P )
| ( epistemic_eval_a @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(3)
thf(fact_323_consequent__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V3 )
=> ( ( member6642669571620171971c_fm_a @ P @ V3 )
=> ( ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ Q ) @ V3 )
=> ( member6642669571620171971c_fm_a @ Q @ V3 ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_324_deriv__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V3 )
=> ( ( epistemic_AK_a @ A @ P )
=> ( member6642669571620171971c_fm_a @ P @ V3 ) ) ) ) ).
% deriv_in_maximal
thf(fact_325_maximal__extension,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V3 )
=> ~ ! [W7: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ V3 @ W7 )
=> ( ( episte2285483198712856226tent_a @ A @ W7 )
=> ~ ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W7 ) ) ) ) ).
% maximal_extension
thf(fact_326_exactly__one__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V3 )
=> ( ( member6642669571620171971c_fm_a @ P @ V3 )
= ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ V3 ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_327_K__DisE,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G5: list_Epistemic_fm_a,R: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G5 ) @ R ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ G5 ) @ R ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ ( epistemic_Dis_a @ P @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ R ) ) ) ) ) ).
% K_DisE
thf(fact_328_fm_Oexhaust,axiom,
! [Y: epistemic_fm_a] :
( ( Y != epistemic_FF_a )
=> ( ! [X22: list_char] :
( Y
!= ( epistemic_Pro_a @ X22 ) )
=> ( ! [X312: epistemic_fm_a,X322: epistemic_fm_a] :
( Y
!= ( epistemic_Dis_a @ X312 @ X322 ) )
=> ( ! [X412: epistemic_fm_a,X422: epistemic_fm_a] :
( Y
!= ( epistemic_Con_a @ X412 @ X422 ) )
=> ( ! [X512: epistemic_fm_a,X522: epistemic_fm_a] :
( Y
!= ( epistemic_Imp_a @ X512 @ X522 ) )
=> ~ ! [X612: a,X622: epistemic_fm_a] :
( Y
!= ( epistemic_K_a @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_329_strong__completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,G5: set_Epistemic_fm_a,P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G5 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W8: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W8 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W8 ) ) )
@ ^ [I2: a,V4: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V4 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V4: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V4 )
@ product_Unity ) ) )
=> ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G5 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs2 @ P ) ) ) ) ) ).
% strong_completeness
thf(fact_330_Euclidean_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( episte2449151000174023629t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W8: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W8 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W8 ) ) )
@ ^ [I2: a,V4: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V4 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V4: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V4 )
@ product_Unity ) ) ) ) ).
% Euclidean\<^sub>K\<^sub>5
thf(fact_331_fm_Oset__cases,axiom,
! [E: set_Epistemic_fm_a,A2: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ A2 ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a] :
( ? [Z22: episte5999154429349999510c_fm_a] :
( A2
= ( episte9190102072588321003c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A2
= ( episte9190102072588321003c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a] :
( ? [Z22: episte5999154429349999510c_fm_a] :
( A2
= ( episte8719284489194433015c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A2
= ( episte8719284489194433015c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a] :
( ? [Z22: episte5999154429349999510c_fm_a] :
( A2
= ( episte5200161025749276397c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A2
= ( episte5200161025749276397c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) )
=> ( ! [Z22: episte5999154429349999510c_fm_a] :
( A2
!= ( episte3013695276341575662c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A2
= ( episte3013695276341575662c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_332_fm_Oset__cases,axiom,
! [E: epistemic_fm_a,A2: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ A2 ) )
=> ( ! [Z1: episte740340785640729014c_fm_a] :
( ? [Z22: episte740340785640729014c_fm_a] :
( A2
= ( episte6088726764479022859c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A2
= ( episte6088726764479022859c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a] :
( ? [Z22: episte740340785640729014c_fm_a] :
( A2
= ( episte3685526487207141399c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A2
= ( episte3685526487207141399c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a] :
( ? [Z22: episte740340785640729014c_fm_a] :
( A2
= ( episte260752218777527565c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A2
= ( episte260752218777527565c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) )
=> ( ! [Z22: episte740340785640729014c_fm_a] :
( A2
!= ( episte5657488632024175118c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A2
= ( episte5657488632024175118c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_333_fm_Oset__cases,axiom,
! [E: a,A2: epistemic_fm_a] :
( ( member_a @ E @ ( epistemic_set_fm_a @ A2 ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Dis_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Dis_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Con_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Con_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Imp_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Imp_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z22: epistemic_fm_a] :
( A2
!= ( epistemic_K_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_K_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( epistemic_set_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_334_symmetric_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( episte5478016696552465318t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W8: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W8 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W8 ) ) )
@ ^ [I2: a,V4: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V4 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V4: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V4 )
@ product_Unity ) ) ) ) ).
% symmetric\<^sub>K\<^sub>B
thf(fact_335_transitive_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( episte8364071018013720454t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W8: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W8 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W8 ) ) )
@ ^ [I2: a,V4: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V4 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V4: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V4 )
@ product_Unity ) ) ) ) ).
% transitive\<^sub>K\<^sub>4
thf(fact_336_fm_Oinject_I1_J,axiom,
! [X23: list_char,Y22: list_char] :
( ( ( epistemic_Pro_a @ X23 )
= ( epistemic_Pro_a @ Y22 ) )
= ( X23 = Y22 ) ) ).
% fm.inject(1)
thf(fact_337_kripke_Ocases,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W5: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi3: set_Epistemic_fm_a > list_char > $o] :
( R
!= ( episte2888590659910966568t_unit @ W5 @ K3 @ ( episte8239586592105053771t_unit @ Pi3 @ product_Unity ) ) ) ).
% kripke.cases
thf(fact_338_fm_Oset__intros_I5_J,axiom,
! [Ye: set_Epistemic_fm_a,X51: episte5999154429349999510c_fm_a,X52: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Ye @ ( episte6227279731293865264c_fm_a @ X51 ) )
=> ( member536094252920883875c_fm_a @ Ye @ ( episte6227279731293865264c_fm_a @ ( episte5200161025749276397c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_339_fm_Oset__intros_I5_J,axiom,
! [Ye: epistemic_fm_a,X51: episte740340785640729014c_fm_a,X52: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Ye @ ( episte9089240958480457552c_fm_a @ X51 ) )
=> ( member6642669571620171971c_fm_a @ Ye @ ( episte9089240958480457552c_fm_a @ ( episte260752218777527565c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_340_fm_Oset__intros_I5_J,axiom,
! [Ye: a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( member_a @ Ye @ ( epistemic_set_fm_a @ X51 ) )
=> ( member_a @ Ye @ ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_341_fm_Oset__intros_I6_J,axiom,
! [Yf: set_Epistemic_fm_a,X52: episte5999154429349999510c_fm_a,X51: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yf @ ( episte6227279731293865264c_fm_a @ X52 ) )
=> ( member536094252920883875c_fm_a @ Yf @ ( episte6227279731293865264c_fm_a @ ( episte5200161025749276397c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_342_fm_Oset__intros_I6_J,axiom,
! [Yf: epistemic_fm_a,X52: episte740340785640729014c_fm_a,X51: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yf @ ( episte9089240958480457552c_fm_a @ X52 ) )
=> ( member6642669571620171971c_fm_a @ Yf @ ( episte9089240958480457552c_fm_a @ ( episte260752218777527565c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_343_fm_Oset__intros_I6_J,axiom,
! [Yf: a,X52: epistemic_fm_a,X51: epistemic_fm_a] :
( ( member_a @ Yf @ ( epistemic_set_fm_a @ X52 ) )
=> ( member_a @ Yf @ ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_344_fm_Oset__intros_I7_J,axiom,
! [X61: set_Epistemic_fm_a,X62: episte5999154429349999510c_fm_a] : ( member536094252920883875c_fm_a @ X61 @ ( episte6227279731293865264c_fm_a @ ( episte3013695276341575662c_fm_a @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_345_fm_Oset__intros_I7_J,axiom,
! [X61: epistemic_fm_a,X62: episte740340785640729014c_fm_a] : ( member6642669571620171971c_fm_a @ X61 @ ( episte9089240958480457552c_fm_a @ ( episte5657488632024175118c_fm_a @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_346_fm_Oset__intros_I7_J,axiom,
! [X61: a,X62: epistemic_fm_a] : ( member_a @ X61 @ ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_347_fm_Oset__intros_I8_J,axiom,
! [Yg: set_Epistemic_fm_a,X62: episte5999154429349999510c_fm_a,X61: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Yg @ ( episte6227279731293865264c_fm_a @ X62 ) )
=> ( member536094252920883875c_fm_a @ Yg @ ( episte6227279731293865264c_fm_a @ ( episte3013695276341575662c_fm_a @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_348_fm_Oset__intros_I8_J,axiom,
! [Yg: epistemic_fm_a,X62: episte740340785640729014c_fm_a,X61: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Yg @ ( episte9089240958480457552c_fm_a @ X62 ) )
=> ( member6642669571620171971c_fm_a @ Yg @ ( episte9089240958480457552c_fm_a @ ( episte5657488632024175118c_fm_a @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_349_fm_Oset__intros_I8_J,axiom,
! [Yg: a,X62: epistemic_fm_a,X61: a] :
( ( member_a @ Yg @ ( epistemic_set_fm_a @ X62 ) )
=> ( member_a @ Yg @ ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_350_fm_Odistinct_I15_J,axiom,
! [X23: list_char,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Pro_a @ X23 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_351_fm_Odistinct_I17_J,axiom,
! [X23: list_char,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Pro_a @ X23 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_352_fm_Odistinct_I1_J,axiom,
! [X23: list_char] :
( epistemic_FF_a
!= ( epistemic_Pro_a @ X23 ) ) ).
% fm.distinct(1)
thf(fact_353_fm_Oset__intros_I3_J,axiom,
! [Yc: set_Epistemic_fm_a,X41: episte5999154429349999510c_fm_a,X42: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yc @ ( episte6227279731293865264c_fm_a @ X41 ) )
=> ( member536094252920883875c_fm_a @ Yc @ ( episte6227279731293865264c_fm_a @ ( episte8719284489194433015c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_354_fm_Oset__intros_I3_J,axiom,
! [Yc: epistemic_fm_a,X41: episte740340785640729014c_fm_a,X42: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yc @ ( episte9089240958480457552c_fm_a @ X41 ) )
=> ( member6642669571620171971c_fm_a @ Yc @ ( episte9089240958480457552c_fm_a @ ( episte3685526487207141399c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_355_fm_Oset__intros_I3_J,axiom,
! [Yc: a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( member_a @ Yc @ ( epistemic_set_fm_a @ X41 ) )
=> ( member_a @ Yc @ ( epistemic_set_fm_a @ ( epistemic_Con_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_356_fm_Oset__intros_I4_J,axiom,
! [Yd: set_Epistemic_fm_a,X42: episte5999154429349999510c_fm_a,X41: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yd @ ( episte6227279731293865264c_fm_a @ X42 ) )
=> ( member536094252920883875c_fm_a @ Yd @ ( episte6227279731293865264c_fm_a @ ( episte8719284489194433015c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_357_fm_Oset__intros_I4_J,axiom,
! [Yd: epistemic_fm_a,X42: episte740340785640729014c_fm_a,X41: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yd @ ( episte9089240958480457552c_fm_a @ X42 ) )
=> ( member6642669571620171971c_fm_a @ Yd @ ( episte9089240958480457552c_fm_a @ ( episte3685526487207141399c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_358_fm_Oset__intros_I4_J,axiom,
! [Yd: a,X42: epistemic_fm_a,X41: epistemic_fm_a] :
( ( member_a @ Yd @ ( epistemic_set_fm_a @ X42 ) )
=> ( member_a @ Yd @ ( epistemic_set_fm_a @ ( epistemic_Con_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_359_semantics_Osimps_I2_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,X3: list_char] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Pro_a @ X3 ) )
= ( episte2398645135750866164t_unit @ M @ W @ X3 ) ) ).
% semantics.simps(2)
thf(fact_360_fm_Oset__intros_I1_J,axiom,
! [Ya: set_Epistemic_fm_a,X31: episte5999154429349999510c_fm_a,X32: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Ya @ ( episte6227279731293865264c_fm_a @ X31 ) )
=> ( member536094252920883875c_fm_a @ Ya @ ( episte6227279731293865264c_fm_a @ ( episte9190102072588321003c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_361_fm_Oset__intros_I1_J,axiom,
! [Ya: epistemic_fm_a,X31: episte740340785640729014c_fm_a,X32: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Ya @ ( episte9089240958480457552c_fm_a @ X31 ) )
=> ( member6642669571620171971c_fm_a @ Ya @ ( episte9089240958480457552c_fm_a @ ( episte6088726764479022859c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_362_fm_Oset__intros_I1_J,axiom,
! [Ya: a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
( ( member_a @ Ya @ ( epistemic_set_fm_a @ X31 ) )
=> ( member_a @ Ya @ ( epistemic_set_fm_a @ ( epistemic_Dis_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_363_fm_Oset__intros_I2_J,axiom,
! [Yb: set_Epistemic_fm_a,X32: episte5999154429349999510c_fm_a,X31: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ X32 ) )
=> ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ ( episte9190102072588321003c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_364_fm_Oset__intros_I2_J,axiom,
! [Yb: epistemic_fm_a,X32: episte740340785640729014c_fm_a,X31: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ X32 ) )
=> ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ ( episte6088726764479022859c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_365_fm_Oset__intros_I2_J,axiom,
! [Yb: a,X32: epistemic_fm_a,X31: epistemic_fm_a] :
( ( member_a @ Yb @ ( epistemic_set_fm_a @ X32 ) )
=> ( member_a @ Yb @ ( epistemic_set_fm_a @ ( epistemic_Dis_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_366_fm_Odistinct_I13_J,axiom,
! [X23: list_char,X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( epistemic_Pro_a @ X23 )
!= ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_367_fm_Opred__mono__strong,axiom,
! [P4: set_Epistemic_fm_a > $o,X3: episte5999154429349999510c_fm_a,Pa: set_Epistemic_fm_a > $o] :
( ( episte1367365995573110961c_fm_a @ P4 @ X3 )
=> ( ! [Z4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ X3 ) )
=> ( ( P4 @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( episte1367365995573110961c_fm_a @ Pa @ X3 ) ) ) ).
% fm.pred_mono_strong
thf(fact_368_fm_Opred__mono__strong,axiom,
! [P4: epistemic_fm_a > $o,X3: episte740340785640729014c_fm_a,Pa: epistemic_fm_a > $o] :
( ( episte3114992851182798545c_fm_a @ P4 @ X3 )
=> ( ! [Z4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( P4 @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( episte3114992851182798545c_fm_a @ Pa @ X3 ) ) ) ).
% fm.pred_mono_strong
thf(fact_369_fm_Opred__mono__strong,axiom,
! [P4: a > $o,X3: epistemic_fm_a,Pa: a > $o] :
( ( epistemic_pred_fm_a @ P4 @ X3 )
=> ( ! [Z4: a] :
( ( member_a @ Z4 @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( P4 @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( epistemic_pred_fm_a @ Pa @ X3 ) ) ) ).
% fm.pred_mono_strong
thf(fact_370_fm_Opred__cong,axiom,
! [X3: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,P4: set_Epistemic_fm_a > $o,Pa: set_Epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ! [Z4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( P4 @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( episte1367365995573110961c_fm_a @ P4 @ X3 )
= ( episte1367365995573110961c_fm_a @ Pa @ Ya ) ) ) ) ).
% fm.pred_cong
thf(fact_371_fm_Opred__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,P4: epistemic_fm_a > $o,Pa: epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ! [Z4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( P4 @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( episte3114992851182798545c_fm_a @ P4 @ X3 )
= ( episte3114992851182798545c_fm_a @ Pa @ Ya ) ) ) ) ).
% fm.pred_cong
thf(fact_372_fm_Opred__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,P4: a > $o,Pa: a > $o] :
( ( X3 = Ya )
=> ( ! [Z4: a] :
( ( member_a @ Z4 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( P4 @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( epistemic_pred_fm_a @ P4 @ X3 )
= ( epistemic_pred_fm_a @ Pa @ Ya ) ) ) ) ).
% fm.pred_cong
thf(fact_373_fm_Odistinct_I11_J,axiom,
! [X23: list_char,X31: epistemic_fm_a,X32: epistemic_fm_a] :
( ( epistemic_Pro_a @ X23 )
!= ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.distinct(11)
thf(fact_374_eval_Osimps_I2_J,axiom,
! [G2: list_char > $o,Uw: epistemic_fm_a > $o,X3: list_char] :
( ( epistemic_eval_a @ G2 @ Uw @ ( epistemic_Pro_a @ X3 ) )
= ( G2 @ X3 ) ) ).
% eval.simps(2)
thf(fact_375_eval__semantics,axiom,
! [Pi4: set_Epistemic_fm_a > list_char > $o,W: set_Epistemic_fm_a,W6: set_se5208064806568342746c_fm_a,R: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,P: epistemic_fm_a] :
( ( epistemic_eval_a @ ( Pi4 @ W ) @ ( episte7081087998767065248c_fm_a @ ( episte2888590659910966568t_unit @ W6 @ R @ ( episte8239586592105053771t_unit @ Pi4 @ product_Unity ) ) @ W ) @ P )
= ( episte7081087998767065248c_fm_a @ ( episte2888590659910966568t_unit @ W6 @ R @ ( episte8239586592105053771t_unit @ Pi4 @ product_Unity ) ) @ W @ P ) ) ).
% eval_semantics
thf(fact_376_truth__lemma,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V3 )
=> ( ( member6642669571620171971c_fm_a @ P @ V3 )
= ( episte7081087998767065248c_fm_a
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W8: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W8 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W8 ) ) )
@ ^ [I2: a,V4: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V4 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V4: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V4 )
@ product_Unity ) )
@ V3
@ P ) ) ) ) ).
% truth_lemma
thf(fact_377_reflexive_092_060_094sub_062T,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( episte5648423998891577755t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W8: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W8 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W8 ) ) )
@ ^ [I2: a,V4: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V4 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V4: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V4 )
@ product_Unity ) ) ) ) ).
% reflexive\<^sub>T
thf(fact_378_completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W8: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W8 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W8 ) ) )
@ ^ [I2: a,V4: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V4 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V4: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V4 )
@ product_Unity ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ) ).
% completeness
thf(fact_379_fm_Orel__induct,axiom,
! [R3: a > a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Q4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( epistemic_rel_fm_a_a @ R3 @ X3 @ Y )
=> ( ( Q4 @ epistemic_FF_a @ epistemic_FF_a )
=> ( ! [A22: list_char,B22: list_char] :
( ( A22 = B22 )
=> ( Q4 @ ( epistemic_Pro_a @ A22 ) @ ( epistemic_Pro_a @ B22 ) ) )
=> ( ! [A31: epistemic_fm_a,A32: epistemic_fm_a,B31: epistemic_fm_a,B32: epistemic_fm_a] :
( ( Q4 @ A31 @ B31 )
=> ( ( Q4 @ A32 @ B32 )
=> ( Q4 @ ( epistemic_Dis_a @ A31 @ A32 ) @ ( epistemic_Dis_a @ B31 @ B32 ) ) ) )
=> ( ! [A41: epistemic_fm_a,A42: epistemic_fm_a,B41: epistemic_fm_a,B42: epistemic_fm_a] :
( ( Q4 @ A41 @ B41 )
=> ( ( Q4 @ A42 @ B42 )
=> ( Q4 @ ( epistemic_Con_a @ A41 @ A42 ) @ ( epistemic_Con_a @ B41 @ B42 ) ) ) )
=> ( ! [A51: epistemic_fm_a,A52: epistemic_fm_a,B51: epistemic_fm_a,B52: epistemic_fm_a] :
( ( Q4 @ A51 @ B51 )
=> ( ( Q4 @ A52 @ B52 )
=> ( Q4 @ ( epistemic_Imp_a @ A51 @ A52 ) @ ( epistemic_Imp_a @ B51 @ B52 ) ) ) )
=> ( ! [A61: a,A62: epistemic_fm_a,B61: a,B62: epistemic_fm_a] :
( ( R3 @ A61 @ B61 )
=> ( ( Q4 @ A62 @ B62 )
=> ( Q4 @ ( epistemic_K_a @ A61 @ A62 ) @ ( epistemic_K_a @ B61 @ B62 ) ) ) )
=> ( Q4 @ X3 @ Y ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_380_fm_Orel__cases,axiom,
! [R3: a > a > $o,A2: epistemic_fm_a,B: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ A2 @ B )
=> ( ( ( A2 = epistemic_FF_a )
=> ( B != epistemic_FF_a ) )
=> ( ! [X2: list_char] :
( ( A2
= ( epistemic_Pro_a @ X2 ) )
=> ! [Y2: list_char] :
( ( B
= ( epistemic_Pro_a @ Y2 ) )
=> ( X2 != Y2 ) ) )
=> ( ! [X1: epistemic_fm_a,X2a: epistemic_fm_a] :
( ( A2
= ( epistemic_Dis_a @ X1 @ X2a ) )
=> ! [Y1: epistemic_fm_a,Y2a: epistemic_fm_a] :
( ( B
= ( epistemic_Dis_a @ Y1 @ Y2a ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2a @ Y2a ) ) ) )
=> ( ! [X1a: epistemic_fm_a,X2b: epistemic_fm_a] :
( ( A2
= ( epistemic_Con_a @ X1a @ X2b ) )
=> ! [Y1a: epistemic_fm_a,Y2b: epistemic_fm_a] :
( ( B
= ( epistemic_Con_a @ Y1a @ Y2b ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2b @ Y2b ) ) ) )
=> ( ! [X1b: epistemic_fm_a,X2c: epistemic_fm_a] :
( ( A2
= ( epistemic_Imp_a @ X1b @ X2c ) )
=> ! [Y1b: epistemic_fm_a,Y2c: epistemic_fm_a] :
( ( B
= ( epistemic_Imp_a @ Y1b @ Y2c ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2c @ Y2c ) ) ) )
=> ~ ! [X1c: a,X2d: epistemic_fm_a] :
( ( A2
= ( epistemic_K_a @ X1c @ X2d ) )
=> ! [Y1c: a,Y2d: epistemic_fm_a] :
( ( B
= ( epistemic_K_a @ Y1c @ Y2d ) )
=> ( ( R3 @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_381_canonical__model_I1_J,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S2 )
=> ( ( member6642669571620171971c_fm_a @ P @ S2 )
=> ( episte7081087998767065248c_fm_a
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W8: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W8 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W8 ) ) )
@ ^ [I2: a,V4: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V4 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V4: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V4 )
@ product_Unity ) )
@ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 )
@ P ) ) ) ).
% canonical_model(1)
thf(fact_382_completeness_092_060_094sub_062A,axiom,
! [P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M2: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ).
% completeness\<^sub>A
thf(fact_383_Powp__mono,axiom,
! [A: epistemic_fm_a > $o,B4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A @ B4 )
=> ( ord_le1568123225352354019fm_a_o @ ( powp_Epistemic_fm_a @ A ) @ ( powp_Epistemic_fm_a @ B4 ) ) ) ).
% Powp_mono
thf(fact_384_K__distrib__K__imp,axiom,
! [A: epistemic_fm_a > $o,I: a,G5: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_K_a @ I @ ( epistemic_imply_a @ G5 @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ I ) @ G5 ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_distrib_K_imp
thf(fact_385_canonical__model_I3_J,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S2 )
=> ( ( member6642669571620171971c_fm_a @ P @ S2 )
=> ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 ) ) ) ) ).
% canonical_model(3)
thf(fact_386_maximal__Extend,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] : ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 ) ) ).
% maximal_Extend
thf(fact_387_consistent__Extend,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S2 )
=> ( episte2285483198712856226tent_a @ A @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 ) ) ) ).
% consistent_Extend
thf(fact_388_canonical__model_I2_J,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S2 )
=> ( ( member6642669571620171971c_fm_a @ P @ S2 )
=> ( episte2285483198712856226tent_a @ A @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 ) ) ) ) ).
% canonical_model(2)
thf(fact_389_fm_Orel__eq,axiom,
( ( epistemic_rel_fm_a_a
@ ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [Y3: epistemic_fm_a,Z: epistemic_fm_a] : ( Y3 = Z ) ) ) ).
% fm.rel_eq
thf(fact_390_fm_Orel__refl,axiom,
! [Ra: a > a > $o,X3: epistemic_fm_a] :
( ! [X2: a] : ( Ra @ X2 @ X2 )
=> ( epistemic_rel_fm_a_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl
thf(fact_391_Extend__subset,axiom,
! [S2: set_Epistemic_fm_a,A: epistemic_fm_a > $o] : ( ord_le3275665582123262618c_fm_a @ S2 @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 ) ) ).
% Extend_subset
thf(fact_392_fm_Orel__mono,axiom,
! [R3: a > a > $o,Ra: a > a > $o] :
( ( ord_less_eq_a_a_o @ R3 @ Ra )
=> ( ord_le3934200179093585166fm_a_o @ ( epistemic_rel_fm_a_a @ R3 ) @ ( epistemic_rel_fm_a_a @ Ra ) ) ) ).
% fm.rel_mono
thf(fact_393_fm_Orel__inject_I5_J,axiom,
! [R3: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) )
= ( ( epistemic_rel_fm_a_a @ R3 @ X51 @ Y51 )
& ( epistemic_rel_fm_a_a @ R3 @ X52 @ Y52 ) ) ) ).
% fm.rel_inject(5)
thf(fact_394_fm_Orel__intros_I5_J,axiom,
! [R3: a > a > $o,X51: epistemic_fm_a,Y51: epistemic_fm_a,X52: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ X51 @ Y51 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X52 @ Y52 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ) ) ).
% fm.rel_intros(5)
thf(fact_395_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_396_bot_Oextremum__uniqueI,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ bot_bo6433428028861825271fm_a_o )
=> ( A2 = bot_bo6433428028861825271fm_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_397_bot_Oextremum__uniqueI,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a )
=> ( A2 = bot_bo3626323581529592678c_fm_a ) ) ).
% bot.extremum_uniqueI
thf(fact_398_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_399_bot_Oextremum__unique,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ bot_bo6433428028861825271fm_a_o )
= ( A2 = bot_bo6433428028861825271fm_a_o ) ) ).
% bot.extremum_unique
thf(fact_400_bot_Oextremum__unique,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a )
= ( A2 = bot_bo3626323581529592678c_fm_a ) ) ).
% bot.extremum_unique
thf(fact_401_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_402_bot_Oextremum,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ bot_bo6433428028861825271fm_a_o @ A2 ) ).
% bot.extremum
thf(fact_403_bot_Oextremum,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ bot_bo3626323581529592678c_fm_a @ A2 ) ).
% bot.extremum
thf(fact_404_fm_Orel__inject_I6_J,axiom,
! [R3: a > a > $o,X61: a,X62: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) )
= ( ( R3 @ X61 @ Y61 )
& ( epistemic_rel_fm_a_a @ R3 @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_405_fm_Orel__intros_I6_J,axiom,
! [R3: a > a > $o,X61: a,Y61: a,X62: epistemic_fm_a,Y62: epistemic_fm_a] :
( ( R3 @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X62 @ Y62 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_406_top_Oextremum__uniqueI,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ top_to6710954824118802707fm_a_o @ A2 )
=> ( A2 = top_to6710954824118802707fm_a_o ) ) ).
% top.extremum_uniqueI
thf(fact_407_top_Oextremum__uniqueI,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ top_to7796028867103199306c_fm_a @ A2 )
=> ( A2 = top_to7796028867103199306c_fm_a ) ) ).
% top.extremum_uniqueI
thf(fact_408_top_Oextremum__unique,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ top_to6710954824118802707fm_a_o @ A2 )
= ( A2 = top_to6710954824118802707fm_a_o ) ) ).
% top.extremum_unique
thf(fact_409_top_Oextremum__unique,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ top_to7796028867103199306c_fm_a @ A2 )
= ( A2 = top_to7796028867103199306c_fm_a ) ) ).
% top.extremum_unique
thf(fact_410_top__greatest,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ top_to6710954824118802707fm_a_o ) ).
% top_greatest
thf(fact_411_top__greatest,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ top_to7796028867103199306c_fm_a ) ).
% top_greatest
thf(fact_412_fm_Octr__transfer_I1_J,axiom,
! [R3: a > a > $o] : ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ epistemic_FF_a ) ).
% fm.ctr_transfer(1)
thf(fact_413_fm_Orel__inject_I4_J,axiom,
! [R3: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) )
= ( ( epistemic_rel_fm_a_a @ R3 @ X41 @ Y41 )
& ( epistemic_rel_fm_a_a @ R3 @ X42 @ Y42 ) ) ) ).
% fm.rel_inject(4)
thf(fact_414_fm_Orel__intros_I4_J,axiom,
! [R3: a > a > $o,X41: epistemic_fm_a,Y41: epistemic_fm_a,X42: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ X41 @ Y41 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X42 @ Y42 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ) ) ).
% fm.rel_intros(4)
thf(fact_415_fm_Orel__cong,axiom,
! [X3: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,Y: episte5999154429349999510c_fm_a,Xa3: episte5999154429349999510c_fm_a,R3: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,Ra: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: set_Epistemic_fm_a,Yb2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ Xa3 ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte8246226516352692192c_fm_a @ R3 @ X3 @ Y )
= ( episte8246226516352692192c_fm_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_416_fm_Orel__cong,axiom,
! [X3: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,Y: episte740340785640729014c_fm_a,Xa3: episte740340785640729014c_fm_a,R3: set_Epistemic_fm_a > epistemic_fm_a > $o,Ra: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: set_Epistemic_fm_a,Yb2: epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa3 ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte6543039790609369600c_fm_a @ R3 @ X3 @ Y )
= ( episte6543039790609369600c_fm_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_417_fm_Orel__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: episte5999154429349999510c_fm_a,Xa3: episte5999154429349999510c_fm_a,R3: epistemic_fm_a > set_Epistemic_fm_a > $o,Ra: epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: epistemic_fm_a,Yb2: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ Xa3 ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte4430291480098810304c_fm_a @ R3 @ X3 @ Y )
= ( episte4430291480098810304c_fm_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_418_fm_Orel__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: episte740340785640729014c_fm_a,Xa3: episte740340785640729014c_fm_a,R3: epistemic_fm_a > epistemic_fm_a > $o,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: epistemic_fm_a,Yb2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa3 ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte7774795710028497888c_fm_a @ R3 @ X3 @ Y )
= ( episte7774795710028497888c_fm_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_419_fm_Orel__cong,axiom,
! [X3: episte5999154429349999510c_fm_a,Ya: episte5999154429349999510c_fm_a,Y: epistemic_fm_a,Xa3: epistemic_fm_a,R3: set_Epistemic_fm_a > a > $o,Ra: set_Epistemic_fm_a > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: set_Epistemic_fm_a,Yb2: a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ Ya ) )
=> ( ( member_a @ Yb2 @ ( epistemic_set_fm_a @ Xa3 ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte7682788968413056228fm_a_a @ R3 @ X3 @ Y )
= ( episte7682788968413056228fm_a_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_420_fm_Orel__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: epistemic_fm_a,Xa3: epistemic_fm_a,R3: epistemic_fm_a > a > $o,Ra: epistemic_fm_a > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: epistemic_fm_a,Yb2: a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member_a @ Yb2 @ ( epistemic_set_fm_a @ Xa3 ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte4428145106359621316fm_a_a @ R3 @ X3 @ Y )
= ( episte4428145106359621316fm_a_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_421_fm_Orel__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,Y: episte5999154429349999510c_fm_a,Xa3: episte5999154429349999510c_fm_a,R3: a > set_Epistemic_fm_a > $o,Ra: a > set_Epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: a,Yb2: set_Epistemic_fm_a] :
( ( member_a @ Z4 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ Xa3 ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte6665775379439401564c_fm_a @ R3 @ X3 @ Y )
= ( episte6665775379439401564c_fm_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_422_fm_Orel__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,Y: episte740340785640729014c_fm_a,Xa3: episte740340785640729014c_fm_a,R3: a > epistemic_fm_a > $o,Ra: a > epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: a,Yb2: epistemic_fm_a] :
( ( member_a @ Z4 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa3 ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte8321036160184370300c_fm_a @ R3 @ X3 @ Y )
= ( episte8321036160184370300c_fm_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_423_fm_Orel__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,Y: epistemic_fm_a,Xa3: epistemic_fm_a,R3: a > a > $o,Ra: a > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z4: a,Yb2: a] :
( ( member_a @ Z4 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member_a @ Yb2 @ ( epistemic_set_fm_a @ Xa3 ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X3 @ Y )
= ( epistemic_rel_fm_a_a @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% fm.rel_cong
thf(fact_424_fm_Orel__mono__strong,axiom,
! [R3: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,X3: episte5999154429349999510c_fm_a,Y: episte5999154429349999510c_fm_a,Ra: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( episte8246226516352692192c_fm_a @ R3 @ X3 @ Y )
=> ( ! [Z4: set_Epistemic_fm_a,Yb2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ X3 ) )
=> ( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte8246226516352692192c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_425_fm_Orel__mono__strong,axiom,
! [R3: set_Epistemic_fm_a > epistemic_fm_a > $o,X3: episte5999154429349999510c_fm_a,Y: episte740340785640729014c_fm_a,Ra: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( episte6543039790609369600c_fm_a @ R3 @ X3 @ Y )
=> ( ! [Z4: set_Epistemic_fm_a,Yb2: epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte6543039790609369600c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_426_fm_Orel__mono__strong,axiom,
! [R3: epistemic_fm_a > set_Epistemic_fm_a > $o,X3: episte740340785640729014c_fm_a,Y: episte5999154429349999510c_fm_a,Ra: epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ( episte4430291480098810304c_fm_a @ R3 @ X3 @ Y )
=> ( ! [Z4: epistemic_fm_a,Yb2: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte4430291480098810304c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_427_fm_Orel__mono__strong,axiom,
! [R3: epistemic_fm_a > epistemic_fm_a > $o,X3: episte740340785640729014c_fm_a,Y: episte740340785640729014c_fm_a,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ( episte7774795710028497888c_fm_a @ R3 @ X3 @ Y )
=> ( ! [Z4: epistemic_fm_a,Yb2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_428_fm_Orel__mono__strong,axiom,
! [R3: set_Epistemic_fm_a > a > $o,X3: episte5999154429349999510c_fm_a,Y: epistemic_fm_a,Ra: set_Epistemic_fm_a > a > $o] :
( ( episte7682788968413056228fm_a_a @ R3 @ X3 @ Y )
=> ( ! [Z4: set_Epistemic_fm_a,Yb2: a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ X3 ) )
=> ( ( member_a @ Yb2 @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte7682788968413056228fm_a_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_429_fm_Orel__mono__strong,axiom,
! [R3: epistemic_fm_a > a > $o,X3: episte740340785640729014c_fm_a,Y: epistemic_fm_a,Ra: epistemic_fm_a > a > $o] :
( ( episte4428145106359621316fm_a_a @ R3 @ X3 @ Y )
=> ( ! [Z4: epistemic_fm_a,Yb2: a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( member_a @ Yb2 @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte4428145106359621316fm_a_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_430_fm_Orel__mono__strong,axiom,
! [R3: a > set_Epistemic_fm_a > $o,X3: epistemic_fm_a,Y: episte5999154429349999510c_fm_a,Ra: a > set_Epistemic_fm_a > $o] :
( ( episte6665775379439401564c_fm_a @ R3 @ X3 @ Y )
=> ( ! [Z4: a,Yb2: set_Epistemic_fm_a] :
( ( member_a @ Z4 @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte6665775379439401564c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_431_fm_Orel__mono__strong,axiom,
! [R3: a > epistemic_fm_a > $o,X3: epistemic_fm_a,Y: episte740340785640729014c_fm_a,Ra: a > epistemic_fm_a > $o] :
( ( episte8321036160184370300c_fm_a @ R3 @ X3 @ Y )
=> ( ! [Z4: a,Yb2: epistemic_fm_a] :
( ( member_a @ Z4 @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte8321036160184370300c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_432_fm_Orel__mono__strong,axiom,
! [R3: a > a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Ra: a > a > $o] :
( ( epistemic_rel_fm_a_a @ R3 @ X3 @ Y )
=> ( ! [Z4: a,Yb2: a] :
( ( member_a @ Z4 @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( member_a @ Yb2 @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( epistemic_rel_fm_a_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_433_fm_Orel__refl__strong,axiom,
! [X3: episte5999154429349999510c_fm_a,Ra: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ! [Z4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z4 @ ( episte6227279731293865264c_fm_a @ X3 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( episte8246226516352692192c_fm_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_434_fm_Orel__refl__strong,axiom,
! [X3: episte740340785640729014c_fm_a,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [Z4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_435_fm_Orel__refl__strong,axiom,
! [X3: epistemic_fm_a,Ra: a > a > $o] :
( ! [Z4: a] :
( ( member_a @ Z4 @ ( epistemic_set_fm_a @ X3 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( epistemic_rel_fm_a_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_436_fm_Orel__inject_I2_J,axiom,
! [R3: a > a > $o,X23: list_char,Y22: list_char] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X23 ) @ ( epistemic_Pro_a @ Y22 ) )
= ( X23 = Y22 ) ) ).
% fm.rel_inject(2)
thf(fact_437_fm_Orel__intros_I2_J,axiom,
! [X23: list_char,Y22: list_char,R3: a > a > $o] :
( ( X23 = Y22 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X23 ) @ ( epistemic_Pro_a @ Y22 ) ) ) ).
% fm.rel_intros(2)
thf(fact_438_fm_Orel__inject_I3_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) )
= ( ( epistemic_rel_fm_a_a @ R3 @ X31 @ Y31 )
& ( epistemic_rel_fm_a_a @ R3 @ X32 @ Y32 ) ) ) ).
% fm.rel_inject(3)
thf(fact_439_fm_Orel__intros_I3_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,Y31: epistemic_fm_a,X32: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ X31 @ Y31 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X32 @ Y32 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ) ) ).
% fm.rel_intros(3)
thf(fact_440_fm_Orel__distinct_I30_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.rel_distinct(30)
thf(fact_441_fm_Orel__distinct_I29_J,axiom,
! [R3: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(29)
thf(fact_442_fm_Orel__distinct_I8_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(8)
thf(fact_443_fm_Orel__distinct_I7_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(7)
thf(fact_444_fm_Orel__distinct_I10_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(10)
thf(fact_445_fm_Orel__distinct_I9_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(9)
thf(fact_446_fm_Orel__distinct_I26_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.rel_distinct(26)
thf(fact_447_fm_Orel__distinct_I25_J,axiom,
! [R3: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(25)
thf(fact_448_fm_Orel__distinct_I28_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.rel_distinct(28)
thf(fact_449_fm_Orel__distinct_I27_J,axiom,
! [R3: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(27)
thf(fact_450_fm_Orel__distinct_I16_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X23: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Pro_a @ X23 ) ) ).
% fm.rel_distinct(16)
thf(fact_451_fm_Orel__distinct_I15_J,axiom,
! [R3: a > a > $o,X23: list_char,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X23 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(15)
thf(fact_452_fm_Orel__distinct_I22_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(22)
thf(fact_453_fm_Orel__distinct_I21_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(21)
thf(fact_454_fm_Orel__distinct_I18_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X23: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Pro_a @ X23 ) ) ).
% fm.rel_distinct(18)
thf(fact_455_fm_Orel__distinct_I17_J,axiom,
! [R3: a > a > $o,X23: list_char,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X23 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(17)
thf(fact_456_fm_Orel__distinct_I24_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(24)
thf(fact_457_fm_Orel__distinct_I23_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(23)
thf(fact_458_fm_Orel__distinct_I6_J,axiom,
! [R3: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(6)
thf(fact_459_fm_Orel__distinct_I5_J,axiom,
! [R3: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(5)
thf(fact_460_fm_Orel__distinct_I2_J,axiom,
! [R3: a > a > $o,Y22: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ Y22 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(2)
thf(fact_461_fm_Orel__distinct_I1_J,axiom,
! [R3: a > a > $o,Y22: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Pro_a @ Y22 ) ) ).
% fm.rel_distinct(1)
thf(fact_462_fm_Orel__distinct_I4_J,axiom,
! [R3: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ Y31 @ Y32 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(4)
thf(fact_463_fm_Orel__distinct_I3_J,axiom,
! [R3: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ).
% fm.rel_distinct(3)
thf(fact_464_fm_Orel__distinct_I14_J,axiom,
! [R3: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a,X23: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ ( epistemic_Pro_a @ X23 ) ) ).
% fm.rel_distinct(14)
thf(fact_465_fm_Orel__distinct_I13_J,axiom,
! [R3: a > a > $o,X23: list_char,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X23 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(13)
thf(fact_466_fm_Orel__distinct_I20_J,axiom,
! [R3: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(20)
thf(fact_467_fm_Orel__distinct_I19_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(19)
thf(fact_468_fm_Orel__distinct_I11_J,axiom,
! [R3: a > a > $o,X23: list_char,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X23 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ).
% fm.rel_distinct(11)
thf(fact_469_fm_Orel__distinct_I12_J,axiom,
! [R3: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a,X23: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ Y31 @ Y32 ) @ ( epistemic_Pro_a @ X23 ) ) ).
% fm.rel_distinct(12)
thf(fact_470_fm_Osimps_I90_J,axiom,
( ( episte9089240958480457552c_fm_a @ episte5073044243917183961c_fm_a )
= bot_bo3626323581529592678c_fm_a ) ).
% fm.simps(90)
thf(fact_471_fm_Osimps_I90_J,axiom,
( ( epistemic_set_fm_a @ epistemic_FF_a )
= bot_bot_set_a ) ).
% fm.simps(90)
thf(fact_472_fm_Osimps_I91_J,axiom,
! [X23: list_char] :
( ( episte9089240958480457552c_fm_a @ ( episte3759128466173231372c_fm_a @ X23 ) )
= bot_bo3626323581529592678c_fm_a ) ).
% fm.simps(91)
thf(fact_473_fm_Osimps_I91_J,axiom,
! [X23: list_char] :
( ( epistemic_set_fm_a @ ( epistemic_Pro_a @ X23 ) )
= bot_bot_set_a ) ).
% fm.simps(91)
thf(fact_474_maximal_H__Extend,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] : ( maxima2621630783216038545c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 ) ) ).
% maximal'_Extend
thf(fact_475_maximal__maximal_H,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S2 )
= ( maxima2621630783216038545c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 ) ) ).
% maximal_maximal'
thf(fact_476_Extend__subset_H,axiom,
! [S2: set_Epistemic_fm_a,A: epistemic_fm_a > $o] :
( ( ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) )
!= bot_bo3626323581529592678c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ S2 @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 ) ) ) ).
% Extend_subset'
thf(fact_477_consistent__extend,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a,N: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S2 )
=> ( episte2285483198712856226tent_a @ A @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 @ N ) ) ) ).
% consistent_extend
thf(fact_478_MCS__Lim__Ord__axioms,axiom,
! [A: epistemic_fm_a > $o] : ( maxima8116622252184345237c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) ) ).
% MCS_Lim_Ord_axioms
thf(fact_479_inconsistent__subset,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P @ bot_bo3626323581529592678c_fm_a ) @ V3 ) )
=> ~ ! [V5: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ V5 ) @ V3 )
=> ~ ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ V5 ) @ epistemic_FF_a ) ) ) ) ) ).
% inconsistent_subset
thf(fact_480_le__sup__iff,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq_set_a @ X3 @ Z2 )
& ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_481_le__sup__iff,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ X3 @ Y ) @ Z2 )
= ( ( ord_le4043730696559282883fm_a_o @ X3 @ Z2 )
& ( ord_le4043730696559282883fm_a_o @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_482_le__sup__iff,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ X3 @ Y ) @ Z2 )
= ( ( ord_le3275665582123262618c_fm_a @ X3 @ Z2 )
& ( ord_le3275665582123262618c_fm_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_483_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_484_sup_Obounded__iff,axiom,
! [B: epistemic_fm_a > $o,C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ B @ C ) @ A2 )
= ( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
& ( ord_le4043730696559282883fm_a_o @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_485_sup_Obounded__iff,axiom,
! [B: set_Epistemic_fm_a,C: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ B @ C ) @ A2 )
= ( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
& ( ord_le3275665582123262618c_fm_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_486_fm_Opred__inject_I2_J,axiom,
! [P4: a > $o,A2: list_char] :
( ( epistemic_pred_fm_a @ P4 @ ( epistemic_Pro_a @ A2 ) )
= ( top_top_list_char_o @ A2 ) ) ).
% fm.pred_inject(2)
thf(fact_487_fm_Osimps_I95_J,axiom,
! [X61: epistemic_fm_a,X62: episte740340785640729014c_fm_a] :
( ( episte9089240958480457552c_fm_a @ ( episte5657488632024175118c_fm_a @ X61 @ X62 ) )
= ( insert7817948963269374698c_fm_a @ X61 @ ( episte9089240958480457552c_fm_a @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_488_fm_Osimps_I95_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) )
= ( insert_a @ X61 @ ( epistemic_set_fm_a @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_489_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X3: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_490_inf__sup__ord_I4_J,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y @ ( sup_su6697315146409554831fm_a_o @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_491_inf__sup__ord_I4_J,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y @ ( sup_su1367922730591523534c_fm_a @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_492_inf__sup__ord_I3_J,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_493_inf__sup__ord_I3_J,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X3 @ ( sup_su6697315146409554831fm_a_o @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_494_inf__sup__ord_I3_J,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X3 @ ( sup_su1367922730591523534c_fm_a @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_495_le__supE,axiom,
! [A2: set_a,B: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ X3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X3 )
=> ~ ( ord_less_eq_set_a @ B @ X3 ) ) ) ).
% le_supE
thf(fact_496_le__supE,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) @ X3 )
=> ~ ( ( ord_le4043730696559282883fm_a_o @ A2 @ X3 )
=> ~ ( ord_le4043730696559282883fm_a_o @ B @ X3 ) ) ) ).
% le_supE
thf(fact_497_le__supE,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) @ X3 )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ A2 @ X3 )
=> ~ ( ord_le3275665582123262618c_fm_a @ B @ X3 ) ) ) ).
% le_supE
thf(fact_498_le__supI,axiom,
! [A2: set_a,X3: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ X3 )
=> ( ( ord_less_eq_set_a @ B @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_499_le__supI,axiom,
! [A2: epistemic_fm_a > $o,X3: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ X3 )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ X3 )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_500_le__supI,axiom,
! [A2: set_Epistemic_fm_a,X3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ X3 )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ X3 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_501_sup__ge1,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_502_sup__ge1,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X3 @ ( sup_su6697315146409554831fm_a_o @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_503_sup__ge1,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X3 @ ( sup_su1367922730591523534c_fm_a @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_504_sup__ge2,axiom,
! [Y: set_a,X3: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_505_sup__ge2,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y @ ( sup_su6697315146409554831fm_a_o @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_506_sup__ge2,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y @ ( sup_su1367922730591523534c_fm_a @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_507_le__supI1,axiom,
! [X3: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_508_le__supI1,axiom,
! [X3: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ A2 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_509_le__supI1,axiom,
! [X3: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ A2 )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_510_le__supI2,axiom,
! [X3: set_a,B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X3 @ B )
=> ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_511_le__supI2,axiom,
! [X3: epistemic_fm_a > $o,B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ B )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_512_le__supI2,axiom,
! [X3: set_Epistemic_fm_a,B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ B )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_513_sup_Omono,axiom,
! [C: set_a,A2: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_514_sup_Omono,axiom,
! [C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,D: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ C @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ D @ B )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ C @ D ) @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_515_sup_Omono,axiom,
! [C: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,D: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ D @ B )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ C @ D ) @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_516_sup__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_517_sup__mono,axiom,
! [A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o,B: epistemic_fm_a > $o,D: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ C )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ D )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) @ ( sup_su6697315146409554831fm_a_o @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_518_sup__mono,axiom,
! [A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a,B: set_Epistemic_fm_a,D: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ C )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ D )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) @ ( sup_su1367922730591523534c_fm_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_519_sup__least,axiom,
! [Y: set_a,X3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X3 )
=> ( ( ord_less_eq_set_a @ Z2 @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_520_sup__least,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( ( ord_le4043730696559282883fm_a_o @ Z2 @ X3 )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_521_sup__least,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( ( ord_le3275665582123262618c_fm_a @ Z2 @ X3 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_522_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_523_le__iff__sup,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] :
( ( sup_su6697315146409554831fm_a_o @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_524_le__iff__sup,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_525_sup_OorderE,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_526_sup_OorderE,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( A2
= ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_527_sup_OorderE,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( A2
= ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_528_sup_OorderI,axiom,
! [A2: set_a,B: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B ) )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% sup.orderI
thf(fact_529_sup_OorderI,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( A2
= ( sup_su6697315146409554831fm_a_o @ A2 @ B ) )
=> ( ord_le4043730696559282883fm_a_o @ B @ A2 ) ) ).
% sup.orderI
thf(fact_530_sup_OorderI,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( A2
= ( sup_su1367922730591523534c_fm_a @ A2 @ B ) )
=> ( ord_le3275665582123262618c_fm_a @ B @ A2 ) ) ).
% sup.orderI
thf(fact_531_sup__unique,axiom,
! [F: set_a > set_a > set_a,X3: set_a,Y: set_a] :
( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_a,Y2: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( ord_less_eq_set_a @ Z4 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y2 @ Z4 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_532_sup__unique,axiom,
! [F: ( epistemic_fm_a > $o ) > ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o,Z4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y2 @ X2 )
=> ( ( ord_le4043730696559282883fm_a_o @ Z4 @ X2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ Y2 @ Z4 ) @ X2 ) ) )
=> ( ( sup_su6697315146409554831fm_a_o @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_533_sup__unique,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y2 @ X2 )
=> ( ( ord_le3275665582123262618c_fm_a @ Z4 @ X2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ Y2 @ Z4 ) @ X2 ) ) )
=> ( ( sup_su1367922730591523534c_fm_a @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_534_sup_Oabsorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_535_sup_Oabsorb1,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( sup_su6697315146409554831fm_a_o @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_536_sup_Oabsorb1,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( sup_su1367922730591523534c_fm_a @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_537_sup_Oabsorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_538_sup_Oabsorb2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( sup_su6697315146409554831fm_a_o @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_539_sup_Oabsorb2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( sup_su1367922730591523534c_fm_a @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_540_sup__absorb1,axiom,
! [Y: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y @ X3 )
=> ( ( sup_sup_set_a @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_541_sup__absorb1,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( ( sup_su6697315146409554831fm_a_o @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_542_sup__absorb1,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( ( sup_su1367922730591523534c_fm_a @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_543_sup__absorb2,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( sup_sup_set_a @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_544_sup__absorb2,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( sup_su6697315146409554831fm_a_o @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_545_sup__absorb2,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( sup_su1367922730591523534c_fm_a @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_546_sup_OboundedE,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_547_sup_OboundedE,axiom,
! [B: epistemic_fm_a > $o,C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ B @ C ) @ A2 )
=> ~ ( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ~ ( ord_le4043730696559282883fm_a_o @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_548_sup_OboundedE,axiom,
! [B: set_Epistemic_fm_a,C: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ B @ C ) @ A2 )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ~ ( ord_le3275665582123262618c_fm_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_549_sup_OboundedI,axiom,
! [B: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_550_sup_OboundedI,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ C @ A2 )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_551_sup_OboundedI,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ C @ A2 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_552_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_553_sup_Oorder__iff,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [B2: epistemic_fm_a > $o,A4: epistemic_fm_a > $o] :
( A4
= ( sup_su6697315146409554831fm_a_o @ A4 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_554_sup_Oorder__iff,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [B2: set_Epistemic_fm_a,A4: set_Epistemic_fm_a] :
( A4
= ( sup_su1367922730591523534c_fm_a @ A4 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_555_sup_Ocobounded1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_556_sup_Ocobounded1,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_557_sup_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_558_sup_Ocobounded2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_559_sup_Ocobounded2,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ B @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_560_sup_Ocobounded2,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_561_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B2 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_562_sup_Oabsorb__iff1,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [B2: epistemic_fm_a > $o,A4: epistemic_fm_a > $o] :
( ( sup_su6697315146409554831fm_a_o @ A4 @ B2 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_563_sup_Oabsorb__iff1,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [B2: set_Epistemic_fm_a,A4: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ A4 @ B2 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_564_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B2: set_a] :
( ( sup_sup_set_a @ A4 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_565_sup_Oabsorb__iff2,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( sup_su6697315146409554831fm_a_o @ A4 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_566_sup_Oabsorb__iff2,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ A4 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_567_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_568_sup_OcoboundedI1,axiom,
! [C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ C @ A2 )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_569_sup_OcoboundedI1,axiom,
! [C: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C @ A2 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_570_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_571_sup_OcoboundedI2,axiom,
! [C: epistemic_fm_a > $o,B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ C @ B )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_572_sup_OcoboundedI2,axiom,
! [C: set_Epistemic_fm_a,B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C @ B )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_573_maximal_H__def,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
( ( maxima2621630783216038545c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) )
=> ( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ X @ bot_bo3626323581529592678c_fm_a ) @ S2 ) )
=> ( member6642669571620171971c_fm_a @ X @ S2 ) ) ) ) ) ).
% maximal'_def
thf(fact_574_consistent__disjuncts,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ( member6642669571620171971c_fm_a @ ( epistemic_Dis_a @ P @ Q ) @ V3 )
=> ( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P @ bot_bo3626323581529592678c_fm_a ) @ V3 ) )
| ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V3 ) ) ) ) ) ).
% consistent_disjuncts
thf(fact_575_maximal__def,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S2 )
= ( ! [P2: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P2 @ bot_bo3626323581529592678c_fm_a ) @ S2 ) )
=> ( member6642669571620171971c_fm_a @ P2 @ S2 ) ) ) ) ).
% maximal_def
thf(fact_576_distrib__inf__le,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ ( inf_inf_set_a @ X3 @ Z2 ) ) @ ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_577_distrib__inf__le,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y ) @ ( inf_in3606484609122063093fm_a_o @ X3 @ Z2 ) ) @ ( inf_in3606484609122063093fm_a_o @ X3 @ ( sup_su6697315146409554831fm_a_o @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_578_distrib__inf__le,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X3 @ Y ) @ ( inf_in2882328776738922472c_fm_a @ X3 @ Z2 ) ) @ ( inf_in2882328776738922472c_fm_a @ X3 @ ( sup_su1367922730591523534c_fm_a @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_579_distrib__sup__le,axiom,
! [X3: set_a,Y: set_a,Z2: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y @ Z2 ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y ) @ ( sup_sup_set_a @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_580_distrib__sup__le,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ X3 @ ( inf_in3606484609122063093fm_a_o @ Y @ Z2 ) ) @ ( inf_in3606484609122063093fm_a_o @ ( sup_su6697315146409554831fm_a_o @ X3 @ Y ) @ ( sup_su6697315146409554831fm_a_o @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_581_distrib__sup__le,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ X3 @ ( inf_in2882328776738922472c_fm_a @ Y @ Z2 ) ) @ ( inf_in2882328776738922472c_fm_a @ ( sup_su1367922730591523534c_fm_a @ X3 @ Y ) @ ( sup_su1367922730591523534c_fm_a @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_582_consistent__consequent,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ( member6642669571620171971c_fm_a @ P @ V3 )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V3 ) ) ) ) ) ).
% consistent_consequent
thf(fact_583_consistent__consequent_H,axiom,
! [A: epistemic_fm_a > $o,V3: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V3 )
=> ( ( member6642669571620171971c_fm_a @ P @ V3 )
=> ( ! [G3: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H3 @ ( epistemic_Imp_a @ P @ Q ) )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V3 ) ) ) ) ) ).
% consistent_consequent'
thf(fact_584_fm_Osimps_I94_J,axiom,
! [X51: episte740340785640729014c_fm_a,X52: episte740340785640729014c_fm_a] :
( ( episte9089240958480457552c_fm_a @ ( episte260752218777527565c_fm_a @ X51 @ X52 ) )
= ( sup_su1367922730591523534c_fm_a @ ( episte9089240958480457552c_fm_a @ X51 ) @ ( episte9089240958480457552c_fm_a @ X52 ) ) ) ).
% fm.simps(94)
thf(fact_585_fm_Osimps_I94_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) )
= ( sup_sup_set_a @ ( epistemic_set_fm_a @ X51 ) @ ( epistemic_set_fm_a @ X52 ) ) ) ).
% fm.simps(94)
thf(fact_586_fm_Osimps_I93_J,axiom,
! [X41: episte740340785640729014c_fm_a,X42: episte740340785640729014c_fm_a] :
( ( episte9089240958480457552c_fm_a @ ( episte3685526487207141399c_fm_a @ X41 @ X42 ) )
= ( sup_su1367922730591523534c_fm_a @ ( episte9089240958480457552c_fm_a @ X41 ) @ ( episte9089240958480457552c_fm_a @ X42 ) ) ) ).
% fm.simps(93)
thf(fact_587_fm_Osimps_I93_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_Con_a @ X41 @ X42 ) )
= ( sup_sup_set_a @ ( epistemic_set_fm_a @ X41 ) @ ( epistemic_set_fm_a @ X42 ) ) ) ).
% fm.simps(93)
thf(fact_588_fm_Osimps_I92_J,axiom,
! [X31: episte740340785640729014c_fm_a,X32: episte740340785640729014c_fm_a] :
( ( episte9089240958480457552c_fm_a @ ( episte6088726764479022859c_fm_a @ X31 @ X32 ) )
= ( sup_su1367922730591523534c_fm_a @ ( episte9089240958480457552c_fm_a @ X31 ) @ ( episte9089240958480457552c_fm_a @ X32 ) ) ) ).
% fm.simps(92)
thf(fact_589_fm_Osimps_I92_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_Dis_a @ X31 @ X32 ) )
= ( sup_sup_set_a @ ( epistemic_set_fm_a @ X31 ) @ ( epistemic_set_fm_a @ X32 ) ) ) ).
% fm.simps(92)
thf(fact_590_extend__subset,axiom,
! [N: epistemic_fm_a,S2: set_Epistemic_fm_a,A: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ N @ ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) )
=> ( ord_le3275665582123262618c_fm_a @ S2 @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 @ N ) ) ) ).
% extend_subset
thf(fact_591_Extend__bound,axiom,
! [N: epistemic_fm_a,A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ N @ ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 @ N ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 ) ) ) ).
% Extend_bound
thf(fact_592_inconsistent__imply,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G5: list_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ ( set_Epistemic_fm_a2 @ G5 ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G5 @ P ) ) ) ).
% inconsistent_imply
thf(fact_593_extendS__def,axiom,
! [A: epistemic_fm_a > $o,N: epistemic_fm_a,Prev: set_Epistemic_fm_a,S2: set_Epistemic_fm_a] :
( ( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ N @ bot_bo3626323581529592678c_fm_a ) @ Prev ) )
=> ( ( maxima5458213586468375476c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S2 @ N @ Prev )
= ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ N @ bot_bo3626323581529592678c_fm_a ) @ Prev ) ) )
& ( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ N @ bot_bo3626323581529592678c_fm_a ) @ Prev ) )
=> ( ( maxima5458213586468375476c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S2 @ N @ Prev )
= Prev ) ) ) ).
% extendS_def
thf(fact_594_exists__finite__inconsistent,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,V3: set_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V3 ) )
=> ~ ! [W7: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W7 ) @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V3 ) )
=> ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ W7 )
=> ( ( finite3304564945125563331c_fm_a @ W7 )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W7 ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_595_extend__under,axiom,
! [M5: epistemic_fm_a,N: epistemic_fm_a,A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ M5 @ ( order_1815523470270482687c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ N ) )
=> ( ord_le3275665582123262618c_fm_a @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 @ M5 ) @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 @ N ) ) ) ).
% extend_under
thf(fact_596_extend__underS,axiom,
! [M5: epistemic_fm_a,N: epistemic_fm_a,A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ M5 @ ( order_1288088311422721504c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ N ) )
=> ( ord_le3275665582123262618c_fm_a @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 @ M5 ) @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 @ N ) ) ) ).
% extend_underS
thf(fact_597_isLimOrd__r,axiom,
wellor5744531135217078139c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ).
% isLimOrd_r
thf(fact_598_Extend__def,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
( ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 )
= ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 ) @ ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) ) ) ) ).
% Extend_def
thf(fact_599_infinite__UNIV__fm,axiom,
~ ( finite3304564945125563331c_fm_a @ top_to7796028867103199306c_fm_a ) ).
% infinite_UNIV_fm
thf(fact_600_inconsistent__finite,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A @ S2 )
=> ? [S4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ S4 @ S2 )
& ( finite3304564945125563331c_fm_a @ S4 )
& ~ ( episte2285483198712856226tent_a @ A @ S4 ) ) ) ).
% inconsistent_finite
thf(fact_601_extendL__def,axiom,
! [Rec: epistemic_fm_a > set_Epistemic_fm_a,N: epistemic_fm_a] :
( ( maxima7895811076573853563c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ Rec @ N )
= ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ Rec @ ( order_1288088311422721504c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ N ) ) ) ) ).
% extendL_def
thf(fact_602_SUP__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,G2: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_603_SUP__subset__mono,axiom,
! [A: set_a,B4: set_a,F: a > epistemic_fm_a > $o,G2: a > epistemic_fm_a > $o] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_604_SUP__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,G2: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_605_SUP__subset__mono,axiom,
! [A: set_a,B4: set_a,F: a > set_Epistemic_fm_a,G2: a > set_Epistemic_fm_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_606_SUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,G2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_607_SUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,G2: epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_608_Sup__inter__less__eq,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] : ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( inf_in8896176524268821035fm_a_o @ A @ B4 ) ) @ ( inf_in3606484609122063093fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B4 ) ) ) ).
% Sup_inter_less_eq
thf(fact_609_Sup__inter__less__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( inf_in1884693029477671368c_fm_a @ A @ B4 ) ) @ ( inf_in2882328776738922472c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B4 ) ) ) ).
% Sup_inter_less_eq
thf(fact_610_SUP__eq__iff,axiom,
! [I4: set_se5208064806568342746c_fm_a,C: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( I4 != bot_bo1868452413526482246c_fm_a )
=> ( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ I4 )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( F @ I3 ) ) )
=> ( ( ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ I4 ) )
= C )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_611_SUP__eq__iff,axiom,
! [I4: set_Epistemic_fm_a,C: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( I4 != bot_bo3626323581529592678c_fm_a )
=> ( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ I4 )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( F @ I3 ) ) )
=> ( ( ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ I4 ) )
= C )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_612_SUP__eq__iff,axiom,
! [I4: set_a,C: epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ( I4 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I4 )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( F @ I3 ) ) )
=> ( ( ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ I4 ) )
= C )
= ( ! [X: a] :
( ( member_a @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_613_SUP__eq__iff,axiom,
! [I4: set_se5208064806568342746c_fm_a,C: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( I4 != bot_bo1868452413526482246c_fm_a )
=> ( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ I4 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ I4 ) )
= C )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_614_SUP__eq__iff,axiom,
! [I4: set_Epistemic_fm_a,C: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( I4 != bot_bo3626323581529592678c_fm_a )
=> ( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ I4 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ I4 ) )
= C )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_615_SUP__eq__iff,axiom,
! [I4: set_a,C: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ( I4 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I4 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ I4 ) )
= C )
= ( ! [X: a] :
( ( member_a @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_616_cSUP__least,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,M: epistemic_fm_a > $o] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ M ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_617_cSUP__least,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,M: epistemic_fm_a > $o] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ M ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_618_cSUP__least,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,M: epistemic_fm_a > $o] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ M ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_619_cSUP__least,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,M: set_Epistemic_fm_a] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ M ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_620_cSUP__least,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,M: set_Epistemic_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ M ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_621_cSUP__least,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,M: set_Epistemic_fm_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ M ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_622_Sup__eqI,axiom,
! [A: set_Epistemic_fm_a_o,X3: epistemic_fm_a > $o] :
( ! [Y2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Y2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [Z5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Z5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ Z5 @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_623_Sup__eqI,axiom,
! [A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ! [Y2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X3 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [Z5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ Z5 @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_624_Sup__mono,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ! [A5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A5 @ A )
=> ? [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ B4 )
& ( ord_le4043730696559282883fm_a_o @ A5 @ X5 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B4 ) ) ) ).
% Sup_mono
thf(fact_625_Sup__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ! [A5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A5 @ A )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ A5 @ X5 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B4 ) ) ) ).
% Sup_mono
thf(fact_626_Sup__least,axiom,
! [A: set_Epistemic_fm_a_o,Z2: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Z2 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_627_Sup__least,axiom,
! [A: set_se5208064806568342746c_fm_a,Z2: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Z2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_628_Sup__upper,axiom,
! [X3: epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( member4486839677911940090fm_a_o @ X3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( comple310706794042174646fm_a_o @ A ) ) ) ).
% Sup_upper
thf(fact_629_Sup__upper,axiom,
! [X3: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( comple7868773486375872231c_fm_a @ A ) ) ) ).
% Sup_upper
thf(fact_630_Sup__le__iff,axiom,
! [A: set_Epistemic_fm_a_o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ B )
= ( ! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ A )
=> ( ord_le4043730696559282883fm_a_o @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_631_Sup__le__iff,axiom,
! [A: set_se5208064806568342746c_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ B )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
=> ( ord_le3275665582123262618c_fm_a @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_632_Sup__upper2,axiom,
! [U2: epistemic_fm_a > $o,A: set_Epistemic_fm_a_o,V6: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ U2 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ V6 @ U2 )
=> ( ord_le4043730696559282883fm_a_o @ V6 @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_633_Sup__upper2,axiom,
! [U2: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,V6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ U2 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ V6 @ U2 )
=> ( ord_le3275665582123262618c_fm_a @ V6 @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_634_cSup__eq__maximum,axiom,
! [Z2: epistemic_fm_a > $o,X6: set_Epistemic_fm_a_o] :
( ( member4486839677911940090fm_a_o @ Z2 @ X6 )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Z2 ) )
=> ( ( comple310706794042174646fm_a_o @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_635_cSup__eq__maximum,axiom,
! [Z2: set_Epistemic_fm_a,X6: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ Z2 @ X6 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Z2 ) )
=> ( ( comple7868773486375872231c_fm_a @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_636_SUP__eq,axiom,
! [A: set_a,B4: set_a,F: a > set_Epistemic_fm_a,G2: a > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G2 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_637_SUP__eq,axiom,
! [A: set_a,B4: set_a,F: a > epistemic_fm_a > $o,G2: a > epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ ( G2 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
= ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_638_SUP__eq,axiom,
! [A: set_Epistemic_fm_a,B4: set_a,F: epistemic_fm_a > set_Epistemic_fm_a,G2: a > set_Epistemic_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G2 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_639_SUP__eq,axiom,
! [A: set_a,B4: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a,G2: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J @ B4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G2 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_640_SUP__eq,axiom,
! [A: set_Epistemic_fm_a,B4: set_a,F: epistemic_fm_a > epistemic_fm_a > $o,G2: a > epistemic_fm_a > $o] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ ( G2 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
= ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_641_SUP__eq,axiom,
! [A: set_a,B4: set_Epistemic_fm_a,F: a > epistemic_fm_a > $o,G2: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ B4 )
& ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J @ B4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ ( G2 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
= ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_642_SUP__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,G2: a > set_Epistemic_fm_a] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G2 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_643_SUP__eq,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,G2: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J @ B4 )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G2 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_644_SUP__eq,axiom,
! [A: set_a,B4: set_se5208064806568342746c_fm_a,F: a > set_Epistemic_fm_a,G2: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ J @ B4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G2 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_645_SUP__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,G2: a > epistemic_fm_a > $o] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ ( G2 @ J ) @ ( F @ X5 ) ) ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) )
= ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_646_cSup__eq__non__empty,axiom,
! [X6: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( X6 != bot_bo4908879468608929837fm_a_o )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ A2 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ X5 @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_647_cSup__eq__non__empty,axiom,
! [X6: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( X6 != bot_bo1868452413526482246c_fm_a )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ A2 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ X5 @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_648_cSup__least,axiom,
! [X6: set_Epistemic_fm_a_o,Z2: epistemic_fm_a > $o] :
( ( X6 != bot_bo4908879468608929837fm_a_o )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Z2 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_649_cSup__least,axiom,
! [X6: set_se5208064806568342746c_fm_a,Z2: set_Epistemic_fm_a] :
( ( X6 != bot_bo1868452413526482246c_fm_a )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Z2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_650_less__eq__Sup,axiom,
! [A: set_Epistemic_fm_a_o,U2: epistemic_fm_a > $o] :
( ! [V2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ V2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ V2 ) )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_651_less__eq__Sup,axiom,
! [A: set_se5208064806568342746c_fm_a,U2: set_Epistemic_fm_a] :
( ! [V2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ V2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ V2 ) )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_652_le__cSup__finite,axiom,
! [X6: set_Epistemic_fm_a_o,X3: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ X6 )
=> ( ( member4486839677911940090fm_a_o @ X3 @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( comple310706794042174646fm_a_o @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_653_le__cSup__finite,axiom,
! [X6: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ X6 )
=> ( ( member536094252920883875c_fm_a @ X3 @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( comple7868773486375872231c_fm_a @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_654_Sup__subset__mono,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ( ord_le4525777775358908921fm_a_o @ A @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B4 ) ) ) ).
% Sup_subset_mono
thf(fact_655_Sup__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B4 ) ) ) ).
% Sup_subset_mono
thf(fact_656_SUP__eqI,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [I5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I5 ) @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_657_SUP__eqI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [I5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I5 ) @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_658_SUP__eqI,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [I5: a] :
( ( member_a @ I5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I5 ) @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_659_SUP__eqI,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [I5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I5 ) @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_660_SUP__eqI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [I5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I5 ) @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_661_SUP__eqI,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [I5: a] :
( ( member_a @ I5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I5 ) @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_662_SUP__least,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_663_SUP__least,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_664_SUP__least,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_665_SUP__least,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ U2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_666_SUP__least,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ U2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_667_SUP__least,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ U2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_668_SUP__upper,axiom,
! [I: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I ) @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_669_SUP__upper,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I ) @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_670_SUP__upper,axiom,
! [I: a,A: set_a,F: a > epistemic_fm_a > $o] :
( ( member_a @ I @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I ) @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_671_SUP__upper,axiom,
! [I: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_672_SUP__upper,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_673_SUP__upper,axiom,
! [I: a,A: set_a,F: a > set_Epistemic_fm_a] :
( ( member_a @ I @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_674_SUP__upper2,axiom,
! [I: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,U2: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ I ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_675_SUP__upper2,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,U2: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ I ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_676_SUP__upper2,axiom,
! [I: a,A: set_a,U2: epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ( member_a @ I @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ I ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_677_SUP__upper2,axiom,
! [I: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,U2: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ I ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_678_SUP__upper2,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,U2: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ I ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_679_SUP__upper2,axiom,
! [I: a,A: set_a,U2: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ( member_a @ I @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ I ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_680_Epistemic__Logic_Oextend__def,axiom,
! [A: epistemic_fm_a > $o,S2: set_Epistemic_fm_a,N: epistemic_fm_a] :
( ( maxima1168816336865142315c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S2 @ N )
= ( wellor7234309849078405374c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ S2 @ ( maxima5458213586468375476c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S2 ) @ ( maxima7895811076573853563c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) @ N ) ) ).
% Epistemic_Logic.extend_def
thf(fact_681_finite__has__minimal,axiom,
! [A: set_Epistemic_fm_a_o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ? [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
& ! [Xa2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa2 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_682_finite__has__minimal,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ! [Xa2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa2 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_683_finite__has__maximal,axiom,
! [A: set_Epistemic_fm_a_o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ? [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
& ! [Xa2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa2 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_684_finite__has__maximal,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ! [Xa2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa2 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_685_finite__has__maximal2,axiom,
! [A: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( member4486839677911940090fm_a_o @ A2 @ A )
=> ? [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
& ( ord_le4043730696559282883fm_a_o @ A2 @ X2 )
& ! [Xa2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa2 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_686_finite__has__maximal2,axiom,
! [A: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( member536094252920883875c_fm_a @ A2 @ A )
=> ? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ( ord_le3275665582123262618c_fm_a @ A2 @ X2 )
& ! [Xa2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa2 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_687_finite__has__minimal2,axiom,
! [A: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( member4486839677911940090fm_a_o @ A2 @ A )
=> ? [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
& ( ord_le4043730696559282883fm_a_o @ X2 @ A2 )
& ! [Xa2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa2 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_688_finite__has__minimal2,axiom,
! [A: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( member536094252920883875c_fm_a @ A2 @ A )
=> ? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ( ord_le3275665582123262618c_fm_a @ X2 @ A2 )
& ! [Xa2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa2 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_689_adm__woL__extendL,axiom,
wellor8531372491315211020c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( maxima7895811076573853563c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) ).
% adm_woL_extendL
thf(fact_690_cSup__inter__less__eq,axiom,
! [A: set_set_a,B4: set_set_a] :
( ( condit3373647341569784514_set_a @ A )
=> ( ( condit3373647341569784514_set_a @ B4 )
=> ( ( ( inf_inf_set_set_a @ A @ B4 )
!= bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A @ B4 ) ) @ ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B4 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_691_cSup__inter__less__eq,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ( condit6663374420580310783fm_a_o @ A )
=> ( ( condit6663374420580310783fm_a_o @ B4 )
=> ( ( ( inf_in8896176524268821035fm_a_o @ A @ B4 )
!= bot_bo4908879468608929837fm_a_o )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( inf_in8896176524268821035fm_a_o @ A @ B4 ) ) @ ( sup_su6697315146409554831fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B4 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_692_cSup__inter__less__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( condit2203544419062987614c_fm_a @ A )
=> ( ( condit2203544419062987614c_fm_a @ B4 )
=> ( ( ( inf_in1884693029477671368c_fm_a @ A @ B4 )
!= bot_bo1868452413526482246c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( inf_in1884693029477671368c_fm_a @ A @ B4 ) ) @ ( sup_su1367922730591523534c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B4 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_693_cSUP__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,G2: set_Epistemic_fm_a > epistemic_fm_a > $o,B4: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( condit6663374420580310783fm_a_o @ ( image_6902499629341727256fm_a_o @ G2 @ B4 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ G2 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_694_cSUP__subset__mono,axiom,
! [A: set_a,G2: a > epistemic_fm_a > $o,B4: set_a,F: a > epistemic_fm_a > $o] :
( ( A != bot_bot_set_a )
=> ( ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ G2 @ B4 ) )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ G2 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_695_cSUP__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,G2: set_Epistemic_fm_a > set_Epistemic_fm_a,B4: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ ( image_8582343580727714565c_fm_a @ G2 @ B4 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ G2 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_696_cSUP__subset__mono,axiom,
! [A: set_a,G2: a > set_Epistemic_fm_a,B4: set_a,F: a > set_Epistemic_fm_a] :
( ( A != bot_bot_set_a )
=> ( ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ G2 @ B4 ) )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G2 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_697_cSUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,G2: epistemic_fm_a > epistemic_fm_a > $o,B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit6663374420580310783fm_a_o @ ( image_2381060403230254136fm_a_o @ G2 @ B4 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ G2 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_698_cSUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,G2: epistemic_fm_a > set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ ( image_1263732536703906021c_fm_a @ G2 @ B4 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G2 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_699_Inf__fin_Osubset__imp,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ( ord_le4525777775358908921fm_a_o @ A @ B4 )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( finite5407549452671652090fm_a_o @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( lattic8370817070757793286fm_a_o @ B4 ) @ ( lattic8370817070757793286fm_a_o @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_700_Inf__fin_Osubset__imp,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( finite7364866884270929315c_fm_a @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic7927669795667837335c_fm_a @ B4 ) @ ( lattic7927669795667837335c_fm_a @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_701_bdd__above_OI,axiom,
! [A: set_Epistemic_fm_a_o,M: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ M ) )
=> ( condit6663374420580310783fm_a_o @ A ) ) ).
% bdd_above.I
thf(fact_702_bdd__above_OI,axiom,
! [A: set_se5208064806568342746c_fm_a,M: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ M ) )
=> ( condit2203544419062987614c_fm_a @ A ) ) ).
% bdd_above.I
thf(fact_703_bdd__above_Ounfold,axiom,
( condit6663374420580310783fm_a_o
= ( ^ [A3: set_Epistemic_fm_a_o] :
? [M3: epistemic_fm_a > $o] :
! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ A3 )
=> ( ord_le4043730696559282883fm_a_o @ X @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_704_bdd__above_Ounfold,axiom,
( condit2203544419062987614c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a] :
? [M3: set_Epistemic_fm_a] :
! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ X @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_705_bdd__above_OE,axiom,
! [A: set_Epistemic_fm_a_o] :
( ( condit6663374420580310783fm_a_o @ A )
=> ~ ! [M2: epistemic_fm_a > $o] :
~ ! [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X5 @ M2 ) ) ) ).
% bdd_above.E
thf(fact_706_bdd__above_OE,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( condit2203544419062987614c_fm_a @ A )
=> ~ ! [M2: set_Epistemic_fm_a] :
~ ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X5 @ M2 ) ) ) ).
% bdd_above.E
thf(fact_707_bdd__above_OI2,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,M: epistemic_fm_a > $o] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ M ) )
=> ( condit6663374420580310783fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_708_bdd__above_OI2,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,M: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ M ) )
=> ( condit6663374420580310783fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_709_bdd__above_OI2,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,M: epistemic_fm_a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ M ) )
=> ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_710_bdd__above_OI2,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,M: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ M ) )
=> ( condit2203544419062987614c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_711_bdd__above_OI2,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,M: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ M ) )
=> ( condit2203544419062987614c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_712_bdd__above_OI2,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,M: set_Epistemic_fm_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ M ) )
=> ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_713_cSup__upper2,axiom,
! [X3: epistemic_fm_a > $o,X6: set_Epistemic_fm_a_o,Y: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X3 @ X6 )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( ( condit6663374420580310783fm_a_o @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ Y @ ( comple310706794042174646fm_a_o @ X6 ) ) ) ) ) ).
% cSup_upper2
thf(fact_714_cSup__upper2,axiom,
! [X3: set_Epistemic_fm_a,X6: set_se5208064806568342746c_fm_a,Y: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ X6 )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( ( condit2203544419062987614c_fm_a @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ Y @ ( comple7868773486375872231c_fm_a @ X6 ) ) ) ) ) ).
% cSup_upper2
thf(fact_715_cSup__upper,axiom,
! [X3: epistemic_fm_a > $o,X6: set_Epistemic_fm_a_o] :
( ( member4486839677911940090fm_a_o @ X3 @ X6 )
=> ( ( condit6663374420580310783fm_a_o @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( comple310706794042174646fm_a_o @ X6 ) ) ) ) ).
% cSup_upper
thf(fact_716_cSup__upper,axiom,
! [X3: set_Epistemic_fm_a,X6: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ X6 )
=> ( ( condit2203544419062987614c_fm_a @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( comple7868773486375872231c_fm_a @ X6 ) ) ) ) ).
% cSup_upper
thf(fact_717_Inf__fin_OcoboundedI,axiom,
! [A: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( member4486839677911940090fm_a_o @ A2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( lattic8370817070757793286fm_a_o @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_718_Inf__fin_OcoboundedI,axiom,
! [A: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( member536094252920883875c_fm_a @ A2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic7927669795667837335c_fm_a @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_719_cSUP__upper2,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a,U2: epistemic_fm_a > $o] :
( ( condit6663374420580310783fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) )
=> ( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ X3 ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_720_cSUP__upper2,axiom,
! [F: epistemic_fm_a > epistemic_fm_a > $o,A: set_Epistemic_fm_a,X3: epistemic_fm_a,U2: epistemic_fm_a > $o] :
( ( condit6663374420580310783fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
=> ( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ X3 ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_721_cSUP__upper2,axiom,
! [F: a > epistemic_fm_a > $o,A: set_a,X3: a,U2: epistemic_fm_a > $o] :
( ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
=> ( ( member_a @ X3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ X3 ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_722_cSUP__upper2,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( condit2203544419062987614c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
=> ( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ X3 ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_723_cSUP__upper2,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a,X3: epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( condit2203544419062987614c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
=> ( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ X3 ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_724_cSUP__upper2,axiom,
! [F: a > set_Epistemic_fm_a,A: set_a,X3: a,U2: set_Epistemic_fm_a] :
( ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
=> ( ( member_a @ X3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ X3 ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_725_cSUP__upper,axiom,
! [X3: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ( condit6663374420580310783fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_726_cSUP__upper,axiom,
! [X3: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( condit6663374420580310783fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_727_cSUP__upper,axiom,
! [X3: a,A: set_a,F: a > epistemic_fm_a > $o] :
( ( member_a @ X3 @ A )
=> ( ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_728_cSUP__upper,axiom,
! [X3: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ( condit2203544419062987614c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_729_cSUP__upper,axiom,
! [X3: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( condit2203544419062987614c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_730_cSUP__upper,axiom,
! [X3: a,A: set_a,F: a > set_Epistemic_fm_a] :
( ( member_a @ X3 @ A )
=> ( ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_731_cSup__le__iff,axiom,
! [S2: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( S2 != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6663374420580310783fm_a_o @ S2 )
=> ( ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ S2 ) @ A2 )
= ( ! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ S2 )
=> ( ord_le4043730696559282883fm_a_o @ X @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_732_cSup__le__iff,axiom,
! [S2: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( S2 != bot_bo1868452413526482246c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ S2 )
=> ( ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ S2 ) @ A2 )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ S2 )
=> ( ord_le3275665582123262618c_fm_a @ X @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_733_cSup__mono,axiom,
! [B4: set_Epistemic_fm_a_o,A: set_Epistemic_fm_a_o] :
( ( B4 != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6663374420580310783fm_a_o @ A )
=> ( ! [B5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ B5 @ B4 )
=> ? [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ B5 @ X5 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ B4 ) @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_734_cSup__mono,axiom,
! [B4: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( B4 != bot_bo1868452413526482246c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ A )
=> ( ! [B5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B5 @ B4 )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ B5 @ X5 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ B4 ) @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_735_Inf__fin_Obounded__iff,axiom,
! [A: set_Epistemic_fm_a_o,X3: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ ( lattic8370817070757793286fm_a_o @ A ) )
= ( ! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ A )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ X ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_736_Inf__fin_Obounded__iff,axiom,
! [A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ ( lattic7927669795667837335c_fm_a @ A ) )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ X ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_737_Inf__fin_OboundedI,axiom,
! [A: set_Epistemic_fm_a_o,X3: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ! [A5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ A5 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( lattic8370817070757793286fm_a_o @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_738_Inf__fin_OboundedI,axiom,
! [A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ! [A5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ A5 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( lattic7927669795667837335c_fm_a @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_739_Inf__fin_OboundedE,axiom,
! [A: set_Epistemic_fm_a_o,X3: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ ( lattic8370817070757793286fm_a_o @ A ) )
=> ! [A6: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A6 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ A6 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_740_Inf__fin_OboundedE,axiom,
! [A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ ( lattic7927669795667837335c_fm_a @ A ) )
=> ! [A6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A6 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ A6 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_741_cSUP__le__iff,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit6663374420580310783fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
=> ( ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ U2 )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_742_cSUP__le__iff,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ( A != bot_bot_set_a )
=> ( ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
=> ( ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ U2 )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_743_cSUP__le__iff,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
=> ( ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ U2 )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_744_cSUP__le__iff,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( A != bot_bot_set_a )
=> ( ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
=> ( ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ U2 )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_745_cSup__subset__mono,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6663374420580310783fm_a_o @ B4 )
=> ( ( ord_le4525777775358908921fm_a_o @ A @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_746_cSup__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ B4 )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_747_mono__cSUP,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: epistemic_fm_a > epistemic_fm_a > $o,I4: set_Epistemic_fm_a] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit6663374420580310783fm_a_o @ ( image_2381060403230254136fm_a_o @ A @ I4 ) )
=> ( ( I4 != bot_bo3626323581529592678c_fm_a )
=> ( ord_le4043730696559282883fm_a_o
@ ( comple310706794042174646fm_a_o
@ ( image_2381060403230254136fm_a_o
@ ^ [X: epistemic_fm_a] : ( F @ ( A @ X ) )
@ I4 ) )
@ ( F @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ A @ I4 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_748_mono__cSUP,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: a > epistemic_fm_a > $o,I4: set_a] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ A @ I4 ) )
=> ( ( I4 != bot_bot_set_a )
=> ( ord_le4043730696559282883fm_a_o
@ ( comple310706794042174646fm_a_o
@ ( image_3809950421912819996fm_a_o
@ ^ [X: a] : ( F @ ( A @ X ) )
@ I4 ) )
@ ( F @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ A @ I4 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_749_mono__cSUP,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: epistemic_fm_a > epistemic_fm_a > $o,I4: set_Epistemic_fm_a] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit6663374420580310783fm_a_o @ ( image_2381060403230254136fm_a_o @ A @ I4 ) )
=> ( ( I4 != bot_bo3626323581529592678c_fm_a )
=> ( ord_le3275665582123262618c_fm_a
@ ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X: epistemic_fm_a] : ( F @ ( A @ X ) )
@ I4 ) )
@ ( F @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ A @ I4 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_750_mono__cSUP,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: a > epistemic_fm_a > $o,I4: set_a] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ A @ I4 ) )
=> ( ( I4 != bot_bot_set_a )
=> ( ord_le3275665582123262618c_fm_a
@ ( comple7868773486375872231c_fm_a
@ ( image_3480842985853347585c_fm_a
@ ^ [X: a] : ( F @ ( A @ X ) )
@ I4 ) )
@ ( F @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ A @ I4 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_751_mono__cSUP,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: epistemic_fm_a > set_Epistemic_fm_a,I4: set_Epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit2203544419062987614c_fm_a @ ( image_1263732536703906021c_fm_a @ A @ I4 ) )
=> ( ( I4 != bot_bo3626323581529592678c_fm_a )
=> ( ord_le4043730696559282883fm_a_o
@ ( comple310706794042174646fm_a_o
@ ( image_2381060403230254136fm_a_o
@ ^ [X: epistemic_fm_a] : ( F @ ( A @ X ) )
@ I4 ) )
@ ( F @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ A @ I4 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_752_mono__cSUP,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: a > set_Epistemic_fm_a,I4: set_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ A @ I4 ) )
=> ( ( I4 != bot_bot_set_a )
=> ( ord_le4043730696559282883fm_a_o
@ ( comple310706794042174646fm_a_o
@ ( image_3809950421912819996fm_a_o
@ ^ [X: a] : ( F @ ( A @ X ) )
@ I4 ) )
@ ( F @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ A @ I4 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_753_mono__cSUP,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: epistemic_fm_a > set_Epistemic_fm_a,I4: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit2203544419062987614c_fm_a @ ( image_1263732536703906021c_fm_a @ A @ I4 ) )
=> ( ( I4 != bot_bo3626323581529592678c_fm_a )
=> ( ord_le3275665582123262618c_fm_a
@ ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X: epistemic_fm_a] : ( F @ ( A @ X ) )
@ I4 ) )
@ ( F @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ A @ I4 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_754_mono__cSUP,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: a > set_Epistemic_fm_a,I4: set_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ A @ I4 ) )
=> ( ( I4 != bot_bot_set_a )
=> ( ord_le3275665582123262618c_fm_a
@ ( comple7868773486375872231c_fm_a
@ ( image_3480842985853347585c_fm_a
@ ^ [X: a] : ( F @ ( A @ X ) )
@ I4 ) )
@ ( F @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ A @ I4 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_755_mono__cSup,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit6663374420580310783fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_6282542973537860301fm_a_o @ F @ A ) ) @ ( F @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ) ) ).
% mono_cSup
thf(fact_756_mono__cSup,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: set_Epistemic_fm_a_o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit6663374420580310783fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3065101469317257744c_fm_a @ F @ A ) ) @ ( F @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ) ) ).
% mono_cSup
thf(fact_757_mono__cSup,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_se5208064806568342746c_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit2203544419062987614c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ ( F @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ) ) ).
% mono_cSup
thf(fact_758_mono__cSup,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit2203544419062987614c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ ( F @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ) ) ).
% mono_cSup
thf(fact_759_Inf__fin__le__Sup__fin,axiom,
! [A: set_Epistemic_fm_a_o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ord_le4043730696559282883fm_a_o @ ( lattic8370817070757793286fm_a_o @ A ) @ ( lattic18181228415053292fm_a_o @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_760_Inf__fin__le__Sup__fin,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic7927669795667837335c_fm_a @ A ) @ ( lattic3176627891459829297c_fm_a @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_761_Sup__fin_Osubset__imp,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ( ord_le4525777775358908921fm_a_o @ A @ B4 )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( finite5407549452671652090fm_a_o @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( lattic18181228415053292fm_a_o @ A ) @ ( lattic18181228415053292fm_a_o @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_762_Sup__fin_Osubset__imp,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( finite7364866884270929315c_fm_a @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic3176627891459829297c_fm_a @ A ) @ ( lattic3176627891459829297c_fm_a @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_763_Sup__fin_OcoboundedI,axiom,
! [A: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( member4486839677911940090fm_a_o @ A2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( lattic18181228415053292fm_a_o @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_764_Sup__fin_OcoboundedI,axiom,
! [A: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( member536094252920883875c_fm_a @ A2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( lattic3176627891459829297c_fm_a @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_765_Sup__fin_OboundedE,axiom,
! [A: set_Epistemic_fm_a_o,X3: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( ord_le4043730696559282883fm_a_o @ ( lattic18181228415053292fm_a_o @ A ) @ X3 )
=> ! [A6: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A6 @ A )
=> ( ord_le4043730696559282883fm_a_o @ A6 @ X3 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_766_Sup__fin_OboundedE,axiom,
! [A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ ( lattic3176627891459829297c_fm_a @ A ) @ X3 )
=> ! [A6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A6 @ A )
=> ( ord_le3275665582123262618c_fm_a @ A6 @ X3 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_767_Sup__fin_OboundedI,axiom,
! [A: set_Epistemic_fm_a_o,X3: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ! [A5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ A5 @ X3 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( lattic18181228415053292fm_a_o @ A ) @ X3 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_768_Sup__fin_OboundedI,axiom,
! [A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ! [A5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ A5 @ X3 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( lattic3176627891459829297c_fm_a @ A ) @ X3 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_769_Sup__fin_Obounded__iff,axiom,
! [A: set_Epistemic_fm_a_o,X3: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( ord_le4043730696559282883fm_a_o @ ( lattic18181228415053292fm_a_o @ A ) @ X3 )
= ( ! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ A )
=> ( ord_le4043730696559282883fm_a_o @ X @ X3 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_770_Sup__fin_Obounded__iff,axiom,
! [A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( ord_le3275665582123262618c_fm_a @ ( lattic3176627891459829297c_fm_a @ A ) @ X3 )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
=> ( ord_le3275665582123262618c_fm_a @ X @ X3 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_771_bdd__above__image__mono,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit6663374420580310783fm_a_o @ A )
=> ( condit6663374420580310783fm_a_o @ ( image_6282542973537860301fm_a_o @ F @ A ) ) ) ) ).
% bdd_above_image_mono
thf(fact_772_bdd__above__image__mono,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: set_Epistemic_fm_a_o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit6663374420580310783fm_a_o @ A )
=> ( condit2203544419062987614c_fm_a @ ( image_3065101469317257744c_fm_a @ F @ A ) ) ) ) ).
% bdd_above_image_mono
thf(fact_773_bdd__above__image__mono,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_se5208064806568342746c_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit2203544419062987614c_fm_a @ A )
=> ( condit6663374420580310783fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ).
% bdd_above_image_mono
thf(fact_774_bdd__above__image__mono,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit2203544419062987614c_fm_a @ A )
=> ( condit2203544419062987614c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ).
% bdd_above_image_mono
thf(fact_775_mono__Sup,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_6282542973537860301fm_a_o @ F @ A ) ) @ ( F @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ).
% mono_Sup
thf(fact_776_mono__Sup,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: set_Epistemic_fm_a_o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3065101469317257744c_fm_a @ F @ A ) ) @ ( F @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ).
% mono_Sup
thf(fact_777_mono__Sup,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_se5208064806568342746c_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ ( F @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ).
% mono_Sup
thf(fact_778_mono__Sup,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ ( F @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ).
% mono_Sup
thf(fact_779_coinduct3__mono__lemma,axiom,
! [F: ( epistemic_fm_a > $o ) > set_a,X6: set_a,B4: set_a] :
( ( monoto2780732390255651730_set_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_less_eq_set_a @ F )
=> ( monoto2780732390255651730_set_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_less_eq_set_a
@ ^ [X: epistemic_fm_a > $o] : ( sup_sup_set_a @ ( sup_sup_set_a @ ( F @ X ) @ X6 ) @ B4 ) ) ) ).
% coinduct3_mono_lemma
thf(fact_780_coinduct3__mono__lemma,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,X6: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a
@ ^ [X: epistemic_fm_a > $o] : ( sup_su1367922730591523534c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( F @ X ) @ X6 ) @ B4 ) ) ) ).
% coinduct3_mono_lemma
thf(fact_781_coinduct3__mono__lemma,axiom,
! [F: set_Epistemic_fm_a > set_a,X6: set_a,B4: set_a] :
( ( monoto8272269529309540939_set_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_less_eq_set_a @ F )
=> ( monoto8272269529309540939_set_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_less_eq_set_a
@ ^ [X: set_Epistemic_fm_a] : ( sup_sup_set_a @ ( sup_sup_set_a @ ( F @ X ) @ X6 ) @ B4 ) ) ) ).
% coinduct3_mono_lemma
thf(fact_782_coinduct3__mono__lemma,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,X6: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( sup_su1367922730591523534c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( F @ X ) @ X6 ) @ B4 ) ) ) ).
% coinduct3_mono_lemma
thf(fact_783_mono__sup,axiom,
! [F: set_a > set_a,A: set_a,B4: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( F @ A ) @ ( F @ B4 ) ) @ ( F @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% mono_sup
thf(fact_784_mono__sup,axiom,
! [F: set_a > epistemic_fm_a > $o,A: set_a,B4: set_a] :
( ( monoto4680196969853040026fm_a_o @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ ( F @ A ) @ ( F @ B4 ) ) @ ( F @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% mono_sup
thf(fact_785_mono__sup,axiom,
! [F: set_a > set_Epistemic_fm_a,A: set_a,B4: set_a] :
( ( monoto6782103247751874819c_fm_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( F @ A ) @ ( F @ B4 ) ) @ ( F @ ( sup_sup_set_a @ A @ B4 ) ) ) ) ).
% mono_sup
thf(fact_786_mono__sup,axiom,
! [F: ( epistemic_fm_a > $o ) > set_a,A: epistemic_fm_a > $o,B4: epistemic_fm_a > $o] :
( ( monoto2780732390255651730_set_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( F @ A ) @ ( F @ B4 ) ) @ ( F @ ( sup_su6697315146409554831fm_a_o @ A @ B4 ) ) ) ) ).
% mono_sup
thf(fact_787_mono__sup,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: epistemic_fm_a > $o,B4: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ ( F @ A ) @ ( F @ B4 ) ) @ ( F @ ( sup_su6697315146409554831fm_a_o @ A @ B4 ) ) ) ) ).
% mono_sup
thf(fact_788_mono__sup,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: epistemic_fm_a > $o,B4: epistemic_fm_a > $o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( F @ A ) @ ( F @ B4 ) ) @ ( F @ ( sup_su6697315146409554831fm_a_o @ A @ B4 ) ) ) ) ).
% mono_sup
thf(fact_789_mono__sup,axiom,
! [F: set_Epistemic_fm_a > set_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( monoto8272269529309540939_set_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( F @ A ) @ ( F @ B4 ) ) @ ( F @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) ) ) ) ).
% mono_sup
thf(fact_790_mono__sup,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ ( F @ A ) @ ( F @ B4 ) ) @ ( F @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) ) ) ) ).
% mono_sup
thf(fact_791_mono__sup,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( F @ A ) @ ( F @ B4 ) ) @ ( F @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) ) ) ) ).
% mono_sup
thf(fact_792_mono__inf,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: epistemic_fm_a > $o,B4: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ ( inf_in3606484609122063093fm_a_o @ A @ B4 ) ) @ ( inf_in3606484609122063093fm_a_o @ ( F @ A ) @ ( F @ B4 ) ) ) ) ).
% mono_inf
thf(fact_793_mono__inf,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: epistemic_fm_a > $o,B4: epistemic_fm_a > $o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ ( inf_in3606484609122063093fm_a_o @ A @ B4 ) ) @ ( inf_in2882328776738922472c_fm_a @ ( F @ A ) @ ( F @ B4 ) ) ) ) ).
% mono_inf
thf(fact_794_mono__inf,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ ( inf_in2882328776738922472c_fm_a @ A @ B4 ) ) @ ( inf_in3606484609122063093fm_a_o @ ( F @ A ) @ ( F @ B4 ) ) ) ) ).
% mono_inf
thf(fact_795_mono__inf,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ ( inf_in2882328776738922472c_fm_a @ A @ B4 ) ) @ ( inf_in2882328776738922472c_fm_a @ ( F @ A ) @ ( F @ B4 ) ) ) ) ).
% mono_inf
thf(fact_796_mono__compose,axiom,
! [Q4: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ Q4 )
=> ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o
@ ^ [I2: epistemic_fm_a > $o,X: epistemic_fm_a] : ( Q4 @ I2 @ ( F @ X ) ) ) ) ).
% mono_compose
thf(fact_797_mono__compose,axiom,
! [Q4: set_Epistemic_fm_a > epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ Q4 )
=> ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o
@ ^ [I2: set_Epistemic_fm_a,X: epistemic_fm_a] : ( Q4 @ I2 @ ( F @ X ) ) ) ) ).
% mono_compose
thf(fact_798_antimonoI,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ Y2 ) @ ( F @ X2 ) ) )
=> ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ F ) ) ).
% antimonoI
thf(fact_799_antimonoI,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ Y2 ) @ ( F @ X2 ) ) )
=> ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ F ) ) ).
% antimonoI
thf(fact_800_antimonoI,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ Y2 ) @ ( F @ X2 ) ) )
=> ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ F ) ) ).
% antimonoI
thf(fact_801_antimonoI,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ Y2 ) @ ( F @ X2 ) ) )
=> ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ F ) ) ).
% antimonoI
thf(fact_802_mono__onI,axiom,
! [A: set_Epistemic_fm_a_o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ! [R4: epistemic_fm_a > $o,S5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ R4 @ A )
=> ( ( member4486839677911940090fm_a_o @ S5 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ R4 @ S5 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R4 ) @ ( F @ S5 ) ) ) ) )
=> ( monoto1601457452331330095fm_a_o @ A @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F ) ) ).
% mono_onI
thf(fact_803_mono__onI,axiom,
! [A: set_Epistemic_fm_a_o,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a] :
( ! [R4: epistemic_fm_a > $o,S5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ R4 @ A )
=> ( ( member4486839677911940090fm_a_o @ S5 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ R4 @ S5 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R4 ) @ ( F @ S5 ) ) ) ) )
=> ( monoto766724803074413614c_fm_a @ A @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F ) ) ).
% mono_onI
thf(fact_804_mono__onI,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ! [R4: set_Epistemic_fm_a,S5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ R4 @ A )
=> ( ( member536094252920883875c_fm_a @ S5 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ R4 @ S5 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R4 ) @ ( F @ S5 ) ) ) ) )
=> ( monoto4604122963098883126fm_a_o @ A @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F ) ) ).
% mono_onI
thf(fact_805_mono__onI,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [R4: set_Epistemic_fm_a,S5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ R4 @ A )
=> ( ( member536094252920883875c_fm_a @ S5 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ R4 @ S5 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R4 ) @ ( F @ S5 ) ) ) ) )
=> ( monoto5809647437951573863c_fm_a @ A @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F ) ) ).
% mono_onI
thf(fact_806_mono__onD,axiom,
! [A: set_Epistemic_fm_a_o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,R: epistemic_fm_a > $o,S6: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ A @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( member4486839677911940090fm_a_o @ R @ A )
=> ( ( member4486839677911940090fm_a_o @ S6 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ R @ S6 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R ) @ ( F @ S6 ) ) ) ) ) ) ).
% mono_onD
thf(fact_807_mono__onD,axiom,
! [A: set_Epistemic_fm_a_o,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,R: epistemic_fm_a > $o,S6: epistemic_fm_a > $o] :
( ( monoto766724803074413614c_fm_a @ A @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( member4486839677911940090fm_a_o @ R @ A )
=> ( ( member4486839677911940090fm_a_o @ S6 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ R @ S6 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R ) @ ( F @ S6 ) ) ) ) ) ) ).
% mono_onD
thf(fact_808_mono__onD,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,R: set_Epistemic_fm_a,S6: set_Epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ A @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( member536094252920883875c_fm_a @ R @ A )
=> ( ( member536094252920883875c_fm_a @ S6 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ R @ S6 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R ) @ ( F @ S6 ) ) ) ) ) ) ).
% mono_onD
thf(fact_809_mono__onD,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,R: set_Epistemic_fm_a,S6: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ A @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( member536094252920883875c_fm_a @ R @ A )
=> ( ( member536094252920883875c_fm_a @ S6 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ R @ S6 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R ) @ ( F @ S6 ) ) ) ) ) ) ).
% mono_onD
thf(fact_810_ord_Omono__on__def,axiom,
! [A: set_se5208064806568342746c_fm_a,Less_eq: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( monoto4604122963098883126fm_a_o @ A @ Less_eq @ ord_le4043730696559282883fm_a_o @ F )
= ( ! [R5: set_Epistemic_fm_a,S7: set_Epistemic_fm_a] :
( ( ( member536094252920883875c_fm_a @ R5 @ A )
& ( member536094252920883875c_fm_a @ S7 @ A )
& ( Less_eq @ R5 @ S7 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R5 ) @ ( F @ S7 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_811_ord_Omono__on__def,axiom,
! [A: set_Epistemic_fm_a,Less_eq: epistemic_fm_a > epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( monoto335617853130380886fm_a_o @ A @ Less_eq @ ord_le4043730696559282883fm_a_o @ F )
= ( ! [R5: epistemic_fm_a,S7: epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ R5 @ A )
& ( member6642669571620171971c_fm_a @ S7 @ A )
& ( Less_eq @ R5 @ S7 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R5 ) @ ( F @ S7 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_812_ord_Omono__on__def,axiom,
! [A: set_a,Less_eq: a > a > $o,F: a > epistemic_fm_a > $o] :
( ( monoto6269740591336016954fm_a_o @ A @ Less_eq @ ord_le4043730696559282883fm_a_o @ F )
= ( ! [R5: a,S7: a] :
( ( ( member_a @ R5 @ A )
& ( member_a @ S7 @ A )
& ( Less_eq @ R5 @ S7 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R5 ) @ ( F @ S7 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_813_ord_Omono__on__def,axiom,
! [A: set_se5208064806568342746c_fm_a,Less_eq: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ A @ Less_eq @ ord_le3275665582123262618c_fm_a @ F )
= ( ! [R5: set_Epistemic_fm_a,S7: set_Epistemic_fm_a] :
( ( ( member536094252920883875c_fm_a @ R5 @ A )
& ( member536094252920883875c_fm_a @ S7 @ A )
& ( Less_eq @ R5 @ S7 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R5 ) @ ( F @ S7 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_814_ord_Omono__on__def,axiom,
! [A: set_Epistemic_fm_a,Less_eq: epistemic_fm_a > epistemic_fm_a > $o,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto6375051269373228359c_fm_a @ A @ Less_eq @ ord_le3275665582123262618c_fm_a @ F )
= ( ! [R5: epistemic_fm_a,S7: epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ R5 @ A )
& ( member6642669571620171971c_fm_a @ S7 @ A )
& ( Less_eq @ R5 @ S7 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R5 ) @ ( F @ S7 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_815_ord_Omono__on__def,axiom,
! [A: set_a,Less_eq: a > a > $o,F: a > set_Epistemic_fm_a] :
( ( monoto6536077295608523363c_fm_a @ A @ Less_eq @ ord_le3275665582123262618c_fm_a @ F )
= ( ! [R5: a,S7: a] :
( ( ( member_a @ R5 @ A )
& ( member_a @ S7 @ A )
& ( Less_eq @ R5 @ S7 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R5 ) @ ( F @ S7 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_816_ord_Omono__onI,axiom,
! [A: set_se5208064806568342746c_fm_a,Less_eq: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ! [R4: set_Epistemic_fm_a,S5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ R4 @ A )
=> ( ( member536094252920883875c_fm_a @ S5 @ A )
=> ( ( Less_eq @ R4 @ S5 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R4 ) @ ( F @ S5 ) ) ) ) )
=> ( monoto4604122963098883126fm_a_o @ A @ Less_eq @ ord_le4043730696559282883fm_a_o @ F ) ) ).
% ord.mono_onI
thf(fact_817_ord_Omono__onI,axiom,
! [A: set_Epistemic_fm_a,Less_eq: epistemic_fm_a > epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [R4: epistemic_fm_a,S5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ R4 @ A )
=> ( ( member6642669571620171971c_fm_a @ S5 @ A )
=> ( ( Less_eq @ R4 @ S5 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R4 ) @ ( F @ S5 ) ) ) ) )
=> ( monoto335617853130380886fm_a_o @ A @ Less_eq @ ord_le4043730696559282883fm_a_o @ F ) ) ).
% ord.mono_onI
thf(fact_818_ord_Omono__onI,axiom,
! [A: set_a,Less_eq: a > a > $o,F: a > epistemic_fm_a > $o] :
( ! [R4: a,S5: a] :
( ( member_a @ R4 @ A )
=> ( ( member_a @ S5 @ A )
=> ( ( Less_eq @ R4 @ S5 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R4 ) @ ( F @ S5 ) ) ) ) )
=> ( monoto6269740591336016954fm_a_o @ A @ Less_eq @ ord_le4043730696559282883fm_a_o @ F ) ) ).
% ord.mono_onI
thf(fact_819_ord_Omono__onI,axiom,
! [A: set_se5208064806568342746c_fm_a,Less_eq: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [R4: set_Epistemic_fm_a,S5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ R4 @ A )
=> ( ( member536094252920883875c_fm_a @ S5 @ A )
=> ( ( Less_eq @ R4 @ S5 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R4 ) @ ( F @ S5 ) ) ) ) )
=> ( monoto5809647437951573863c_fm_a @ A @ Less_eq @ ord_le3275665582123262618c_fm_a @ F ) ) ).
% ord.mono_onI
thf(fact_820_ord_Omono__onI,axiom,
! [A: set_Epistemic_fm_a,Less_eq: epistemic_fm_a > epistemic_fm_a > $o,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [R4: epistemic_fm_a,S5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ R4 @ A )
=> ( ( member6642669571620171971c_fm_a @ S5 @ A )
=> ( ( Less_eq @ R4 @ S5 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R4 ) @ ( F @ S5 ) ) ) ) )
=> ( monoto6375051269373228359c_fm_a @ A @ Less_eq @ ord_le3275665582123262618c_fm_a @ F ) ) ).
% ord.mono_onI
thf(fact_821_ord_Omono__onI,axiom,
! [A: set_a,Less_eq: a > a > $o,F: a > set_Epistemic_fm_a] :
( ! [R4: a,S5: a] :
( ( member_a @ R4 @ A )
=> ( ( member_a @ S5 @ A )
=> ( ( Less_eq @ R4 @ S5 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R4 ) @ ( F @ S5 ) ) ) ) )
=> ( monoto6536077295608523363c_fm_a @ A @ Less_eq @ ord_le3275665582123262618c_fm_a @ F ) ) ).
% ord.mono_onI
thf(fact_822_ord_Omono__onD,axiom,
! [A: set_se5208064806568342746c_fm_a,Less_eq: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o,R: set_Epistemic_fm_a,S6: set_Epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ A @ Less_eq @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( member536094252920883875c_fm_a @ R @ A )
=> ( ( member536094252920883875c_fm_a @ S6 @ A )
=> ( ( Less_eq @ R @ S6 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R ) @ ( F @ S6 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_823_ord_Omono__onD,axiom,
! [A: set_Epistemic_fm_a,Less_eq: epistemic_fm_a > epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o,R: epistemic_fm_a,S6: epistemic_fm_a] :
( ( monoto335617853130380886fm_a_o @ A @ Less_eq @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( member6642669571620171971c_fm_a @ R @ A )
=> ( ( member6642669571620171971c_fm_a @ S6 @ A )
=> ( ( Less_eq @ R @ S6 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R ) @ ( F @ S6 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_824_ord_Omono__onD,axiom,
! [A: set_a,Less_eq: a > a > $o,F: a > epistemic_fm_a > $o,R: a,S6: a] :
( ( monoto6269740591336016954fm_a_o @ A @ Less_eq @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( member_a @ R @ A )
=> ( ( member_a @ S6 @ A )
=> ( ( Less_eq @ R @ S6 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ R ) @ ( F @ S6 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_825_ord_Omono__onD,axiom,
! [A: set_se5208064806568342746c_fm_a,Less_eq: set_Epistemic_fm_a > set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > set_Epistemic_fm_a,R: set_Epistemic_fm_a,S6: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ A @ Less_eq @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( member536094252920883875c_fm_a @ R @ A )
=> ( ( member536094252920883875c_fm_a @ S6 @ A )
=> ( ( Less_eq @ R @ S6 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R ) @ ( F @ S6 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_826_ord_Omono__onD,axiom,
! [A: set_Epistemic_fm_a,Less_eq: epistemic_fm_a > epistemic_fm_a > $o,F: epistemic_fm_a > set_Epistemic_fm_a,R: epistemic_fm_a,S6: epistemic_fm_a] :
( ( monoto6375051269373228359c_fm_a @ A @ Less_eq @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( member6642669571620171971c_fm_a @ R @ A )
=> ( ( member6642669571620171971c_fm_a @ S6 @ A )
=> ( ( Less_eq @ R @ S6 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R ) @ ( F @ S6 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_827_ord_Omono__onD,axiom,
! [A: set_a,Less_eq: a > a > $o,F: a > set_Epistemic_fm_a,R: a,S6: a] :
( ( monoto6536077295608523363c_fm_a @ A @ Less_eq @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( member_a @ R @ A )
=> ( ( member_a @ S6 @ A )
=> ( ( Less_eq @ R @ S6 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ R ) @ ( F @ S6 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_828_mono__imp__mono__on,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( monoto1601457452331330095fm_a_o @ A @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F ) ) ).
% mono_imp_mono_on
thf(fact_829_mono__imp__mono__on,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: set_Epistemic_fm_a_o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( monoto766724803074413614c_fm_a @ A @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_830_mono__imp__mono__on,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_se5208064806568342746c_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( monoto4604122963098883126fm_a_o @ A @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F ) ) ).
% mono_imp_mono_on
thf(fact_831_mono__imp__mono__on,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( monoto5809647437951573863c_fm_a @ A @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_832_monoI,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F ) ) ).
% monoI
thf(fact_833_monoI,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F ) ) ).
% monoI
thf(fact_834_monoI,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F ) ) ).
% monoI
thf(fact_835_monoI,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F ) ) ).
% monoI
thf(fact_836_monoE,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_837_monoE,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_838_monoE,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_839_monoE,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_840_monoD,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_841_monoD,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_842_monoD,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_843_monoD,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_844_ord_Omono__on__subset,axiom,
! [A: set_Epistemic_fm_a,Less_eq: epistemic_fm_a > epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o,B4: set_Epistemic_fm_a] :
( ( monoto335617853130380886fm_a_o @ A @ Less_eq @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( monoto335617853130380886fm_a_o @ B4 @ Less_eq @ ord_le4043730696559282883fm_a_o @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_845_ord_Omono__on__subset,axiom,
! [A: set_Epistemic_fm_a,Less_eq: epistemic_fm_a > epistemic_fm_a > $o,F: epistemic_fm_a > set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( monoto6375051269373228359c_fm_a @ A @ Less_eq @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( monoto6375051269373228359c_fm_a @ B4 @ Less_eq @ ord_le3275665582123262618c_fm_a @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_846_mono__on__subset,axiom,
! [A: set_Epistemic_fm_a_o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,B4: set_Epistemic_fm_a_o] :
( ( monoto1601457452331330095fm_a_o @ A @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ord_le4525777775358908921fm_a_o @ B4 @ A )
=> ( monoto1601457452331330095fm_a_o @ B4 @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F ) ) ) ).
% mono_on_subset
thf(fact_847_mono__on__subset,axiom,
! [A: set_Epistemic_fm_a_o,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,B4: set_Epistemic_fm_a_o] :
( ( monoto766724803074413614c_fm_a @ A @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ord_le4525777775358908921fm_a_o @ B4 @ A )
=> ( monoto766724803074413614c_fm_a @ B4 @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F ) ) ) ).
% mono_on_subset
thf(fact_848_mono__on__subset,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,B4: set_se5208064806568342746c_fm_a] :
( ( monoto4604122963098883126fm_a_o @ A @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ A )
=> ( monoto4604122963098883126fm_a_o @ B4 @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F ) ) ) ).
% mono_on_subset
thf(fact_849_mono__on__subset,axiom,
! [A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( monoto5809647437951573863c_fm_a @ A @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ord_le7112219575281605754c_fm_a @ B4 @ A )
=> ( monoto5809647437951573863c_fm_a @ B4 @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F ) ) ) ).
% mono_on_subset
thf(fact_850_antimonoD,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ F )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoD
thf(fact_851_antimonoD,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ F )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoD
thf(fact_852_antimonoD,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ F )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoD
thf(fact_853_antimonoD,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ F )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoD
thf(fact_854_antimonoE,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ F )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoE
thf(fact_855_antimonoE,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ F )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoE
thf(fact_856_antimonoE,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ F )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoE
thf(fact_857_antimonoE,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ F )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoE
thf(fact_858_Least__mono,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,S2: set_Epistemic_fm_a_o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ? [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ S2 )
& ! [Xa: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa @ S2 )
=> ( ord_le4043730696559282883fm_a_o @ X5 @ Xa ) ) )
=> ( ( ord_Le8586633845379911507fm_a_o
@ ^ [Y4: epistemic_fm_a > $o] : ( member4486839677911940090fm_a_o @ Y4 @ ( image_6282542973537860301fm_a_o @ F @ S2 ) ) )
= ( F
@ ( ord_Le8586633845379911507fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( member4486839677911940090fm_a_o @ X @ S2 ) ) ) ) ) ) ).
% Least_mono
thf(fact_859_Least__mono,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,S2: set_Epistemic_fm_a_o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ? [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ S2 )
& ! [Xa: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Xa @ S2 )
=> ( ord_le4043730696559282883fm_a_o @ X5 @ Xa ) ) )
=> ( ( ord_Le4844491855390411274c_fm_a
@ ^ [Y4: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ Y4 @ ( image_3065101469317257744c_fm_a @ F @ S2 ) ) )
= ( F
@ ( ord_Le8586633845379911507fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( member4486839677911940090fm_a_o @ X @ S2 ) ) ) ) ) ) ).
% Least_mono
thf(fact_860_Least__mono,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,S2: set_se5208064806568342746c_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ S2 )
& ! [Xa: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa @ S2 )
=> ( ord_le3275665582123262618c_fm_a @ X5 @ Xa ) ) )
=> ( ( ord_Le8586633845379911507fm_a_o
@ ^ [Y4: epistemic_fm_a > $o] : ( member4486839677911940090fm_a_o @ Y4 @ ( image_6902499629341727256fm_a_o @ F @ S2 ) ) )
= ( F
@ ( ord_Le4844491855390411274c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ S2 ) ) ) ) ) ) ).
% Least_mono
thf(fact_861_Least__mono,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,S2: set_se5208064806568342746c_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ S2 )
& ! [Xa: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Xa @ S2 )
=> ( ord_le3275665582123262618c_fm_a @ X5 @ Xa ) ) )
=> ( ( ord_Le4844491855390411274c_fm_a
@ ^ [Y4: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ Y4 @ ( image_8582343580727714565c_fm_a @ F @ S2 ) ) )
= ( F
@ ( ord_Le4844491855390411274c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ S2 ) ) ) ) ) ) ).
% Least_mono
thf(fact_862_bdd__above__image__antimono,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ F )
=> ( ( condit6597432229783682067fm_a_o @ A )
=> ( condit6663374420580310783fm_a_o @ ( image_6282542973537860301fm_a_o @ F @ A ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_863_bdd__above__image__antimono,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: set_Epistemic_fm_a_o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ F )
=> ( ( condit6597432229783682067fm_a_o @ A )
=> ( condit2203544419062987614c_fm_a @ ( image_3065101469317257744c_fm_a @ F @ A ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_864_bdd__above__image__antimono,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_se5208064806568342746c_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ F )
=> ( ( condit4283900098890313034c_fm_a @ A )
=> ( condit6663374420580310783fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_865_bdd__above__image__antimono,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ F )
=> ( ( condit4283900098890313034c_fm_a @ A )
=> ( condit2203544419062987614c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_866_bdd__belowI,axiom,
! [A: set_Epistemic_fm_a_o,M5: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ M5 @ X2 ) )
=> ( condit6597432229783682067fm_a_o @ A ) ) ).
% bdd_belowI
thf(fact_867_bdd__belowI,axiom,
! [A: set_se5208064806568342746c_fm_a,M5: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ M5 @ X2 ) )
=> ( condit4283900098890313034c_fm_a @ A ) ) ).
% bdd_belowI
thf(fact_868_bdd__below_OI,axiom,
! [A: set_Epistemic_fm_a_o,M: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ M @ X2 ) )
=> ( condit6597432229783682067fm_a_o @ A ) ) ).
% bdd_below.I
thf(fact_869_bdd__below_OI,axiom,
! [A: set_se5208064806568342746c_fm_a,M: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ M @ X2 ) )
=> ( condit4283900098890313034c_fm_a @ A ) ) ).
% bdd_below.I
thf(fact_870_bdd__below_OE,axiom,
! [A: set_Epistemic_fm_a_o] :
( ( condit6597432229783682067fm_a_o @ A )
=> ~ ! [M2: epistemic_fm_a > $o] :
~ ! [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ M2 @ X5 ) ) ) ).
% bdd_below.E
thf(fact_871_bdd__below_OE,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( condit4283900098890313034c_fm_a @ A )
=> ~ ! [M2: set_Epistemic_fm_a] :
~ ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ M2 @ X5 ) ) ) ).
% bdd_below.E
thf(fact_872_bdd__below_Ounfold,axiom,
( condit6597432229783682067fm_a_o
= ( ^ [A3: set_Epistemic_fm_a_o] :
? [M3: epistemic_fm_a > $o] :
! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ A3 )
=> ( ord_le4043730696559282883fm_a_o @ M3 @ X ) ) ) ) ).
% bdd_below.unfold
thf(fact_873_bdd__below_Ounfold,axiom,
( condit4283900098890313034c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a] :
? [M3: set_Epistemic_fm_a] :
! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ M3 @ X ) ) ) ) ).
% bdd_below.unfold
thf(fact_874_Least__equality,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,X3: epistemic_fm_a > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( ord_Le8586633845379911507fm_a_o @ P4 )
= X3 ) ) ) ).
% Least_equality
thf(fact_875_Least__equality,axiom,
! [P4: set_Epistemic_fm_a > $o,X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( ord_Le4844491855390411274c_fm_a @ P4 )
= X3 ) ) ) ).
% Least_equality
thf(fact_876_LeastI2__order,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,X3: epistemic_fm_a > $o,Q4: ( epistemic_fm_a > $o ) > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( P4 @ X2 )
=> ( ! [Y5: epistemic_fm_a > $o] :
( ( P4 @ Y5 )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Y5 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( ord_Le8586633845379911507fm_a_o @ P4 ) ) ) ) ) ).
% LeastI2_order
thf(fact_877_LeastI2__order,axiom,
! [P4: set_Epistemic_fm_a > $o,X3: set_Epistemic_fm_a,Q4: set_Epistemic_fm_a > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( ! [Y5: set_Epistemic_fm_a] :
( ( P4 @ Y5 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Y5 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( ord_Le4844491855390411274c_fm_a @ P4 ) ) ) ) ) ).
% LeastI2_order
thf(fact_878_Least1__le,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,Z2: epistemic_fm_a > $o] :
( ? [X5: epistemic_fm_a > $o] :
( ( P4 @ X5 )
& ! [Y2: epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ X5 @ Y2 ) )
& ! [Y2: epistemic_fm_a > $o] :
( ( ( P4 @ Y2 )
& ! [Ya2: epistemic_fm_a > $o] :
( ( P4 @ Ya2 )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ Ya2 ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P4 @ Z2 )
=> ( ord_le4043730696559282883fm_a_o @ ( ord_Le8586633845379911507fm_a_o @ P4 ) @ Z2 ) ) ) ).
% Least1_le
thf(fact_879_Least1__le,axiom,
! [P4: set_Epistemic_fm_a > $o,Z2: set_Epistemic_fm_a] :
( ? [X5: set_Epistemic_fm_a] :
( ( P4 @ X5 )
& ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ X5 @ Y2 ) )
& ! [Y2: set_Epistemic_fm_a] :
( ( ( P4 @ Y2 )
& ! [Ya2: set_Epistemic_fm_a] :
( ( P4 @ Ya2 )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ Ya2 ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P4 @ Z2 )
=> ( ord_le3275665582123262618c_fm_a @ ( ord_Le4844491855390411274c_fm_a @ P4 ) @ Z2 ) ) ) ).
% Least1_le
thf(fact_880_Least1I,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o] :
( ? [X5: epistemic_fm_a > $o] :
( ( P4 @ X5 )
& ! [Y2: epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ X5 @ Y2 ) )
& ! [Y2: epistemic_fm_a > $o] :
( ( ( P4 @ Y2 )
& ! [Ya2: epistemic_fm_a > $o] :
( ( P4 @ Ya2 )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ Ya2 ) ) )
=> ( Y2 = X5 ) ) )
=> ( P4 @ ( ord_Le8586633845379911507fm_a_o @ P4 ) ) ) ).
% Least1I
thf(fact_881_Least1I,axiom,
! [P4: set_Epistemic_fm_a > $o] :
( ? [X5: set_Epistemic_fm_a] :
( ( P4 @ X5 )
& ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ X5 @ Y2 ) )
& ! [Y2: set_Epistemic_fm_a] :
( ( ( P4 @ Y2 )
& ! [Ya2: set_Epistemic_fm_a] :
( ( P4 @ Ya2 )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ Ya2 ) ) )
=> ( Y2 = X5 ) ) )
=> ( P4 @ ( ord_Le4844491855390411274c_fm_a @ P4 ) ) ) ).
% Least1I
thf(fact_882_bdd__belowI2,axiom,
! [A: set_se5208064806568342746c_fm_a,M5: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ M5 @ ( F @ X2 ) ) )
=> ( condit6597432229783682067fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_883_bdd__belowI2,axiom,
! [A: set_Epistemic_fm_a,M5: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ M5 @ ( F @ X2 ) ) )
=> ( condit6597432229783682067fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_884_bdd__belowI2,axiom,
! [A: set_a,M5: epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ M5 @ ( F @ X2 ) ) )
=> ( condit6597432229783682067fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_885_bdd__belowI2,axiom,
! [A: set_se5208064806568342746c_fm_a,M5: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ M5 @ ( F @ X2 ) ) )
=> ( condit4283900098890313034c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_886_bdd__belowI2,axiom,
! [A: set_Epistemic_fm_a,M5: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ M5 @ ( F @ X2 ) ) )
=> ( condit4283900098890313034c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_887_bdd__belowI2,axiom,
! [A: set_a,M5: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ M5 @ ( F @ X2 ) ) )
=> ( condit4283900098890313034c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ).
% bdd_belowI2
thf(fact_888_bdd__below_OI2,axiom,
! [A: set_se5208064806568342746c_fm_a,M: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ M @ ( F @ X2 ) ) )
=> ( condit6597432229783682067fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_889_bdd__below_OI2,axiom,
! [A: set_Epistemic_fm_a,M: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ M @ ( F @ X2 ) ) )
=> ( condit6597432229783682067fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_890_bdd__below_OI2,axiom,
! [A: set_a,M: epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ M @ ( F @ X2 ) ) )
=> ( condit6597432229783682067fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_891_bdd__below_OI2,axiom,
! [A: set_se5208064806568342746c_fm_a,M: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ M @ ( F @ X2 ) ) )
=> ( condit4283900098890313034c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_892_bdd__below_OI2,axiom,
! [A: set_Epistemic_fm_a,M: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ M @ ( F @ X2 ) ) )
=> ( condit4283900098890313034c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_893_bdd__below_OI2,axiom,
! [A: set_a,M: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ M @ ( F @ X2 ) ) )
=> ( condit4283900098890313034c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ).
% bdd_below.I2
thf(fact_894_bdd__below__image__mono,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit6597432229783682067fm_a_o @ A )
=> ( condit6597432229783682067fm_a_o @ ( image_6282542973537860301fm_a_o @ F @ A ) ) ) ) ).
% bdd_below_image_mono
thf(fact_895_bdd__below__image__mono,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: set_Epistemic_fm_a_o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit6597432229783682067fm_a_o @ A )
=> ( condit4283900098890313034c_fm_a @ ( image_3065101469317257744c_fm_a @ F @ A ) ) ) ) ).
% bdd_below_image_mono
thf(fact_896_bdd__below__image__mono,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_se5208064806568342746c_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit4283900098890313034c_fm_a @ A )
=> ( condit6597432229783682067fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ).
% bdd_below_image_mono
thf(fact_897_bdd__below__image__mono,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit4283900098890313034c_fm_a @ A )
=> ( condit4283900098890313034c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ).
% bdd_below_image_mono
thf(fact_898_bdd__below__image__antimono,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ F )
=> ( ( condit6663374420580310783fm_a_o @ A )
=> ( condit6597432229783682067fm_a_o @ ( image_6282542973537860301fm_a_o @ F @ A ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_899_bdd__below__image__antimono,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: set_Epistemic_fm_a_o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ F )
=> ( ( condit6663374420580310783fm_a_o @ A )
=> ( condit4283900098890313034c_fm_a @ ( image_3065101469317257744c_fm_a @ F @ A ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_900_bdd__below__image__antimono,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_se5208064806568342746c_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ F )
=> ( ( condit2203544419062987614c_fm_a @ A )
=> ( condit6597432229783682067fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_901_bdd__below__image__antimono,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ F )
=> ( ( condit2203544419062987614c_fm_a @ A )
=> ( condit4283900098890313034c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_902_mono__cInf,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit6597432229783682067fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ ( comple6443248293609458716fm_a_o @ A ) ) @ ( comple6443248293609458716fm_a_o @ ( image_6282542973537860301fm_a_o @ F @ A ) ) ) ) ) ) ).
% mono_cInf
thf(fact_903_mono__cInf,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: set_Epistemic_fm_a_o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit6597432229783682067fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ ( comple6443248293609458716fm_a_o @ A ) ) @ ( comple159807495668495361c_fm_a @ ( image_3065101469317257744c_fm_a @ F @ A ) ) ) ) ) ) ).
% mono_cInf
thf(fact_904_mono__cInf,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_se5208064806568342746c_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit4283900098890313034c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ ( comple159807495668495361c_fm_a @ A ) ) @ ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ) ) ).
% mono_cInf
thf(fact_905_mono__cInf,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit4283900098890313034c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ ( comple159807495668495361c_fm_a @ A ) ) @ ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ) ) ).
% mono_cInf
thf(fact_906_mono__cINF,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: epistemic_fm_a > epistemic_fm_a > $o,I4: set_Epistemic_fm_a] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit6597432229783682067fm_a_o @ ( image_2381060403230254136fm_a_o @ A @ I4 ) )
=> ( ( I4 != bot_bo3626323581529592678c_fm_a )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ A @ I4 ) ) )
@ ( comple6443248293609458716fm_a_o
@ ( image_2381060403230254136fm_a_o
@ ^ [X: epistemic_fm_a] : ( F @ ( A @ X ) )
@ I4 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_907_mono__cINF,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: a > epistemic_fm_a > $o,I4: set_a] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit6597432229783682067fm_a_o @ ( image_3809950421912819996fm_a_o @ A @ I4 ) )
=> ( ( I4 != bot_bot_set_a )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ A @ I4 ) ) )
@ ( comple6443248293609458716fm_a_o
@ ( image_3809950421912819996fm_a_o
@ ^ [X: a] : ( F @ ( A @ X ) )
@ I4 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_908_mono__cINF,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: epistemic_fm_a > epistemic_fm_a > $o,I4: set_Epistemic_fm_a] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit6597432229783682067fm_a_o @ ( image_2381060403230254136fm_a_o @ A @ I4 ) )
=> ( ( I4 != bot_bo3626323581529592678c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ A @ I4 ) ) )
@ ( comple159807495668495361c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X: epistemic_fm_a] : ( F @ ( A @ X ) )
@ I4 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_909_mono__cINF,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: a > epistemic_fm_a > $o,I4: set_a] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit6597432229783682067fm_a_o @ ( image_3809950421912819996fm_a_o @ A @ I4 ) )
=> ( ( I4 != bot_bot_set_a )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ A @ I4 ) ) )
@ ( comple159807495668495361c_fm_a
@ ( image_3480842985853347585c_fm_a
@ ^ [X: a] : ( F @ ( A @ X ) )
@ I4 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_910_mono__cINF,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: epistemic_fm_a > set_Epistemic_fm_a,I4: set_Epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit4283900098890313034c_fm_a @ ( image_1263732536703906021c_fm_a @ A @ I4 ) )
=> ( ( I4 != bot_bo3626323581529592678c_fm_a )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ A @ I4 ) ) )
@ ( comple6443248293609458716fm_a_o
@ ( image_2381060403230254136fm_a_o
@ ^ [X: epistemic_fm_a] : ( F @ ( A @ X ) )
@ I4 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_911_mono__cINF,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: a > set_Epistemic_fm_a,I4: set_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( condit4283900098890313034c_fm_a @ ( image_3480842985853347585c_fm_a @ A @ I4 ) )
=> ( ( I4 != bot_bot_set_a )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ A @ I4 ) ) )
@ ( comple6443248293609458716fm_a_o
@ ( image_3809950421912819996fm_a_o
@ ^ [X: a] : ( F @ ( A @ X ) )
@ I4 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_912_mono__cINF,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: epistemic_fm_a > set_Epistemic_fm_a,I4: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit4283900098890313034c_fm_a @ ( image_1263732536703906021c_fm_a @ A @ I4 ) )
=> ( ( I4 != bot_bo3626323581529592678c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ A @ I4 ) ) )
@ ( comple159807495668495361c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X: epistemic_fm_a] : ( F @ ( A @ X ) )
@ I4 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_913_mono__cINF,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: a > set_Epistemic_fm_a,I4: set_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( condit4283900098890313034c_fm_a @ ( image_3480842985853347585c_fm_a @ A @ I4 ) )
=> ( ( I4 != bot_bot_set_a )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ A @ I4 ) ) )
@ ( comple159807495668495361c_fm_a
@ ( image_3480842985853347585c_fm_a
@ ^ [X: a] : ( F @ ( A @ X ) )
@ I4 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_914_cInf__le__cSup,axiom,
! [A: set_Epistemic_fm_a_o] :
( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6663374420580310783fm_a_o @ A )
=> ( ( condit6597432229783682067fm_a_o @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_915_cInf__le__cSup,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ A )
=> ( ( condit4283900098890313034c_fm_a @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_916_cInf__eq__minimum,axiom,
! [Z2: epistemic_fm_a > $o,X6: set_Epistemic_fm_a_o] :
( ( member4486839677911940090fm_a_o @ Z2 @ X6 )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ Z2 @ X2 ) )
=> ( ( comple6443248293609458716fm_a_o @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_917_cInf__eq__minimum,axiom,
! [Z2: set_Epistemic_fm_a,X6: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ Z2 @ X6 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ Z2 @ X2 ) )
=> ( ( comple159807495668495361c_fm_a @ X6 )
= Z2 ) ) ) ).
% cInf_eq_minimum
thf(fact_918_Inf__greatest,axiom,
! [A: set_Epistemic_fm_a_o,Z2: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ Z2 @ X2 ) )
=> ( ord_le4043730696559282883fm_a_o @ Z2 @ ( comple6443248293609458716fm_a_o @ A ) ) ) ).
% Inf_greatest
thf(fact_919_Inf__greatest,axiom,
! [A: set_se5208064806568342746c_fm_a,Z2: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ Z2 @ X2 ) )
=> ( ord_le3275665582123262618c_fm_a @ Z2 @ ( comple159807495668495361c_fm_a @ A ) ) ) ).
% Inf_greatest
thf(fact_920_le__Inf__iff,axiom,
! [B: epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( ord_le4043730696559282883fm_a_o @ B @ ( comple6443248293609458716fm_a_o @ A ) )
= ( ! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ A )
=> ( ord_le4043730696559282883fm_a_o @ B @ X ) ) ) ) ).
% le_Inf_iff
thf(fact_921_le__Inf__iff,axiom,
! [B: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ ( comple159807495668495361c_fm_a @ A ) )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
=> ( ord_le3275665582123262618c_fm_a @ B @ X ) ) ) ) ).
% le_Inf_iff
thf(fact_922_Inf__lower2,axiom,
! [U2: epistemic_fm_a > $o,A: set_Epistemic_fm_a_o,V6: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ U2 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ V6 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ A ) @ V6 ) ) ) ).
% Inf_lower2
thf(fact_923_Inf__lower2,axiom,
! [U2: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,V6: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ U2 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ V6 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ A ) @ V6 ) ) ) ).
% Inf_lower2
thf(fact_924_Inf__lower,axiom,
! [X3: epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( member4486839677911940090fm_a_o @ X3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ A ) @ X3 ) ) ).
% Inf_lower
thf(fact_925_Inf__lower,axiom,
! [X3: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ A ) @ X3 ) ) ).
% Inf_lower
thf(fact_926_Inf__mono,axiom,
! [B4: set_Epistemic_fm_a_o,A: set_Epistemic_fm_a_o] :
( ! [B5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ B5 @ B4 )
=> ? [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ X5 @ B5 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ A ) @ ( comple6443248293609458716fm_a_o @ B4 ) ) ) ).
% Inf_mono
thf(fact_927_Inf__mono,axiom,
! [B4: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a] :
( ! [B5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B5 @ B4 )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ X5 @ B5 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ A ) @ ( comple159807495668495361c_fm_a @ B4 ) ) ) ).
% Inf_mono
thf(fact_928_Inf__eqI,axiom,
! [A: set_Epistemic_fm_a_o,X3: epistemic_fm_a > $o] :
( ! [I3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ I3 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [I5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ I5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ I5 ) )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 ) )
=> ( ( comple6443248293609458716fm_a_o @ A )
= X3 ) ) ) ).
% Inf_eqI
thf(fact_929_Inf__eqI,axiom,
! [A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ I3 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [I5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ I5 ) )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X3 ) )
=> ( ( comple159807495668495361c_fm_a @ A )
= X3 ) ) ) ).
% Inf_eqI
thf(fact_930_INF__eq,axiom,
! [A: set_a,B4: set_a,G2: a > set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( G2 @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( F @ X5 ) @ ( G2 @ J ) ) ) )
=> ( ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_931_INF__eq,axiom,
! [A: set_a,B4: set_a,G2: a > epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le4043730696559282883fm_a_o @ ( G2 @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ ( F @ X5 ) @ ( G2 @ J ) ) ) )
=> ( ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
= ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_932_INF__eq,axiom,
! [A: set_Epistemic_fm_a,B4: set_a,G2: a > set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( G2 @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( F @ X5 ) @ ( G2 @ J ) ) ) )
=> ( ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_933_INF__eq,axiom,
! [A: set_a,B4: set_Epistemic_fm_a,G2: epistemic_fm_a > set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( G2 @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J @ B4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( F @ X5 ) @ ( G2 @ J ) ) ) )
=> ( ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_934_INF__eq,axiom,
! [A: set_Epistemic_fm_a,B4: set_a,G2: a > epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le4043730696559282883fm_a_o @ ( G2 @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ ( F @ X5 ) @ ( G2 @ J ) ) ) )
=> ( ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
= ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_935_INF__eq,axiom,
! [A: set_a,B4: set_Epistemic_fm_a,G2: epistemic_fm_a > epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ B4 )
& ( ord_le4043730696559282883fm_a_o @ ( G2 @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J @ B4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ ( F @ X5 ) @ ( G2 @ J ) ) ) )
=> ( ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
= ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_936_INF__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_a,G2: a > set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( G2 @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( F @ X5 ) @ ( G2 @ J ) ) ) )
=> ( ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
= ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_937_INF__eq,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,G2: epistemic_fm_a > set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( G2 @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J @ B4 )
=> ? [X5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( F @ X5 ) @ ( G2 @ J ) ) ) )
=> ( ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_938_INF__eq,axiom,
! [A: set_a,B4: set_se5208064806568342746c_fm_a,G2: set_Epistemic_fm_a > set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( G2 @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ J @ B4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( F @ X5 ) @ ( G2 @ J ) ) ) )
=> ( ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_939_INF__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_a,G2: a > epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B4 )
& ( ord_le4043730696559282883fm_a_o @ ( G2 @ X5 ) @ ( F @ I3 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B4 )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ ( F @ X5 ) @ ( G2 @ J ) ) ) )
=> ( ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) )
= ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% INF_eq
thf(fact_940_cInf__eq__non__empty,axiom,
! [X6: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( X6 != bot_bo4908879468608929837fm_a_o )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ X2 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X5 ) )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ A2 ) )
=> ( ( comple6443248293609458716fm_a_o @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_941_cInf__eq__non__empty,axiom,
! [X6: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( X6 != bot_bo1868452413526482246c_fm_a )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ X2 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X5 ) )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ A2 ) )
=> ( ( comple159807495668495361c_fm_a @ X6 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_942_cInf__greatest,axiom,
! [X6: set_Epistemic_fm_a_o,Z2: epistemic_fm_a > $o] :
( ( X6 != bot_bo4908879468608929837fm_a_o )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ Z2 @ X2 ) )
=> ( ord_le4043730696559282883fm_a_o @ Z2 @ ( comple6443248293609458716fm_a_o @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_943_cInf__greatest,axiom,
! [X6: set_se5208064806568342746c_fm_a,Z2: set_Epistemic_fm_a] :
( ( X6 != bot_bo1868452413526482246c_fm_a )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ Z2 @ X2 ) )
=> ( ord_le3275665582123262618c_fm_a @ Z2 @ ( comple159807495668495361c_fm_a @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_944_Inf__less__eq,axiom,
! [A: set_Epistemic_fm_a_o,U2: epistemic_fm_a > $o] :
( ! [V2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ V2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ V2 @ U2 ) )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ A ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_945_Inf__less__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,U2: set_Epistemic_fm_a] :
( ! [V2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ V2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ V2 @ U2 ) )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ A ) @ U2 ) ) ) ).
% Inf_less_eq
thf(fact_946_cInf__le__finite,axiom,
! [X6: set_Epistemic_fm_a_o,X3: epistemic_fm_a > $o] :
( ( finite5407549452671652090fm_a_o @ X6 )
=> ( ( member4486839677911940090fm_a_o @ X3 @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ X6 ) @ X3 ) ) ) ).
% cInf_le_finite
thf(fact_947_cInf__le__finite,axiom,
! [X6: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ( finite7364866884270929315c_fm_a @ X6 )
=> ( ( member536094252920883875c_fm_a @ X3 @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ X6 ) @ X3 ) ) ) ).
% cInf_le_finite
thf(fact_948_Inf__superset__mono,axiom,
! [B4: set_Epistemic_fm_a_o,A: set_Epistemic_fm_a_o] :
( ( ord_le4525777775358908921fm_a_o @ B4 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ A ) @ ( comple6443248293609458716fm_a_o @ B4 ) ) ) ).
% Inf_superset_mono
thf(fact_949_Inf__superset__mono,axiom,
! [B4: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ B4 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ A ) @ ( comple159807495668495361c_fm_a @ B4 ) ) ) ).
% Inf_superset_mono
thf(fact_950_cInf__lower,axiom,
! [X3: epistemic_fm_a > $o,X6: set_Epistemic_fm_a_o] :
( ( member4486839677911940090fm_a_o @ X3 @ X6 )
=> ( ( condit6597432229783682067fm_a_o @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ X6 ) @ X3 ) ) ) ).
% cInf_lower
thf(fact_951_cInf__lower,axiom,
! [X3: set_Epistemic_fm_a,X6: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ X6 )
=> ( ( condit4283900098890313034c_fm_a @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ X6 ) @ X3 ) ) ) ).
% cInf_lower
thf(fact_952_cInf__lower2,axiom,
! [X3: epistemic_fm_a > $o,X6: set_Epistemic_fm_a_o,Y: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X3 @ X6 )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( condit6597432229783682067fm_a_o @ X6 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ X6 ) @ Y ) ) ) ) ).
% cInf_lower2
thf(fact_953_cInf__lower2,axiom,
! [X3: set_Epistemic_fm_a,X6: set_se5208064806568342746c_fm_a,Y: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ X6 )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( condit4283900098890313034c_fm_a @ X6 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ X6 ) @ Y ) ) ) ) ).
% cInf_lower2
thf(fact_954_INF__eqI,axiom,
! [A: set_se5208064806568342746c_fm_a,X3: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( F @ I3 ) ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [I5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ ( F @ I5 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 ) )
=> ( ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_955_INF__eqI,axiom,
! [A: set_Epistemic_fm_a,X3: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( F @ I3 ) ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [I5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ ( F @ I5 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 ) )
=> ( ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_956_INF__eqI,axiom,
! [A: set_a,X3: epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( F @ I3 ) ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [I5: a] :
( ( member_a @ I5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ ( F @ I5 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 ) )
=> ( ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_957_INF__eqI,axiom,
! [A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( F @ I3 ) ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [I5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ ( F @ I5 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X3 ) )
=> ( ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_958_INF__eqI,axiom,
! [A: set_Epistemic_fm_a,X3: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( F @ I3 ) ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [I5: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ ( F @ I5 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X3 ) )
=> ( ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_959_INF__eqI,axiom,
! [A: set_a,X3: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( F @ I3 ) ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [I5: a] :
( ( member_a @ I5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ ( F @ I5 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X3 ) )
=> ( ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_960_INF__lower,axiom,
! [I: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_961_INF__lower,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_962_INF__lower,axiom,
! [I: a,A: set_a,F: a > epistemic_fm_a > $o] :
( ( member_a @ I @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_963_INF__lower,axiom,
! [I: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_964_INF__lower,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_965_INF__lower,axiom,
! [I: a,A: set_a,F: a > set_Epistemic_fm_a] :
( ( member_a @ I @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_966_INF__lower2,axiom,
! [I: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ I ) @ U2 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_967_INF__lower2,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ I ) @ U2 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_968_INF__lower2,axiom,
! [I: a,A: set_a,F: a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ( member_a @ I @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ I ) @ U2 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_969_INF__lower2,axiom,
! [I: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ U2 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_970_INF__lower2,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ U2 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_971_INF__lower2,axiom,
! [I: a,A: set_a,F: a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( member_a @ I @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ U2 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ U2 ) ) ) ).
% INF_lower2
thf(fact_972_INF__greatest,axiom,
! [A: set_se5208064806568342746c_fm_a,U2: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ I3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_973_INF__greatest,axiom,
! [A: set_Epistemic_fm_a,U2: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ I3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_974_INF__greatest,axiom,
! [A: set_a,U2: epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ I3 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_975_INF__greatest,axiom,
! [A: set_se5208064806568342746c_fm_a,U2: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ I3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_976_INF__greatest,axiom,
! [A: set_Epistemic_fm_a,U2: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ I3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_977_INF__greatest,axiom,
! [A: set_a,U2: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ I3 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ).
% INF_greatest
thf(fact_978_cINF__greatest,axiom,
! [A: set_se5208064806568342746c_fm_a,M5: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ M5 @ ( F @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ M5 @ ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_979_cINF__greatest,axiom,
! [A: set_Epistemic_fm_a,M5: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ M5 @ ( F @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ M5 @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_980_cINF__greatest,axiom,
! [A: set_a,M5: epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ M5 @ ( F @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ M5 @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_981_cINF__greatest,axiom,
! [A: set_se5208064806568342746c_fm_a,M5: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ M5 @ ( F @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ M5 @ ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_982_cINF__greatest,axiom,
! [A: set_Epistemic_fm_a,M5: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ M5 @ ( F @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ M5 @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_983_cINF__greatest,axiom,
! [A: set_a,M5: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ M5 @ ( F @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ M5 @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_984_INF__eq__iff,axiom,
! [I4: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( I4 != bot_bo1868452413526482246c_fm_a )
=> ( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ I4 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ C ) )
=> ( ( ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ I4 ) )
= C )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_985_INF__eq__iff,axiom,
! [I4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( I4 != bot_bo3626323581529592678c_fm_a )
=> ( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ I4 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ C ) )
=> ( ( ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ I4 ) )
= C )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_986_INF__eq__iff,axiom,
! [I4: set_a,F: a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( I4 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I4 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ C ) )
=> ( ( ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ I4 ) )
= C )
= ( ! [X: a] :
( ( member_a @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_987_INF__eq__iff,axiom,
! [I4: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( I4 != bot_bo1868452413526482246c_fm_a )
=> ( ! [I3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I3 @ I4 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ C ) )
=> ( ( ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ I4 ) )
= C )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_988_INF__eq__iff,axiom,
! [I4: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( I4 != bot_bo3626323581529592678c_fm_a )
=> ( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ I4 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ C ) )
=> ( ( ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ I4 ) )
= C )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_989_INF__eq__iff,axiom,
! [I4: set_a,F: a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( I4 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I4 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ C ) )
=> ( ( ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ I4 ) )
= C )
= ( ! [X: a] :
( ( member_a @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_990_Inf__le__Sup,axiom,
! [A: set_Epistemic_fm_a_o] :
( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ A ) ) ) ).
% Inf_le_Sup
thf(fact_991_Inf__le__Sup,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ A ) ) ) ).
% Inf_le_Sup
thf(fact_992_less__eq__Inf__inter,axiom,
! [A: set_set_a,B4: set_set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( comple6135023378680113637_set_a @ A ) @ ( comple6135023378680113637_set_a @ B4 ) ) @ ( comple6135023378680113637_set_a @ ( inf_inf_set_set_a @ A @ B4 ) ) ) ).
% less_eq_Inf_inter
thf(fact_993_less__eq__Inf__inter,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] : ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ ( comple6443248293609458716fm_a_o @ A ) @ ( comple6443248293609458716fm_a_o @ B4 ) ) @ ( comple6443248293609458716fm_a_o @ ( inf_in8896176524268821035fm_a_o @ A @ B4 ) ) ) ).
% less_eq_Inf_inter
thf(fact_994_less__eq__Inf__inter,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( comple159807495668495361c_fm_a @ A ) @ ( comple159807495668495361c_fm_a @ B4 ) ) @ ( comple159807495668495361c_fm_a @ ( inf_in1884693029477671368c_fm_a @ A @ B4 ) ) ) ).
% less_eq_Inf_inter
thf(fact_995_cINF__lower2,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a,U2: epistemic_fm_a > $o] :
( ( condit6597432229783682067fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) )
=> ( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ U2 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ U2 ) ) ) ) ).
% cINF_lower2
thf(fact_996_cINF__lower2,axiom,
! [F: epistemic_fm_a > epistemic_fm_a > $o,A: set_Epistemic_fm_a,X3: epistemic_fm_a,U2: epistemic_fm_a > $o] :
( ( condit6597432229783682067fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
=> ( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ U2 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ U2 ) ) ) ) ).
% cINF_lower2
thf(fact_997_cINF__lower2,axiom,
! [F: a > epistemic_fm_a > $o,A: set_a,X3: a,U2: epistemic_fm_a > $o] :
( ( condit6597432229783682067fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
=> ( ( member_a @ X3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ U2 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ U2 ) ) ) ) ).
% cINF_lower2
thf(fact_998_cINF__lower2,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( condit4283900098890313034c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
=> ( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ U2 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ U2 ) ) ) ) ).
% cINF_lower2
thf(fact_999_cINF__lower2,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a,X3: epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( condit4283900098890313034c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
=> ( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ U2 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ U2 ) ) ) ) ).
% cINF_lower2
thf(fact_1000_cINF__lower2,axiom,
! [F: a > set_Epistemic_fm_a,A: set_a,X3: a,U2: set_Epistemic_fm_a] :
( ( condit4283900098890313034c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
=> ( ( member_a @ X3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ U2 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ U2 ) ) ) ) ).
% cINF_lower2
thf(fact_1001_cINF__lower,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ( condit6597432229783682067fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) )
=> ( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ ( F @ X3 ) ) ) ) ).
% cINF_lower
thf(fact_1002_cINF__lower,axiom,
! [F: epistemic_fm_a > epistemic_fm_a > $o,A: set_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( condit6597432229783682067fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
=> ( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ ( F @ X3 ) ) ) ) ).
% cINF_lower
thf(fact_1003_cINF__lower,axiom,
! [F: a > epistemic_fm_a > $o,A: set_a,X3: a] :
( ( condit6597432229783682067fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
=> ( ( member_a @ X3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ ( F @ X3 ) ) ) ) ).
% cINF_lower
thf(fact_1004_cINF__lower,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ( condit4283900098890313034c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) )
=> ( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ ( F @ X3 ) ) ) ) ).
% cINF_lower
thf(fact_1005_cINF__lower,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( condit4283900098890313034c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
=> ( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ ( F @ X3 ) ) ) ) ).
% cINF_lower
thf(fact_1006_cINF__lower,axiom,
! [F: a > set_Epistemic_fm_a,A: set_a,X3: a] :
( ( condit4283900098890313034c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
=> ( ( member_a @ X3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ ( F @ X3 ) ) ) ) ).
% cINF_lower
thf(fact_1007_cInf__mono,axiom,
! [B4: set_Epistemic_fm_a_o,A: set_Epistemic_fm_a_o] :
( ( B4 != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6597432229783682067fm_a_o @ A )
=> ( ! [B5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ B5 @ B4 )
=> ? [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ A )
& ( ord_le4043730696559282883fm_a_o @ X5 @ B5 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ A ) @ ( comple6443248293609458716fm_a_o @ B4 ) ) ) ) ) ).
% cInf_mono
thf(fact_1008_cInf__mono,axiom,
! [B4: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( B4 != bot_bo1868452413526482246c_fm_a )
=> ( ( condit4283900098890313034c_fm_a @ A )
=> ( ! [B5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B5 @ B4 )
=> ? [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ A )
& ( ord_le3275665582123262618c_fm_a @ X5 @ B5 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ A ) @ ( comple159807495668495361c_fm_a @ B4 ) ) ) ) ) ).
% cInf_mono
thf(fact_1009_le__cInf__iff,axiom,
! [S2: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( S2 != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6597432229783682067fm_a_o @ S2 )
=> ( ( ord_le4043730696559282883fm_a_o @ A2 @ ( comple6443248293609458716fm_a_o @ S2 ) )
= ( ! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ S2 )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ X ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_1010_le__cInf__iff,axiom,
! [S2: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( S2 != bot_bo1868452413526482246c_fm_a )
=> ( ( condit4283900098890313034c_fm_a @ S2 )
=> ( ( ord_le3275665582123262618c_fm_a @ A2 @ ( comple159807495668495361c_fm_a @ S2 ) )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ S2 )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ X ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_1011_INF__superset__mono,axiom,
! [B4: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,G2: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ B4 @ A )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) @ ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_1012_INF__superset__mono,axiom,
! [B4: set_a,A: set_a,F: a > epistemic_fm_a > $o,G2: a > epistemic_fm_a > $o] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_1013_INF__superset__mono,axiom,
! [B4: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,G2: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ B4 @ A )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) @ ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_1014_INF__superset__mono,axiom,
! [B4: set_a,A: set_a,F: a > set_Epistemic_fm_a,G2: a > set_Epistemic_fm_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_1015_INF__superset__mono,axiom,
! [B4: set_Epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,G2: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ G2 @ B4 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_1016_INF__superset__mono,axiom,
! [B4: set_Epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,G2: epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ G2 @ B4 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_1017_INF__le__SUP,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_1018_INF__le__SUP,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o] :
( ( A != bot_bot_set_a )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_1019_INF__le__SUP,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_1020_INF__le__SUP,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a] :
( ( A != bot_bot_set_a )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_1021_le__cINF__iff,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit6597432229783682067fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ X ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_1022_le__cINF__iff,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ( A != bot_bot_set_a )
=> ( ( condit6597432229783682067fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ X ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_1023_le__cINF__iff,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit4283900098890313034c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ X ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_1024_le__cINF__iff,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( A != bot_bot_set_a )
=> ( ( condit4283900098890313034c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ X ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_1025_cInf__superset__mono,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6597432229783682067fm_a_o @ B4 )
=> ( ( ord_le4525777775358908921fm_a_o @ A @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ B4 ) @ ( comple6443248293609458716fm_a_o @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1026_cInf__superset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( condit4283900098890313034c_fm_a @ B4 )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ B4 ) @ ( comple159807495668495361c_fm_a @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1027_mono__Inf,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ ( comple6443248293609458716fm_a_o @ A ) ) @ ( comple6443248293609458716fm_a_o @ ( image_6282542973537860301fm_a_o @ F @ A ) ) ) ) ).
% mono_Inf
thf(fact_1028_mono__Inf,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,A: set_Epistemic_fm_a_o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ ( comple6443248293609458716fm_a_o @ A ) ) @ ( comple159807495668495361c_fm_a @ ( image_3065101469317257744c_fm_a @ F @ A ) ) ) ) ).
% mono_Inf
thf(fact_1029_mono__Inf,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,A: set_se5208064806568342746c_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ ( comple159807495668495361c_fm_a @ A ) ) @ ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ).
% mono_Inf
thf(fact_1030_mono__Inf,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ ( comple159807495668495361c_fm_a @ A ) ) @ ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ).
% mono_Inf
thf(fact_1031_cINF__superset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,G2: set_Epistemic_fm_a > epistemic_fm_a > $o,B4: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( condit6597432229783682067fm_a_o @ ( image_6902499629341727256fm_a_o @ G2 @ B4 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( G2 @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ G2 @ B4 ) ) @ ( comple6443248293609458716fm_a_o @ ( image_6902499629341727256fm_a_o @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1032_cINF__superset__mono,axiom,
! [A: set_a,G2: a > epistemic_fm_a > $o,B4: set_a,F: a > epistemic_fm_a > $o] :
( ( A != bot_bot_set_a )
=> ( ( condit6597432229783682067fm_a_o @ ( image_3809950421912819996fm_a_o @ G2 @ B4 ) )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( G2 @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ G2 @ B4 ) ) @ ( comple6443248293609458716fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1033_cINF__superset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,G2: set_Epistemic_fm_a > set_Epistemic_fm_a,B4: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( condit4283900098890313034c_fm_a @ ( image_8582343580727714565c_fm_a @ G2 @ B4 ) )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( G2 @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ G2 @ B4 ) ) @ ( comple159807495668495361c_fm_a @ ( image_8582343580727714565c_fm_a @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1034_cINF__superset__mono,axiom,
! [A: set_a,G2: a > set_Epistemic_fm_a,B4: set_a,F: a > set_Epistemic_fm_a] :
( ( A != bot_bot_set_a )
=> ( ( condit4283900098890313034c_fm_a @ ( image_3480842985853347585c_fm_a @ G2 @ B4 ) )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( G2 @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ G2 @ B4 ) ) @ ( comple159807495668495361c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1035_cINF__superset__mono,axiom,
! [A: set_Epistemic_fm_a,G2: epistemic_fm_a > epistemic_fm_a > $o,B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit6597432229783682067fm_a_o @ ( image_2381060403230254136fm_a_o @ G2 @ B4 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( G2 @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ G2 @ B4 ) ) @ ( comple6443248293609458716fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1036_cINF__superset__mono,axiom,
! [A: set_Epistemic_fm_a,G2: epistemic_fm_a > set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit4283900098890313034c_fm_a @ ( image_1263732536703906021c_fm_a @ G2 @ B4 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( G2 @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ G2 @ B4 ) ) @ ( comple159807495668495361c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_1037_less__eq__cInf__inter,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ( condit6597432229783682067fm_a_o @ A )
=> ( ( condit6597432229783682067fm_a_o @ B4 )
=> ( ( ( inf_in8896176524268821035fm_a_o @ A @ B4 )
!= bot_bo4908879468608929837fm_a_o )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ ( comple6443248293609458716fm_a_o @ A ) @ ( comple6443248293609458716fm_a_o @ B4 ) ) @ ( comple6443248293609458716fm_a_o @ ( inf_in8896176524268821035fm_a_o @ A @ B4 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1038_less__eq__cInf__inter,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( condit4283900098890313034c_fm_a @ A )
=> ( ( condit4283900098890313034c_fm_a @ B4 )
=> ( ( ( inf_in1884693029477671368c_fm_a @ A @ B4 )
!= bot_bo1868452413526482246c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ ( comple159807495668495361c_fm_a @ A ) @ ( comple159807495668495361c_fm_a @ B4 ) ) @ ( comple159807495668495361c_fm_a @ ( inf_in1884693029477671368c_fm_a @ A @ B4 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1039_cSup__cInf,axiom,
! [S2: set_Epistemic_fm_a_o] :
( ( S2 != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6663374420580310783fm_a_o @ S2 )
=> ( ( comple310706794042174646fm_a_o @ S2 )
= ( comple6443248293609458716fm_a_o
@ ( collec3226785624747288888fm_a_o
@ ^ [X: epistemic_fm_a > $o] :
! [Y4: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Y4 @ S2 )
=> ( ord_le4043730696559282883fm_a_o @ Y4 @ X ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_1040_cSup__cInf,axiom,
! [S2: set_se5208064806568342746c_fm_a] :
( ( S2 != bot_bo1868452413526482246c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ S2 )
=> ( ( comple7868773486375872231c_fm_a @ S2 )
= ( comple159807495668495361c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ S2 )
=> ( ord_le3275665582123262618c_fm_a @ Y4 @ X ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_1041_cInf__cSup,axiom,
! [S2: set_Epistemic_fm_a_o] :
( ( S2 != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6597432229783682067fm_a_o @ S2 )
=> ( ( comple6443248293609458716fm_a_o @ S2 )
= ( comple310706794042174646fm_a_o
@ ( collec3226785624747288888fm_a_o
@ ^ [X: epistemic_fm_a > $o] :
! [Y4: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Y4 @ S2 )
=> ( ord_le4043730696559282883fm_a_o @ X @ Y4 ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_1042_cInf__cSup,axiom,
! [S2: set_se5208064806568342746c_fm_a] :
( ( S2 != bot_bo1868452413526482246c_fm_a )
=> ( ( condit4283900098890313034c_fm_a @ S2 )
=> ( ( comple159807495668495361c_fm_a @ S2 )
= ( comple7868773486375872231c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ S2 )
=> ( ord_le3275665582123262618c_fm_a @ X @ Y4 ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_1043_lfp__ordinal__induct,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,P4: ( epistemic_fm_a > $o ) > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ! [S8: epistemic_fm_a > $o] :
( ( P4 @ S8 )
=> ( ( ord_le4043730696559282883fm_a_o @ S8 @ ( comple3933800597814182506fm_a_o @ F ) )
=> ( P4 @ ( F @ S8 ) ) ) )
=> ( ! [M2: set_Epistemic_fm_a_o] :
( ! [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ M2 )
=> ( P4 @ X5 ) )
=> ( P4 @ ( comple310706794042174646fm_a_o @ M2 ) ) )
=> ( P4 @ ( comple3933800597814182506fm_a_o @ F ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_1044_lfp__ordinal__induct,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ! [S8: set_Epistemic_fm_a] :
( ( P4 @ S8 )
=> ( ( ord_le3275665582123262618c_fm_a @ S8 @ ( comple693844135356717107c_fm_a @ F ) )
=> ( P4 @ ( F @ S8 ) ) ) )
=> ( ! [M2: set_se5208064806568342746c_fm_a] :
( ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ M2 )
=> ( P4 @ X5 ) )
=> ( P4 @ ( comple7868773486375872231c_fm_a @ M2 ) ) )
=> ( P4 @ ( comple693844135356717107c_fm_a @ F ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_1045_lfp__induct,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,P4: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ ( inf_in3606484609122063093fm_a_o @ ( comple3933800597814182506fm_a_o @ F ) @ P4 ) ) @ P4 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple3933800597814182506fm_a_o @ F ) @ P4 ) ) ) ).
% lfp_induct
thf(fact_1046_lfp__induct,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,P4: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ ( inf_in2882328776738922472c_fm_a @ ( comple693844135356717107c_fm_a @ F ) @ P4 ) ) @ P4 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple693844135356717107c_fm_a @ F ) @ P4 ) ) ) ).
% lfp_induct
thf(fact_1047_lfp__greatest,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: epistemic_fm_a > $o] :
( ! [U3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( F @ U3 ) @ U3 )
=> ( ord_le4043730696559282883fm_a_o @ A @ U3 ) )
=> ( ord_le4043730696559282883fm_a_o @ A @ ( comple3933800597814182506fm_a_o @ F ) ) ) ).
% lfp_greatest
thf(fact_1048_lfp__greatest,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ! [U3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( F @ U3 ) @ U3 )
=> ( ord_le3275665582123262618c_fm_a @ A @ U3 ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( comple693844135356717107c_fm_a @ F ) ) ) ).
% lfp_greatest
thf(fact_1049_lfp__lowerbound,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( F @ A ) @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( comple3933800597814182506fm_a_o @ F ) @ A ) ) ).
% lfp_lowerbound
thf(fact_1050_lfp__lowerbound,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( F @ A ) @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( comple693844135356717107c_fm_a @ F ) @ A ) ) ).
% lfp_lowerbound
thf(fact_1051_lfp__mono,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,G2: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ! [Z6: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( F @ Z6 ) @ ( G2 @ Z6 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple3933800597814182506fm_a_o @ F ) @ ( comple3933800597814182506fm_a_o @ G2 ) ) ) ).
% lfp_mono
thf(fact_1052_lfp__mono,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,G2: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [Z6: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( F @ Z6 ) @ ( G2 @ Z6 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple693844135356717107c_fm_a @ F ) @ ( comple693844135356717107c_fm_a @ G2 ) ) ) ).
% lfp_mono
thf(fact_1053_lfp__lfp,axiom,
! [F: ( epistemic_fm_a > $o ) > ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o,W2: epistemic_fm_a > $o,Z4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ( ord_le4043730696559282883fm_a_o @ W2 @ Z4 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 @ W2 ) @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( comple3933800597814182506fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( comple3933800597814182506fm_a_o @ ( F @ X ) ) )
= ( comple3933800597814182506fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( F @ X @ X ) ) ) ) ).
% lfp_lfp
thf(fact_1054_lfp__lfp,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a,W2: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ( ord_le3275665582123262618c_fm_a @ W2 @ Z4 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 @ W2 ) @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( comple693844135356717107c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( comple693844135356717107c_fm_a @ ( F @ X ) ) )
= ( comple693844135356717107c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( F @ X @ X ) ) ) ) ).
% lfp_lfp
thf(fact_1055_fm_Opred__set,axiom,
( epistemic_pred_fm_a
= ( ^ [P6: a > $o,X: epistemic_fm_a] :
! [Y4: a] :
( ( member_a @ Y4 @ ( epistemic_set_fm_a @ X ) )
=> ( P6 @ Y4 ) ) ) ) ).
% fm.pred_set
thf(fact_1056_lfp__def,axiom,
( comple3933800597814182506fm_a_o
= ( ^ [F2: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( comple6443248293609458716fm_a_o
@ ( collec3226785624747288888fm_a_o
@ ^ [U4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( F2 @ U4 ) @ U4 ) ) ) ) ) ).
% lfp_def
thf(fact_1057_lfp__def,axiom,
( comple693844135356717107c_fm_a
= ( ^ [F2: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( comple159807495668495361c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [U4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( F2 @ U4 ) @ U4 ) ) ) ) ) ).
% lfp_def
thf(fact_1058_Sup__eq__Inf,axiom,
( comple310706794042174646fm_a_o
= ( ^ [A3: set_Epistemic_fm_a_o] :
( comple6443248293609458716fm_a_o
@ ( collec3226785624747288888fm_a_o
@ ^ [B2: epistemic_fm_a > $o] :
! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ A3 )
=> ( ord_le4043730696559282883fm_a_o @ X @ B2 ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_1059_Sup__eq__Inf,axiom,
( comple7868773486375872231c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a] :
( comple159807495668495361c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [B2: set_Epistemic_fm_a] :
! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ X @ B2 ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_1060_Inf__eq__Sup,axiom,
( comple6443248293609458716fm_a_o
= ( ^ [A3: set_Epistemic_fm_a_o] :
( comple310706794042174646fm_a_o
@ ( collec3226785624747288888fm_a_o
@ ^ [B2: epistemic_fm_a > $o] :
! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ A3 )
=> ( ord_le4043730696559282883fm_a_o @ B2 @ X ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_1061_Inf__eq__Sup,axiom,
( comple159807495668495361c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a] :
( comple7868773486375872231c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [B2: set_Epistemic_fm_a] :
! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ B2 @ X ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_1062_def__lfp__unfold,axiom,
! [H2: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( H2
= ( comple3933800597814182506fm_a_o @ F ) )
=> ( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( H2
= ( F @ H2 ) ) ) ) ).
% def_lfp_unfold
thf(fact_1063_def__lfp__unfold,axiom,
! [H2: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( H2
= ( comple693844135356717107c_fm_a @ F ) )
=> ( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( H2
= ( F @ H2 ) ) ) ) ).
% def_lfp_unfold
thf(fact_1064_lfp__fixpoint,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( F @ ( comple3933800597814182506fm_a_o @ F ) )
= ( comple3933800597814182506fm_a_o @ F ) ) ) ).
% lfp_fixpoint
thf(fact_1065_lfp__fixpoint,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( F @ ( comple693844135356717107c_fm_a @ F ) )
= ( comple693844135356717107c_fm_a @ F ) ) ) ).
% lfp_fixpoint
thf(fact_1066_lfp__unfold,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( comple3933800597814182506fm_a_o @ F )
= ( F @ ( comple3933800597814182506fm_a_o @ F ) ) ) ) ).
% lfp_unfold
thf(fact_1067_lfp__unfold,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( comple693844135356717107c_fm_a @ F )
= ( F @ ( comple693844135356717107c_fm_a @ F ) ) ) ) ).
% lfp_unfold
thf(fact_1068_lfp__eqI,axiom,
! [F4: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F4 )
=> ( ( ( F4 @ X3 )
= X3 )
=> ( ! [Z4: epistemic_fm_a > $o] :
( ( ( F4 @ Z4 )
= Z4 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Z4 ) )
=> ( ( comple3933800597814182506fm_a_o @ F4 )
= X3 ) ) ) ) ).
% lfp_eqI
thf(fact_1069_lfp__eqI,axiom,
! [F4: set_Epistemic_fm_a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F4 )
=> ( ( ( F4 @ X3 )
= X3 )
=> ( ! [Z4: set_Epistemic_fm_a] :
( ( ( F4 @ Z4 )
= Z4 )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Z4 ) )
=> ( ( comple693844135356717107c_fm_a @ F4 )
= X3 ) ) ) ) ).
% lfp_eqI
thf(fact_1070_lfp__rolling,axiom,
! [G2: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ G2 )
=> ( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( G2
@ ( comple3933800597814182506fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( F @ ( G2 @ X ) ) ) )
= ( comple3933800597814182506fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( G2 @ ( F @ X ) ) ) ) ) ) ).
% lfp_rolling
thf(fact_1071_lfp__rolling,axiom,
! [G2: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ G2 )
=> ( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( G2
@ ( comple3933800597814182506fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( F @ ( G2 @ X ) ) ) )
= ( comple693844135356717107c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( G2 @ ( F @ X ) ) ) ) ) ) ).
% lfp_rolling
thf(fact_1072_lfp__rolling,axiom,
! [G2: set_Epistemic_fm_a > epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ G2 )
=> ( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( G2
@ ( comple693844135356717107c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( F @ ( G2 @ X ) ) ) )
= ( comple3933800597814182506fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( G2 @ ( F @ X ) ) ) ) ) ) ).
% lfp_rolling
thf(fact_1073_lfp__rolling,axiom,
! [G2: set_Epistemic_fm_a > set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ G2 )
=> ( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( G2
@ ( comple693844135356717107c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( F @ ( G2 @ X ) ) ) )
= ( comple693844135356717107c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( G2 @ ( F @ X ) ) ) ) ) ) ).
% lfp_rolling
thf(fact_1074_def__lfp__induct,axiom,
! [A: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,P4: epistemic_fm_a > $o] :
( ( A
= ( comple3933800597814182506fm_a_o @ F ) )
=> ( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ ( inf_in3606484609122063093fm_a_o @ A @ P4 ) ) @ P4 )
=> ( ord_le4043730696559282883fm_a_o @ A @ P4 ) ) ) ) ).
% def_lfp_induct
thf(fact_1075_def__lfp__induct,axiom,
! [A: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,P4: set_Epistemic_fm_a] :
( ( A
= ( comple693844135356717107c_fm_a @ F ) )
=> ( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ ( inf_in2882328776738922472c_fm_a @ A @ P4 ) ) @ P4 )
=> ( ord_le3275665582123262618c_fm_a @ A @ P4 ) ) ) ) ).
% def_lfp_induct
thf(fact_1076_lfp__le__gfp,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ord_le4043730696559282883fm_a_o @ ( comple3933800597814182506fm_a_o @ F ) @ ( comple2680801738399552997fm_a_o @ F ) ) ) ).
% lfp_le_gfp
thf(fact_1077_lfp__le__gfp,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ord_le3275665582123262618c_fm_a @ ( comple693844135356717107c_fm_a @ F ) @ ( comple8088461926768726264c_fm_a @ F ) ) ) ).
% lfp_le_gfp
thf(fact_1078_lfp__eq__fixp,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( comple3933800597814182506fm_a_o @ F )
= ( comple4896442438279371158fm_a_o @ F ) ) ) ).
% lfp_eq_fixp
thf(fact_1079_lfp__eq__fixp,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( comple693844135356717107c_fm_a @ F )
= ( comple6395132419548045831c_fm_a @ F ) ) ) ).
% lfp_eq_fixp
thf(fact_1080_gfp__ordinal__induct,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,P4: ( epistemic_fm_a > $o ) > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ! [S8: epistemic_fm_a > $o] :
( ( P4 @ S8 )
=> ( ( ord_le4043730696559282883fm_a_o @ ( comple2680801738399552997fm_a_o @ F ) @ S8 )
=> ( P4 @ ( F @ S8 ) ) ) )
=> ( ! [M2: set_Epistemic_fm_a_o] :
( ! [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ M2 )
=> ( P4 @ X5 ) )
=> ( P4 @ ( comple6443248293609458716fm_a_o @ M2 ) ) )
=> ( P4 @ ( comple2680801738399552997fm_a_o @ F ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_1081_gfp__ordinal__induct,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ! [S8: set_Epistemic_fm_a] :
( ( P4 @ S8 )
=> ( ( ord_le3275665582123262618c_fm_a @ ( comple8088461926768726264c_fm_a @ F ) @ S8 )
=> ( P4 @ ( F @ S8 ) ) ) )
=> ( ! [M2: set_se5208064806568342746c_fm_a] :
( ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ M2 )
=> ( P4 @ X5 ) )
=> ( P4 @ ( comple159807495668495361c_fm_a @ M2 ) ) )
=> ( P4 @ ( comple8088461926768726264c_fm_a @ F ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_1082_gfp__mono,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,G2: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ! [Z6: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( F @ Z6 ) @ ( G2 @ Z6 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple2680801738399552997fm_a_o @ F ) @ ( comple2680801738399552997fm_a_o @ G2 ) ) ) ).
% gfp_mono
thf(fact_1083_gfp__mono,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,G2: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [Z6: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( F @ Z6 ) @ ( G2 @ Z6 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple8088461926768726264c_fm_a @ F ) @ ( comple8088461926768726264c_fm_a @ G2 ) ) ) ).
% gfp_mono
thf(fact_1084_gfp__upperbound,axiom,
! [X6: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X6 @ ( F @ X6 ) )
=> ( ord_le4043730696559282883fm_a_o @ X6 @ ( comple2680801738399552997fm_a_o @ F ) ) ) ).
% gfp_upperbound
thf(fact_1085_gfp__upperbound,axiom,
! [X6: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X6 @ ( F @ X6 ) )
=> ( ord_le3275665582123262618c_fm_a @ X6 @ ( comple8088461926768726264c_fm_a @ F ) ) ) ).
% gfp_upperbound
thf(fact_1086_gfp__least,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X6: epistemic_fm_a > $o] :
( ! [U3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ U3 @ ( F @ U3 ) )
=> ( ord_le4043730696559282883fm_a_o @ U3 @ X6 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple2680801738399552997fm_a_o @ F ) @ X6 ) ) ).
% gfp_least
thf(fact_1087_gfp__least,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,X6: set_Epistemic_fm_a] :
( ! [U3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ U3 @ ( F @ U3 ) )
=> ( ord_le3275665582123262618c_fm_a @ U3 @ X6 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple8088461926768726264c_fm_a @ F ) @ X6 ) ) ).
% gfp_least
thf(fact_1088_gfp__gfp,axiom,
! [F: ( epistemic_fm_a > $o ) > ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o,W2: epistemic_fm_a > $o,Z4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ( ord_le4043730696559282883fm_a_o @ W2 @ Z4 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 @ W2 ) @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( comple2680801738399552997fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( comple2680801738399552997fm_a_o @ ( F @ X ) ) )
= ( comple2680801738399552997fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( F @ X @ X ) ) ) ) ).
% gfp_gfp
thf(fact_1089_gfp__gfp,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a,W2: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ( ord_le3275665582123262618c_fm_a @ W2 @ Z4 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 @ W2 ) @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( comple8088461926768726264c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( comple8088461926768726264c_fm_a @ ( F @ X ) ) )
= ( comple8088461926768726264c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( F @ X @ X ) ) ) ) ).
% gfp_gfp
thf(fact_1090_gfp__def,axiom,
( comple2680801738399552997fm_a_o
= ( ^ [F2: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( comple310706794042174646fm_a_o
@ ( collec3226785624747288888fm_a_o
@ ^ [U4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ U4 @ ( F2 @ U4 ) ) ) ) ) ) ).
% gfp_def
thf(fact_1091_gfp__def,axiom,
( comple8088461926768726264c_fm_a
= ( ^ [F2: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( comple7868773486375872231c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [U4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ U4 @ ( F2 @ U4 ) ) ) ) ) ) ).
% gfp_def
thf(fact_1092_fixp__lowerbound,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ Z2 ) @ Z2 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple4896442438279371158fm_a_o @ F ) @ Z2 ) ) ) ).
% fixp_lowerbound
thf(fact_1093_fixp__lowerbound,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ Z2 ) @ Z2 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple6395132419548045831c_fm_a @ F ) @ Z2 ) ) ) ).
% fixp_lowerbound
thf(fact_1094_fixp__unfold,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( comple4896442438279371158fm_a_o @ F )
= ( F @ ( comple4896442438279371158fm_a_o @ F ) ) ) ) ).
% fixp_unfold
thf(fact_1095_fixp__unfold,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( comple6395132419548045831c_fm_a @ F )
= ( F @ ( comple6395132419548045831c_fm_a @ F ) ) ) ) ).
% fixp_unfold
thf(fact_1096_gfp__eqI,axiom,
! [F4: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F4 )
=> ( ( ( F4 @ X3 )
= X3 )
=> ( ! [Z4: epistemic_fm_a > $o] :
( ( ( F4 @ Z4 )
= Z4 )
=> ( ord_le4043730696559282883fm_a_o @ Z4 @ X3 ) )
=> ( ( comple2680801738399552997fm_a_o @ F4 )
= X3 ) ) ) ) ).
% gfp_eqI
thf(fact_1097_gfp__eqI,axiom,
! [F4: set_Epistemic_fm_a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F4 )
=> ( ( ( F4 @ X3 )
= X3 )
=> ( ! [Z4: set_Epistemic_fm_a] :
( ( ( F4 @ Z4 )
= Z4 )
=> ( ord_le3275665582123262618c_fm_a @ Z4 @ X3 ) )
=> ( ( comple8088461926768726264c_fm_a @ F4 )
= X3 ) ) ) ) ).
% gfp_eqI
thf(fact_1098_gfp__unfold,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( comple2680801738399552997fm_a_o @ F )
= ( F @ ( comple2680801738399552997fm_a_o @ F ) ) ) ) ).
% gfp_unfold
thf(fact_1099_gfp__unfold,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( comple8088461926768726264c_fm_a @ F )
= ( F @ ( comple8088461926768726264c_fm_a @ F ) ) ) ) ).
% gfp_unfold
thf(fact_1100_gfp__fixpoint,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( F @ ( comple2680801738399552997fm_a_o @ F ) )
= ( comple2680801738399552997fm_a_o @ F ) ) ) ).
% gfp_fixpoint
thf(fact_1101_gfp__fixpoint,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( F @ ( comple8088461926768726264c_fm_a @ F ) )
= ( comple8088461926768726264c_fm_a @ F ) ) ) ).
% gfp_fixpoint
thf(fact_1102_def__gfp__unfold,axiom,
! [A: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( A
= ( comple2680801738399552997fm_a_o @ F ) )
=> ( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( A
= ( F @ A ) ) ) ) ).
% def_gfp_unfold
thf(fact_1103_def__gfp__unfold,axiom,
! [A: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( A
= ( comple8088461926768726264c_fm_a @ F ) )
=> ( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( A
= ( F @ A ) ) ) ) ).
% def_gfp_unfold
thf(fact_1104_gfp__rolling,axiom,
! [G2: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ G2 )
=> ( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( G2
@ ( comple2680801738399552997fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( F @ ( G2 @ X ) ) ) )
= ( comple2680801738399552997fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( G2 @ ( F @ X ) ) ) ) ) ) ).
% gfp_rolling
thf(fact_1105_gfp__rolling,axiom,
! [G2: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ G2 )
=> ( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( G2
@ ( comple2680801738399552997fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( F @ ( G2 @ X ) ) ) )
= ( comple8088461926768726264c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( G2 @ ( F @ X ) ) ) ) ) ) ).
% gfp_rolling
thf(fact_1106_gfp__rolling,axiom,
! [G2: set_Epistemic_fm_a > epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ G2 )
=> ( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( G2
@ ( comple8088461926768726264c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( F @ ( G2 @ X ) ) ) )
= ( comple2680801738399552997fm_a_o
@ ^ [X: epistemic_fm_a > $o] : ( G2 @ ( F @ X ) ) ) ) ) ) ).
% gfp_rolling
thf(fact_1107_gfp__rolling,axiom,
! [G2: set_Epistemic_fm_a > set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ G2 )
=> ( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( G2
@ ( comple8088461926768726264c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( F @ ( G2 @ X ) ) ) )
= ( comple8088461926768726264c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( G2 @ ( F @ X ) ) ) ) ) ) ).
% gfp_rolling
thf(fact_1108_coinduct,axiom,
! [F: set_a > set_a,X6: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ X6 @ ( F @ ( sup_sup_set_a @ X6 @ ( comple3341859861669737308_set_a @ F ) ) ) )
=> ( ord_less_eq_set_a @ X6 @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% coinduct
thf(fact_1109_coinduct,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X6: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ord_le4043730696559282883fm_a_o @ X6 @ ( F @ ( sup_su6697315146409554831fm_a_o @ X6 @ ( comple2680801738399552997fm_a_o @ F ) ) ) )
=> ( ord_le4043730696559282883fm_a_o @ X6 @ ( comple2680801738399552997fm_a_o @ F ) ) ) ) ).
% coinduct
thf(fact_1110_coinduct,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,X6: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ord_le3275665582123262618c_fm_a @ X6 @ ( F @ ( sup_su1367922730591523534c_fm_a @ X6 @ ( comple8088461926768726264c_fm_a @ F ) ) ) )
=> ( ord_le3275665582123262618c_fm_a @ X6 @ ( comple8088461926768726264c_fm_a @ F ) ) ) ) ).
% coinduct
thf(fact_1111_def__coinduct,axiom,
! [A: set_a,F: set_a > set_a,X6: set_a] :
( ( A
= ( comple3341859861669737308_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( ord_less_eq_set_a @ X6 @ ( F @ ( sup_sup_set_a @ X6 @ A ) ) )
=> ( ord_less_eq_set_a @ X6 @ A ) ) ) ) ).
% def_coinduct
thf(fact_1112_def__coinduct,axiom,
! [A: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X6: epistemic_fm_a > $o] :
( ( A
= ( comple2680801738399552997fm_a_o @ F ) )
=> ( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ord_le4043730696559282883fm_a_o @ X6 @ ( F @ ( sup_su6697315146409554831fm_a_o @ X6 @ A ) ) )
=> ( ord_le4043730696559282883fm_a_o @ X6 @ A ) ) ) ) ).
% def_coinduct
thf(fact_1113_def__coinduct,axiom,
! [A: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,X6: set_Epistemic_fm_a] :
( ( A
= ( comple8088461926768726264c_fm_a @ F ) )
=> ( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ord_le3275665582123262618c_fm_a @ X6 @ ( F @ ( sup_su1367922730591523534c_fm_a @ X6 @ A ) ) )
=> ( ord_le3275665582123262618c_fm_a @ X6 @ A ) ) ) ) ).
% def_coinduct
thf(fact_1114_coinduct__lemma,axiom,
! [X6: set_a,F: set_a > set_a] :
( ( ord_less_eq_set_a @ X6 @ ( F @ ( sup_sup_set_a @ X6 @ ( comple3341859861669737308_set_a @ F ) ) ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ X6 @ ( comple3341859861669737308_set_a @ F ) ) @ ( F @ ( sup_sup_set_a @ X6 @ ( comple3341859861669737308_set_a @ F ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_1115_coinduct__lemma,axiom,
! [X6: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X6 @ ( F @ ( sup_su6697315146409554831fm_a_o @ X6 @ ( comple2680801738399552997fm_a_o @ F ) ) ) )
=> ( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ X6 @ ( comple2680801738399552997fm_a_o @ F ) ) @ ( F @ ( sup_su6697315146409554831fm_a_o @ X6 @ ( comple2680801738399552997fm_a_o @ F ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_1116_coinduct__lemma,axiom,
! [X6: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X6 @ ( F @ ( sup_su1367922730591523534c_fm_a @ X6 @ ( comple8088461926768726264c_fm_a @ F ) ) ) )
=> ( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ X6 @ ( comple8088461926768726264c_fm_a @ F ) ) @ ( F @ ( sup_su1367922730591523534c_fm_a @ X6 @ ( comple8088461926768726264c_fm_a @ F ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_1117_fixp__induct,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( comple7605158736722780118fm_a_o @ comple310706794042174646fm_a_o @ ord_le4043730696559282883fm_a_o @ P4 )
=> ( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( P4 @ ( comple310706794042174646fm_a_o @ bot_bo4908879468608929837fm_a_o ) )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( P4 @ X2 )
=> ( P4 @ ( F @ X2 ) ) )
=> ( P4 @ ( comple4896442438279371158fm_a_o @ F ) ) ) ) ) ) ).
% fixp_induct
thf(fact_1118_fixp__induct,axiom,
! [P4: set_Epistemic_fm_a > $o,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( comple5991965312082234311c_fm_a @ comple7868773486375872231c_fm_a @ ord_le3275665582123262618c_fm_a @ P4 )
=> ( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( P4 @ ( comple7868773486375872231c_fm_a @ bot_bo1868452413526482246c_fm_a ) )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( P4 @ ( F @ X2 ) ) )
=> ( P4 @ ( comple6395132419548045831c_fm_a @ F ) ) ) ) ) ) ).
% fixp_induct
thf(fact_1119_iterates__fixp,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( member4486839677911940090fm_a_o @ ( comple4896442438279371158fm_a_o @ F ) @ ( comple854385128145777876fm_a_o @ F ) ) ) ).
% iterates_fixp
thf(fact_1120_iterates__fixp,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( member536094252920883875c_fm_a @ ( comple6395132419548045831c_fm_a @ F ) @ ( comple8889762090083046729c_fm_a @ F ) ) ) ).
% iterates_fixp
thf(fact_1121_iterates__le__f,axiom,
! [X3: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X3 @ ( comple854385128145777876fm_a_o @ F ) )
=> ( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( F @ X3 ) ) ) ) ).
% iterates_le_f
thf(fact_1122_iterates__le__f,axiom,
! [X3: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( comple8889762090083046729c_fm_a @ F ) )
=> ( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( F @ X3 ) ) ) ) ).
% iterates_le_f
thf(fact_1123_admissible__disj,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,Q4: ( epistemic_fm_a > $o ) > $o] :
( ( comple7605158736722780118fm_a_o @ comple310706794042174646fm_a_o @ ord_le4043730696559282883fm_a_o @ P4 )
=> ( ( comple7605158736722780118fm_a_o @ comple310706794042174646fm_a_o @ ord_le4043730696559282883fm_a_o @ Q4 )
=> ( comple7605158736722780118fm_a_o @ comple310706794042174646fm_a_o @ ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a > $o] :
( ( P4 @ X )
| ( Q4 @ X ) ) ) ) ) ).
% admissible_disj
thf(fact_1124_admissible__disj,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ( comple5991965312082234311c_fm_a @ comple7868773486375872231c_fm_a @ ord_le3275665582123262618c_fm_a @ P4 )
=> ( ( comple5991965312082234311c_fm_a @ comple7868773486375872231c_fm_a @ ord_le3275665582123262618c_fm_a @ Q4 )
=> ( comple5991965312082234311c_fm_a @ comple7868773486375872231c_fm_a @ ord_le3275665582123262618c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( P4 @ X )
| ( Q4 @ X ) ) ) ) ) ).
% admissible_disj
thf(fact_1125_chain__iterates,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ ( comple854385128145777876fm_a_o @ F ) ) ) ).
% chain_iterates
thf(fact_1126_chain__iterates,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ ( comple8889762090083046729c_fm_a @ F ) ) ) ).
% chain_iterates
thf(fact_1127_bdd__below__primitive__def,axiom,
( condit6597432229783682067fm_a_o
= ( condit8300200636528278574fm_a_o
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X ) ) ) ).
% bdd_below_primitive_def
thf(fact_1128_bdd__below__primitive__def,axiom,
( condit4283900098890313034c_fm_a
= ( condit2964900197203784047c_fm_a
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X ) ) ) ).
% bdd_below_primitive_def
thf(fact_1129_WELL,axiom,
order_6624744507358568120c_fm_a @ ( field_Epistemic_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ) @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) ).
% WELL
thf(fact_1130_ccpo__Sup__least,axiom,
! [A: set_Epistemic_fm_a_o,Z2: epistemic_fm_a > $o] :
( ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ A )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Z2 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ Z2 ) ) ) ).
% ccpo_Sup_least
thf(fact_1131_ccpo__Sup__least,axiom,
! [A: set_se5208064806568342746c_fm_a,Z2: set_Epistemic_fm_a] :
( ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ A )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Z2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ Z2 ) ) ) ).
% ccpo_Sup_least
thf(fact_1132_ccpo__Sup__upper,axiom,
! [A: set_Epistemic_fm_a_o,X3: epistemic_fm_a > $o] :
( ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ A )
=> ( ( member4486839677911940090fm_a_o @ X3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_1133_ccpo__Sup__upper,axiom,
! [A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ A )
=> ( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_1134_chain__singleton,axiom,
! [X3: epistemic_fm_a > $o] : ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ ( insert6817254513119271059fm_a_o @ X3 @ bot_bo4908879468608929837fm_a_o ) ) ).
% chain_singleton
thf(fact_1135_chain__singleton,axiom,
! [X3: set_Epistemic_fm_a] : ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ ( insert7129737658439086282c_fm_a @ X3 @ bot_bo1868452413526482246c_fm_a ) ) ).
% chain_singleton
thf(fact_1136_bdd__above__primitive__def,axiom,
( condit6663374420580310783fm_a_o
= ( condit8300200636528278574fm_a_o @ ord_le4043730696559282883fm_a_o ) ) ).
% bdd_above_primitive_def
thf(fact_1137_bdd__above__primitive__def,axiom,
( condit2203544419062987614c_fm_a
= ( condit2964900197203784047c_fm_a @ ord_le3275665582123262618c_fm_a ) ) ).
% bdd_above_primitive_def
thf(fact_1138_iterates_OSup,axiom,
! [M: set_Epistemic_fm_a_o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ M )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ M )
=> ( member4486839677911940090fm_a_o @ X2 @ ( comple854385128145777876fm_a_o @ F ) ) )
=> ( member4486839677911940090fm_a_o @ ( comple310706794042174646fm_a_o @ M ) @ ( comple854385128145777876fm_a_o @ F ) ) ) ) ).
% iterates.Sup
thf(fact_1139_iterates_OSup,axiom,
! [M: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ M )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ M )
=> ( member536094252920883875c_fm_a @ X2 @ ( comple8889762090083046729c_fm_a @ F ) ) )
=> ( member536094252920883875c_fm_a @ ( comple7868773486375872231c_fm_a @ M ) @ ( comple8889762090083046729c_fm_a @ F ) ) ) ) ).
% iterates.Sup
thf(fact_1140_iterates_Ocases,axiom,
! [A2: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A2 @ ( comple854385128145777876fm_a_o @ F ) )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( A2
= ( F @ X2 ) )
=> ~ ( member4486839677911940090fm_a_o @ X2 @ ( comple854385128145777876fm_a_o @ F ) ) )
=> ~ ! [M2: set_Epistemic_fm_a_o] :
( ( A2
= ( comple310706794042174646fm_a_o @ M2 ) )
=> ( ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ M2 )
=> ~ ! [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ M2 )
=> ( member4486839677911940090fm_a_o @ X5 @ ( comple854385128145777876fm_a_o @ F ) ) ) ) ) ) ) ).
% iterates.cases
thf(fact_1141_iterates_Ocases,axiom,
! [A2: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A2 @ ( comple8889762090083046729c_fm_a @ F ) )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( A2
= ( F @ X2 ) )
=> ~ ( member536094252920883875c_fm_a @ X2 @ ( comple8889762090083046729c_fm_a @ F ) ) )
=> ~ ! [M2: set_se5208064806568342746c_fm_a] :
( ( A2
= ( comple7868773486375872231c_fm_a @ M2 ) )
=> ( ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ M2 )
=> ~ ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ M2 )
=> ( member536094252920883875c_fm_a @ X5 @ ( comple8889762090083046729c_fm_a @ F ) ) ) ) ) ) ) ).
% iterates.cases
thf(fact_1142_iterates_Osimps,axiom,
! [A2: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A2 @ ( comple854385128145777876fm_a_o @ F ) )
= ( ? [X: epistemic_fm_a > $o] :
( ( A2
= ( F @ X ) )
& ( member4486839677911940090fm_a_o @ X @ ( comple854385128145777876fm_a_o @ F ) ) )
| ? [M3: set_Epistemic_fm_a_o] :
( ( A2
= ( comple310706794042174646fm_a_o @ M3 ) )
& ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ M3 )
& ! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ M3 )
=> ( member4486839677911940090fm_a_o @ X @ ( comple854385128145777876fm_a_o @ F ) ) ) ) ) ) ).
% iterates.simps
thf(fact_1143_iterates_Osimps,axiom,
! [A2: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A2 @ ( comple8889762090083046729c_fm_a @ F ) )
= ( ? [X: set_Epistemic_fm_a] :
( ( A2
= ( F @ X ) )
& ( member536094252920883875c_fm_a @ X @ ( comple8889762090083046729c_fm_a @ F ) ) )
| ? [M3: set_se5208064806568342746c_fm_a] :
( ( A2
= ( comple7868773486375872231c_fm_a @ M3 ) )
& ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ M3 )
& ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ M3 )
=> ( member536094252920883875c_fm_a @ X @ ( comple8889762090083046729c_fm_a @ F ) ) ) ) ) ) ).
% iterates.simps
thf(fact_1144_admissible__chfin,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o] :
( ! [S8: set_Epistemic_fm_a_o] :
( ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ S8 )
=> ( finite5407549452671652090fm_a_o @ S8 ) )
=> ( comple7605158736722780118fm_a_o @ comple310706794042174646fm_a_o @ ord_le4043730696559282883fm_a_o @ P4 ) ) ).
% admissible_chfin
thf(fact_1145_admissible__chfin,axiom,
! [P4: set_Epistemic_fm_a > $o] :
( ! [S8: set_se5208064806568342746c_fm_a] :
( ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ S8 )
=> ( finite7364866884270929315c_fm_a @ S8 ) )
=> ( comple5991965312082234311c_fm_a @ comple7868773486375872231c_fm_a @ ord_le3275665582123262618c_fm_a @ P4 ) ) ).
% admissible_chfin
thf(fact_1146_in__chain__finite,axiom,
! [A: set_Epistemic_fm_a_o] :
( ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ A )
=> ( ( finite5407549452671652090fm_a_o @ A )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( member4486839677911940090fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ A ) ) ) ) ).
% in_chain_finite
thf(fact_1147_in__chain__finite,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ A )
=> ( ( finite7364866884270929315c_fm_a @ A )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( member536094252920883875c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ A ) ) ) ) ).
% in_chain_finite
thf(fact_1148_iteratesp_OSup,axiom,
! [M: set_Epistemic_fm_a_o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ M )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ M )
=> ( comple2170954288673221540fm_a_o @ F @ X2 ) )
=> ( comple2170954288673221540fm_a_o @ F @ ( comple310706794042174646fm_a_o @ M ) ) ) ) ).
% iteratesp.Sup
thf(fact_1149_iteratesp_OSup,axiom,
! [M: set_se5208064806568342746c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ M )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ M )
=> ( comple3273429119107037817c_fm_a @ F @ X2 ) )
=> ( comple3273429119107037817c_fm_a @ F @ ( comple7868773486375872231c_fm_a @ M ) ) ) ) ).
% iteratesp.Sup
thf(fact_1150_iteratesp_Osimps,axiom,
( comple2170954288673221540fm_a_o
= ( ^ [F2: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A4: epistemic_fm_a > $o] :
( ? [X: epistemic_fm_a > $o] :
( ( A4
= ( F2 @ X ) )
& ( comple2170954288673221540fm_a_o @ F2 @ X ) )
| ? [M3: set_Epistemic_fm_a_o] :
( ( A4
= ( comple310706794042174646fm_a_o @ M3 ) )
& ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ M3 )
& ! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ M3 )
=> ( comple2170954288673221540fm_a_o @ F2 @ X ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_1151_iteratesp_Osimps,axiom,
( comple3273429119107037817c_fm_a
= ( ^ [F2: set_Epistemic_fm_a > set_Epistemic_fm_a,A4: set_Epistemic_fm_a] :
( ? [X: set_Epistemic_fm_a] :
( ( A4
= ( F2 @ X ) )
& ( comple3273429119107037817c_fm_a @ F2 @ X ) )
| ? [M3: set_se5208064806568342746c_fm_a] :
( ( A4
= ( comple7868773486375872231c_fm_a @ M3 ) )
& ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ M3 )
& ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ M3 )
=> ( comple3273429119107037817c_fm_a @ F2 @ X ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_1152_iteratesp_Ocases,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( comple2170954288673221540fm_a_o @ F @ A2 )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( A2
= ( F @ X2 ) )
=> ~ ( comple2170954288673221540fm_a_o @ F @ X2 ) )
=> ~ ! [M2: set_Epistemic_fm_a_o] :
( ( A2
= ( comple310706794042174646fm_a_o @ M2 ) )
=> ( ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ M2 )
=> ~ ! [X5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X5 @ M2 )
=> ( comple2170954288673221540fm_a_o @ F @ X5 ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_1153_iteratesp_Ocases,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( comple3273429119107037817c_fm_a @ F @ A2 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( A2
= ( F @ X2 ) )
=> ~ ( comple3273429119107037817c_fm_a @ F @ X2 ) )
=> ~ ! [M2: set_se5208064806568342746c_fm_a] :
( ( A2
= ( comple7868773486375872231c_fm_a @ M2 ) )
=> ( ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ M2 )
=> ~ ! [X5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X5 @ M2 )
=> ( comple3273429119107037817c_fm_a @ F @ X5 ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_1154_Inf__Sup__le,axiom,
! [A: set_se3897318249199329743fm_a_o] :
( ord_le4043730696559282883fm_a_o @ ( comple6443248293609458716fm_a_o @ ( image_3680702584007291907fm_a_o @ comple310706794042174646fm_a_o @ A ) )
@ ( comple310706794042174646fm_a_o
@ ( image_3680702584007291907fm_a_o @ comple6443248293609458716fm_a_o
@ ( collec2512887859873555566fm_a_o
@ ^ [Uu2: set_Epistemic_fm_a_o] :
? [F2: set_Epistemic_fm_a_o > epistemic_fm_a > $o] :
( ( Uu2
= ( image_3680702584007291907fm_a_o @ F2 @ A ) )
& ! [X: set_Epistemic_fm_a_o] :
( ( member3894352381434343216fm_a_o @ X @ A )
=> ( member4486839677911940090fm_a_o @ ( F2 @ X ) @ X ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_1155_Inf__Sup__le,axiom,
! [A: set_se8609989359752948410c_fm_a] :
( ord_le3275665582123262618c_fm_a @ ( comple159807495668495361c_fm_a @ ( image_114381943869839653c_fm_a @ comple7868773486375872231c_fm_a @ A ) )
@ ( comple7868773486375872231c_fm_a
@ ( image_114381943869839653c_fm_a @ comple159807495668495361c_fm_a
@ ( collec4484617205158951365c_fm_a
@ ^ [Uu2: set_se5208064806568342746c_fm_a] :
? [F2: set_se5208064806568342746c_fm_a > set_Epistemic_fm_a] :
( ( Uu2
= ( image_114381943869839653c_fm_a @ F2 @ A ) )
& ! [X: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X @ A )
=> ( member536094252920883875c_fm_a @ ( F2 @ X ) @ X ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_1156_Sup__Inf__le,axiom,
! [A: set_se3897318249199329743fm_a_o] :
( ord_le4043730696559282883fm_a_o
@ ( comple310706794042174646fm_a_o
@ ( image_3680702584007291907fm_a_o @ comple6443248293609458716fm_a_o
@ ( collec2512887859873555566fm_a_o
@ ^ [Uu2: set_Epistemic_fm_a_o] :
? [F2: set_Epistemic_fm_a_o > epistemic_fm_a > $o] :
( ( Uu2
= ( image_3680702584007291907fm_a_o @ F2 @ A ) )
& ! [X: set_Epistemic_fm_a_o] :
( ( member3894352381434343216fm_a_o @ X @ A )
=> ( member4486839677911940090fm_a_o @ ( F2 @ X ) @ X ) ) ) ) ) )
@ ( comple6443248293609458716fm_a_o @ ( image_3680702584007291907fm_a_o @ comple310706794042174646fm_a_o @ A ) ) ) ).
% Sup_Inf_le
thf(fact_1157_Sup__Inf__le,axiom,
! [A: set_se8609989359752948410c_fm_a] :
( ord_le3275665582123262618c_fm_a
@ ( comple7868773486375872231c_fm_a
@ ( image_114381943869839653c_fm_a @ comple159807495668495361c_fm_a
@ ( collec4484617205158951365c_fm_a
@ ^ [Uu2: set_se5208064806568342746c_fm_a] :
? [F2: set_se5208064806568342746c_fm_a > set_Epistemic_fm_a] :
( ( Uu2
= ( image_114381943869839653c_fm_a @ F2 @ A ) )
& ! [X: set_se5208064806568342746c_fm_a] :
( ( member3183810400728440963c_fm_a @ X @ A )
=> ( member536094252920883875c_fm_a @ ( F2 @ X ) @ X ) ) ) ) ) )
@ ( comple159807495668495361c_fm_a @ ( image_114381943869839653c_fm_a @ comple7868773486375872231c_fm_a @ A ) ) ) ).
% Sup_Inf_le
thf(fact_1158_iteratesp_Omono,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( monoto1304637729855739751_a_o_o @ top_to8237543284498297820_a_o_o @ ord_le3153299366064206284_a_o_o @ ord_le3153299366064206284_a_o_o
@ ^ [P2: ( epistemic_fm_a > $o ) > $o,X: epistemic_fm_a > $o] :
( ? [Y4: epistemic_fm_a > $o] :
( ( X
= ( F @ Y4 ) )
& ( P2 @ Y4 ) )
| ? [M3: set_Epistemic_fm_a_o] :
( ( X
= ( comple310706794042174646fm_a_o @ M3 ) )
& ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ M3 )
& ! [Y4: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Y4 @ M3 )
=> ( P2 @ Y4 ) ) ) ) ) ).
% iteratesp.mono
thf(fact_1159_iteratesp_Omono,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( monoto1683349816274310767fm_a_o @ top_to155495981955717737fm_a_o @ ord_le1568123225352354019fm_a_o @ ord_le1568123225352354019fm_a_o
@ ^ [P2: set_Epistemic_fm_a > $o,X: set_Epistemic_fm_a] :
( ? [Y4: set_Epistemic_fm_a] :
( ( X
= ( F @ Y4 ) )
& ( P2 @ Y4 ) )
| ? [M3: set_se5208064806568342746c_fm_a] :
( ( X
= ( comple7868773486375872231c_fm_a @ M3 ) )
& ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ M3 )
& ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ M3 )
=> ( P2 @ Y4 ) ) ) ) ) ).
% iteratesp.mono
thf(fact_1160_iteratesp__def,axiom,
( comple2170954288673221540fm_a_o
= ( ^ [F2: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( comple7029256114438256869_a_o_o
@ ^ [P2: ( epistemic_fm_a > $o ) > $o,X: epistemic_fm_a > $o] :
( ? [Y4: epistemic_fm_a > $o] :
( ( X
= ( F2 @ Y4 ) )
& ( P2 @ Y4 ) )
| ? [M3: set_Epistemic_fm_a_o] :
( ( X
= ( comple310706794042174646fm_a_o @ M3 ) )
& ( comple4220757189516020531fm_a_o @ ord_le4043730696559282883fm_a_o @ M3 )
& ! [Y4: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Y4 @ M3 )
=> ( P2 @ Y4 ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_1161_iteratesp__def,axiom,
( comple3273429119107037817c_fm_a
= ( ^ [F2: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( comple2002745723559171210fm_a_o
@ ^ [P2: set_Epistemic_fm_a > $o,X: set_Epistemic_fm_a] :
( ? [Y4: set_Epistemic_fm_a] :
( ( X
= ( F2 @ Y4 ) )
& ( P2 @ Y4 ) )
| ? [M3: set_se5208064806568342746c_fm_a] :
( ( X
= ( comple7868773486375872231c_fm_a @ M3 ) )
& ( comple2266461377711601194c_fm_a @ ord_le3275665582123262618c_fm_a @ M3 )
& ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ M3 )
=> ( P2 @ Y4 ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_1162_less__fun__def,axiom,
( ord_le8771080710536341687fm_a_o
= ( ^ [F2: epistemic_fm_a > $o,G4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ F2 @ G4 )
& ~ ( ord_le4043730696559282883fm_a_o @ G4 @ F2 ) ) ) ) ).
% less_fun_def
thf(fact_1163_bot_Onot__eq__extremum,axiom,
! [A2: set_Epistemic_fm_a] :
( ( A2 != bot_bo3626323581529592678c_fm_a )
= ( ord_le2139327010432383910c_fm_a @ bot_bo3626323581529592678c_fm_a @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1164_bot_Onot__eq__extremum,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1165_bot_Oextremum__strict,axiom,
! [A2: set_Epistemic_fm_a] :
~ ( ord_le2139327010432383910c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a ) ).
% bot.extremum_strict
thf(fact_1166_bot_Oextremum__strict,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_1167_leD,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ~ ( ord_le8771080710536341687fm_a_o @ X3 @ Y ) ) ).
% leD
thf(fact_1168_leD,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ~ ( ord_le2139327010432383910c_fm_a @ X3 @ Y ) ) ).
% leD
thf(fact_1169_nless__le,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ~ ( ord_le8771080710536341687fm_a_o @ A2 @ B ) )
= ( ~ ( ord_le4043730696559282883fm_a_o @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_1170_nless__le,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ~ ( ord_le2139327010432383910c_fm_a @ A2 @ B ) )
= ( ~ ( ord_le3275665582123262618c_fm_a @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_1171_antisym__conv1,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ~ ( ord_le8771080710536341687fm_a_o @ X3 @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_1172_antisym__conv1,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ~ ( ord_le2139327010432383910c_fm_a @ X3 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_1173_antisym__conv2,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ~ ( ord_le8771080710536341687fm_a_o @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_1174_antisym__conv2,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ~ ( ord_le2139327010432383910c_fm_a @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_1175_less__le__not__le,axiom,
( ord_le8771080710536341687fm_a_o
= ( ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y4 )
& ~ ( ord_le4043730696559282883fm_a_o @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_1176_less__le__not__le,axiom,
( ord_le2139327010432383910c_fm_a
= ( ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y4 )
& ~ ( ord_le3275665582123262618c_fm_a @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_1177_order_Oorder__iff__strict,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le8771080710536341687fm_a_o @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1178_order_Oorder__iff__strict,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( ord_le2139327010432383910c_fm_a @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1179_order_Ostrict__iff__order,axiom,
( ord_le8771080710536341687fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1180_order_Ostrict__iff__order,axiom,
( ord_le2139327010432383910c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1181_order_Ostrict__trans1,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le8771080710536341687fm_a_o @ B @ C )
=> ( ord_le8771080710536341687fm_a_o @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1182_order_Ostrict__trans1,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le2139327010432383910c_fm_a @ B @ C )
=> ( ord_le2139327010432383910c_fm_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1183_order_Ostrict__trans2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le8771080710536341687fm_a_o @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ord_le8771080710536341687fm_a_o @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1184_order_Ostrict__trans2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le2139327010432383910c_fm_a @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ord_le2139327010432383910c_fm_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1185_order_Ostrict__iff__not,axiom,
( ord_le8771080710536341687fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A4 @ B2 )
& ~ ( ord_le4043730696559282883fm_a_o @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1186_order_Ostrict__iff__not,axiom,
( ord_le2139327010432383910c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A4 @ B2 )
& ~ ( ord_le3275665582123262618c_fm_a @ B2 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1187_dual__order_Oorder__iff__strict,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [B2: epistemic_fm_a > $o,A4: epistemic_fm_a > $o] :
( ( ord_le8771080710536341687fm_a_o @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1188_dual__order_Oorder__iff__strict,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [B2: set_Epistemic_fm_a,A4: set_Epistemic_fm_a] :
( ( ord_le2139327010432383910c_fm_a @ B2 @ A4 )
| ( A4 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1189_dual__order_Ostrict__iff__order,axiom,
( ord_le8771080710536341687fm_a_o
= ( ^ [B2: epistemic_fm_a > $o,A4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1190_dual__order_Ostrict__iff__order,axiom,
( ord_le2139327010432383910c_fm_a
= ( ^ [B2: set_Epistemic_fm_a,A4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B2 @ A4 )
& ( A4 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1191_dual__order_Ostrict__trans1,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( ord_le8771080710536341687fm_a_o @ C @ B )
=> ( ord_le8771080710536341687fm_a_o @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1192_dual__order_Ostrict__trans1,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( ord_le2139327010432383910c_fm_a @ C @ B )
=> ( ord_le2139327010432383910c_fm_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1193_dual__order_Ostrict__trans2,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le8771080710536341687fm_a_o @ B @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ C @ B )
=> ( ord_le8771080710536341687fm_a_o @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1194_dual__order_Ostrict__trans2,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le2139327010432383910c_fm_a @ B @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ C @ B )
=> ( ord_le2139327010432383910c_fm_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1195_dual__order_Ostrict__iff__not,axiom,
( ord_le8771080710536341687fm_a_o
= ( ^ [B2: epistemic_fm_a > $o,A4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B2 @ A4 )
& ~ ( ord_le4043730696559282883fm_a_o @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1196_dual__order_Ostrict__iff__not,axiom,
( ord_le2139327010432383910c_fm_a
= ( ^ [B2: set_Epistemic_fm_a,A4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B2 @ A4 )
& ~ ( ord_le3275665582123262618c_fm_a @ A4 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1197_order_Ostrict__implies__order,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le8771080710536341687fm_a_o @ A2 @ B )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1198_order_Ostrict__implies__order,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le2139327010432383910c_fm_a @ A2 @ B )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1199_dual__order_Ostrict__implies__order,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le8771080710536341687fm_a_o @ B @ A2 )
=> ( ord_le4043730696559282883fm_a_o @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1200_dual__order_Ostrict__implies__order,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le2139327010432383910c_fm_a @ B @ A2 )
=> ( ord_le3275665582123262618c_fm_a @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1201_order__le__less,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] :
( ( ord_le8771080710536341687fm_a_o @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1202_order__le__less,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( ord_le2139327010432383910c_fm_a @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1203_order__less__le,axiom,
( ord_le8771080710536341687fm_a_o
= ( ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1204_order__less__le,axiom,
( ord_le2139327010432383910c_fm_a
= ( ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1205_order__less__imp__le,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( ord_le8771080710536341687fm_a_o @ X3 @ Y )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1206_order__less__imp__le,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le2139327010432383910c_fm_a @ X3 @ Y )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_1207_order__le__neq__trans,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_le8771080710536341687fm_a_o @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1208_order__le__neq__trans,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_le2139327010432383910c_fm_a @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1209_order__neq__le__trans,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( A2 != B )
=> ( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ord_le8771080710536341687fm_a_o @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1210_order__neq__le__trans,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( A2 != B )
=> ( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ord_le2139327010432383910c_fm_a @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1211_order__le__less__trans,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ord_le8771080710536341687fm_a_o @ Y @ Z2 )
=> ( ord_le8771080710536341687fm_a_o @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1212_order__le__less__trans,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ord_le2139327010432383910c_fm_a @ Y @ Z2 )
=> ( ord_le2139327010432383910c_fm_a @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1213_order__less__le__trans,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le8771080710536341687fm_a_o @ X3 @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ Z2 )
=> ( ord_le8771080710536341687fm_a_o @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1214_order__less__le__trans,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( ord_le2139327010432383910c_fm_a @ X3 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ Z2 )
=> ( ord_le2139327010432383910c_fm_a @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1215_order__le__less__subst2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le8771080710536341687fm_a_o @ ( F @ B ) @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8771080710536341687fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1216_order__le__less__subst2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le2139327010432383910c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2139327010432383910c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1217_order__le__less__subst2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le8771080710536341687fm_a_o @ ( F @ B ) @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8771080710536341687fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1218_order__le__less__subst2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le2139327010432383910c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2139327010432383910c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1219_order__less__le__subst1,axiom,
! [A2: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le8771080710536341687fm_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8771080710536341687fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1220_order__less__le__subst1,axiom,
! [A2: set_Epistemic_fm_a,F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le2139327010432383910c_fm_a @ A2 @ ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2139327010432383910c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1221_order__less__le__subst1,axiom,
! [A2: epistemic_fm_a > $o,F: set_Epistemic_fm_a > epistemic_fm_a > $o,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le8771080710536341687fm_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le8771080710536341687fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1222_order__less__le__subst1,axiom,
! [A2: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le2139327010432383910c_fm_a @ A2 @ ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2139327010432383910c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1223_order__le__imp__less__or__eq,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ord_le8771080710536341687fm_a_o @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1224_order__le__imp__less__or__eq,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ord_le2139327010432383910c_fm_a @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1225_top_Onot__eq__extremum,axiom,
! [A2: set_Epistemic_fm_a] :
( ( A2 != top_to7796028867103199306c_fm_a )
= ( ord_le2139327010432383910c_fm_a @ A2 @ top_to7796028867103199306c_fm_a ) ) ).
% top.not_eq_extremum
thf(fact_1226_top_Oextremum__strict,axiom,
! [A2: set_Epistemic_fm_a] :
~ ( ord_le2139327010432383910c_fm_a @ top_to7796028867103199306c_fm_a @ A2 ) ).
% top.extremum_strict
thf(fact_1227_strict__mono__mono,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le8771080710536341687fm_a_o @ ord_le8771080710536341687fm_a_o @ F )
=> ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F ) ) ).
% strict_mono_mono
thf(fact_1228_strict__mono__mono,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le8771080710536341687fm_a_o @ ord_le2139327010432383910c_fm_a @ F )
=> ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F ) ) ).
% strict_mono_mono
thf(fact_1229_strict__mono__mono,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le2139327010432383910c_fm_a @ ord_le8771080710536341687fm_a_o @ F )
=> ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F ) ) ).
% strict_mono_mono
thf(fact_1230_strict__mono__mono,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le2139327010432383910c_fm_a @ ord_le2139327010432383910c_fm_a @ F )
=> ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F ) ) ).
% strict_mono_mono
thf(fact_1231_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila2496817875450240012_set_a @ sup_sup_set_a @ bot_bot_set_a
@ ^ [X: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ X )
@ ^ [X: set_a,Y4: set_a] : ( ord_less_set_a @ Y4 @ X ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_1232_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila3938086466787414325fm_a_o @ sup_su6697315146409554831fm_a_o @ bot_bo6433428028861825271fm_a_o
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le8771080710536341687fm_a_o @ Y4 @ X ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_1233_sup__bot_Osemilattice__neutr__order__axioms,axiom,
( semila1768514948596158888c_fm_a @ sup_su1367922730591523534c_fm_a @ bot_bo3626323581529592678c_fm_a
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le2139327010432383910c_fm_a @ Y4 @ X ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_1234_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila3938086466787414325fm_a_o @ inf_in3606484609122063093fm_a_o @ top_to6710954824118802707fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le8771080710536341687fm_a_o ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1235_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila1768514948596158888c_fm_a @ inf_in2882328776738922472c_fm_a @ top_to7796028867103199306c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le2139327010432383910c_fm_a ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1236_bot_Oordering__top__axioms,axiom,
( ordering_top_set_a
@ ^ [X: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ X )
@ ^ [X: set_a,Y4: set_a] : ( ord_less_set_a @ Y4 @ X )
@ bot_bot_set_a ) ).
% bot.ordering_top_axioms
thf(fact_1237_bot_Oordering__top__axioms,axiom,
( orderi282552358289776498fm_a_o
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le8771080710536341687fm_a_o @ Y4 @ X )
@ bot_bo6433428028861825271fm_a_o ) ).
% bot.ordering_top_axioms
thf(fact_1238_bot_Oordering__top__axioms,axiom,
( orderi596021265140580651c_fm_a
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le2139327010432383910c_fm_a @ Y4 @ X )
@ bot_bo3626323581529592678c_fm_a ) ).
% bot.ordering_top_axioms
thf(fact_1239_Sup__fin_Osemilattice__order__set__axioms,axiom,
( lattic8986249270076014136_set_a @ sup_sup_set_a
@ ^ [X: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ X )
@ ^ [X: set_a,Y4: set_a] : ( ord_less_set_a @ Y4 @ X ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_1240_Sup__fin_Osemilattice__order__set__axioms,axiom,
( lattic7707135717978707977fm_a_o @ sup_su6697315146409554831fm_a_o
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le8771080710536341687fm_a_o @ Y4 @ X ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_1241_Sup__fin_Osemilattice__order__set__axioms,axiom,
( lattic6029172335137696212c_fm_a @ sup_su1367922730591523534c_fm_a
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le2139327010432383910c_fm_a @ Y4 @ X ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_1242_top_Oordering__top__axioms,axiom,
orderi282552358289776498fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le8771080710536341687fm_a_o @ top_to6710954824118802707fm_a_o ).
% top.ordering_top_axioms
thf(fact_1243_top_Oordering__top__axioms,axiom,
orderi596021265140580651c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le2139327010432383910c_fm_a @ top_to7796028867103199306c_fm_a ).
% top.ordering_top_axioms
thf(fact_1244_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic7707135717978707977fm_a_o @ inf_in3606484609122063093fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le8771080710536341687fm_a_o ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1245_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic6029172335137696212c_fm_a @ inf_in2882328776738922472c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le2139327010432383910c_fm_a ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1246_sup_Osemilattice__order__axioms,axiom,
( semila4706084620769370446_set_a @ sup_sup_set_a
@ ^ [X: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ X )
@ ^ [X: set_a,Y4: set_a] : ( ord_less_set_a @ Y4 @ X ) ) ).
% sup.semilattice_order_axioms
thf(fact_1247_sup_Osemilattice__order__axioms,axiom,
( semila6139244836443138419fm_a_o @ sup_su6697315146409554831fm_a_o
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le8771080710536341687fm_a_o @ Y4 @ X ) ) ).
% sup.semilattice_order_axioms
thf(fact_1248_sup_Osemilattice__order__axioms,axiom,
( semila5664283851796400106c_fm_a @ sup_su1367922730591523534c_fm_a
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le2139327010432383910c_fm_a @ Y4 @ X ) ) ).
% sup.semilattice_order_axioms
thf(fact_1249_bdd__below_Opreordering__bdd__axioms,axiom,
( condit8779728373063496844fm_a_o
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le8771080710536341687fm_a_o @ Y4 @ X ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_1250_bdd__below_Opreordering__bdd__axioms,axiom,
( condit103312184941666193c_fm_a
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le2139327010432383910c_fm_a @ Y4 @ X ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_1251_bdd__above_Opreordering__bdd__axioms,axiom,
condit8779728373063496844fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le8771080710536341687fm_a_o ).
% bdd_above.preordering_bdd_axioms
thf(fact_1252_bdd__above_Opreordering__bdd__axioms,axiom,
condit103312184941666193c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le2139327010432383910c_fm_a ).
% bdd_above.preordering_bdd_axioms
thf(fact_1253_inf_Osemilattice__order__axioms,axiom,
semila6139244836443138419fm_a_o @ inf_in3606484609122063093fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le8771080710536341687fm_a_o ).
% inf.semilattice_order_axioms
thf(fact_1254_inf_Osemilattice__order__axioms,axiom,
semila5664283851796400106c_fm_a @ inf_in2882328776738922472c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le2139327010432383910c_fm_a ).
% inf.semilattice_order_axioms
thf(fact_1255_order_Oordering__axioms,axiom,
orderi7350105783456702155fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le8771080710536341687fm_a_o ).
% order.ordering_axioms
thf(fact_1256_order_Oordering__axioms,axiom,
orderi1643405053430256786c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le2139327010432383910c_fm_a ).
% order.ordering_axioms
thf(fact_1257_dual__order_Oordering__axioms,axiom,
( orderi7350105783456702155fm_a_o
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ Y4 @ X )
@ ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] : ( ord_le8771080710536341687fm_a_o @ Y4 @ X ) ) ).
% dual_order.ordering_axioms
thf(fact_1258_dual__order_Oordering__axioms,axiom,
( orderi1643405053430256786c_fm_a
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ Y4 @ X )
@ ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] : ( ord_le2139327010432383910c_fm_a @ Y4 @ X ) ) ).
% dual_order.ordering_axioms
thf(fact_1259_mono__image__least,axiom,
! [F: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,M5: epistemic_fm_a > $o,N: epistemic_fm_a > $o,M6: epistemic_fm_a > $o,N2: epistemic_fm_a > $o] :
( ( monoto1601457452331330095fm_a_o @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ( image_6282542973537860301fm_a_o @ F @ ( set_or7471044323730626962fm_a_o @ M5 @ N ) )
= ( set_or7471044323730626962fm_a_o @ M6 @ N2 ) )
=> ( ( ord_le8771080710536341687fm_a_o @ M5 @ N )
=> ( ( F @ M5 )
= M6 ) ) ) ) ).
% mono_image_least
thf(fact_1260_mono__image__least,axiom,
! [F: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a,M5: epistemic_fm_a > $o,N: epistemic_fm_a > $o,M6: set_Epistemic_fm_a,N2: set_Epistemic_fm_a] :
( ( monoto766724803074413614c_fm_a @ top_to8436834881581361225fm_a_o @ ord_le4043730696559282883fm_a_o @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ( image_3065101469317257744c_fm_a @ F @ ( set_or7471044323730626962fm_a_o @ M5 @ N ) )
= ( set_or1522782340569636619c_fm_a @ M6 @ N2 ) )
=> ( ( ord_le8771080710536341687fm_a_o @ M5 @ N )
=> ( ( F @ M5 )
= M6 ) ) ) ) ).
% mono_image_least
thf(fact_1261_mono__image__least,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a > $o,M5: set_Epistemic_fm_a,N: set_Epistemic_fm_a,M6: epistemic_fm_a > $o,N2: epistemic_fm_a > $o] :
( ( monoto4604122963098883126fm_a_o @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le4043730696559282883fm_a_o @ F )
=> ( ( ( image_6902499629341727256fm_a_o @ F @ ( set_or1522782340569636619c_fm_a @ M5 @ N ) )
= ( set_or7471044323730626962fm_a_o @ M6 @ N2 ) )
=> ( ( ord_le2139327010432383910c_fm_a @ M5 @ N )
=> ( ( F @ M5 )
= M6 ) ) ) ) ).
% mono_image_least
thf(fact_1262_mono__image__least,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,M5: set_Epistemic_fm_a,N: set_Epistemic_fm_a,M6: set_Epistemic_fm_a,N2: set_Epistemic_fm_a] :
( ( monoto5809647437951573863c_fm_a @ top_to5398033090887092778c_fm_a @ ord_le3275665582123262618c_fm_a @ ord_le3275665582123262618c_fm_a @ F )
=> ( ( ( image_8582343580727714565c_fm_a @ F @ ( set_or1522782340569636619c_fm_a @ M5 @ N ) )
= ( set_or1522782340569636619c_fm_a @ M6 @ N2 ) )
=> ( ( ord_le2139327010432383910c_fm_a @ M5 @ N )
=> ( ( F @ M5 )
= M6 ) ) ) ) ).
% mono_image_least
thf(fact_1263_max__bot2,axiom,
! [X3: set_Epistemic_fm_a] :
( ( ord_ma5084070730493774927c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a )
= X3 ) ).
% max_bot2
thf(fact_1264_max__bot2,axiom,
! [X3: set_a] :
( ( ord_max_set_a @ X3 @ bot_bot_set_a )
= X3 ) ).
% max_bot2
thf(fact_1265_max__bot,axiom,
! [X3: set_Epistemic_fm_a] :
( ( ord_ma5084070730493774927c_fm_a @ bot_bo3626323581529592678c_fm_a @ X3 )
= X3 ) ).
% max_bot
thf(fact_1266_max__bot,axiom,
! [X3: set_a] :
( ( ord_max_set_a @ bot_bot_set_a @ X3 )
= X3 ) ).
% max_bot
thf(fact_1267_max__top2,axiom,
! [X3: set_Epistemic_fm_a] :
( ( ord_ma5084070730493774927c_fm_a @ X3 @ top_to7796028867103199306c_fm_a )
= top_to7796028867103199306c_fm_a ) ).
% max_top2
thf(fact_1268_max__top,axiom,
! [X3: set_Epistemic_fm_a] :
( ( ord_ma5084070730493774927c_fm_a @ top_to7796028867103199306c_fm_a @ X3 )
= top_to7796028867103199306c_fm_a ) ).
% max_top
thf(fact_1269_atLeastLessThan__iff,axiom,
! [I: epistemic_fm_a > $o,L: epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ I @ ( set_or7471044323730626962fm_a_o @ L @ U2 ) )
= ( ( ord_le4043730696559282883fm_a_o @ L @ I )
& ( ord_le8771080710536341687fm_a_o @ I @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_1270_atLeastLessThan__iff,axiom,
! [I: set_Epistemic_fm_a,L: set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ I @ ( set_or1522782340569636619c_fm_a @ L @ U2 ) )
= ( ( ord_le3275665582123262618c_fm_a @ L @ I )
& ( ord_le2139327010432383910c_fm_a @ I @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_1271_atLeastLessThan__empty,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( set_or7471044323730626962fm_a_o @ A2 @ B )
= bot_bo4908879468608929837fm_a_o ) ) ).
% atLeastLessThan_empty
thf(fact_1272_atLeastLessThan__empty,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( set_or1522782340569636619c_fm_a @ A2 @ B )
= bot_bo1868452413526482246c_fm_a ) ) ).
% atLeastLessThan_empty
thf(fact_1273_max__def__raw,axiom,
( ord_ma2191051425148795214fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] : ( if_Epistemic_fm_a_o @ ( ord_le4043730696559282883fm_a_o @ A4 @ B2 ) @ B2 @ A4 ) ) ) ).
% max_def_raw
thf(fact_1274_max__def__raw,axiom,
( ord_ma5084070730493774927c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] : ( if_set2375185401397246656c_fm_a @ ( ord_le3275665582123262618c_fm_a @ A4 @ B2 ) @ B2 @ A4 ) ) ) ).
% max_def_raw
thf(fact_1275_max__def,axiom,
( ord_ma2191051425148795214fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] : ( if_Epistemic_fm_a_o @ ( ord_le4043730696559282883fm_a_o @ A4 @ B2 ) @ B2 @ A4 ) ) ) ).
% max_def
thf(fact_1276_max__def,axiom,
( ord_ma5084070730493774927c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] : ( if_set2375185401397246656c_fm_a @ ( ord_le3275665582123262618c_fm_a @ A4 @ B2 ) @ B2 @ A4 ) ) ) ).
% max_def
thf(fact_1277_max__absorb1,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( ( ord_ma2191051425148795214fm_a_o @ X3 @ Y )
= X3 ) ) ).
% max_absorb1
thf(fact_1278_max__absorb1,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( ( ord_ma5084070730493774927c_fm_a @ X3 @ Y )
= X3 ) ) ).
% max_absorb1
% Helper facts (5)
thf(help_If_2_1_If_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J_T,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( if_Epistemic_fm_a_o @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J_T,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( if_Epistemic_fm_a_o @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [P4: $o] :
( ( P4 = $true )
| ( P4 = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( if_set2375185401397246656c_fm_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( if_set2375185401397246656c_fm_a @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ i @ p ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( epistemic_Imp_a @ q @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( epistemic_Imp_a @ ( epistemic_Con_a @ p @ q ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ).
%------------------------------------------------------------------------------